diff --git a/index.html b/index.html index 7c23336a..70e5ac85 100644 --- a/index.html +++ b/index.html @@ -139,7 +139,7 @@

IMPORTANT: This library is in beta, and should not be used in production software.

Documentation

Minimum Supported Rust Version

-

Requires Rust 1.51 or higher.

+

Requires Rust 1.56.1 or higher.

Minimum supported Rust version can be changed in the future, but it will be done with a minor version bump.

Controlling parallelism

diff --git a/print.html b/print.html index 5d163588..a599a40d 100644 --- a/print.html +++ b/print.html @@ -140,7 +140,7 @@

IMPORTANT: This library is in beta, and should not be used in production software.

Documentation

Minimum Supported Rust Version

-

Requires Rust 1.51 or higher.

+

Requires Rust 1.56.1 or higher.

Minimum supported Rust version can be changed in the future, but it will be done with a minor version bump.

Controlling parallelism

diff --git a/rustdoc/latest/COPYRIGHT.txt b/rustdoc/latest/COPYRIGHT.txt index c2629a83..34e48134 100644 --- a/rustdoc/latest/COPYRIGHT.txt +++ b/rustdoc/latest/COPYRIGHT.txt @@ -2,8 +2,7 @@ These documentation pages include resources by third parties. This copyright file applies only to those resources. The following third party resources are included, and carry their own copyright notices and license terms: -* Fira Sans (FiraSans-Regular.woff2, FiraSans-Medium.woff2, - FiraSans-Regular.woff, FiraSans-Medium.woff): +* Fira Sans (FiraSans-Regular.woff2, FiraSans-Medium.woff2): Copyright (c) 2014, Mozilla Foundation https://mozilla.org/ with Reserved Font Name Fira Sans. @@ -25,9 +24,7 @@ included, and carry their own copyright notices and license terms: Licensed under the MIT license (see LICENSE-MIT.txt). * Source Code Pro (SourceCodePro-Regular.ttf.woff2, - SourceCodePro-Semibold.ttf.woff2, SourceCodePro-It.ttf.woff2, - SourceCodePro-Regular.ttf.woff, SourceCodePro-Semibold.ttf.woff, - SourceCodePro-It.ttf.woff): + SourceCodePro-Semibold.ttf.woff2, SourceCodePro-It.ttf.woff2): Copyright 2010, 2012 Adobe Systems Incorporated (http://www.adobe.com/), with Reserved Font Name 'Source'. All Rights Reserved. Source is a trademark @@ -37,8 +34,7 @@ included, and carry their own copyright notices and license terms: See SourceCodePro-LICENSE.txt. * Source Serif 4 (SourceSerif4-Regular.ttf.woff2, SourceSerif4-Bold.ttf.woff2, - SourceSerif4-It.ttf.woff2, SourceSerif4-Regular.ttf.woff, - SourceSerif4-Bold.ttf.woff, SourceSerif4-It.ttf.woff): + SourceSerif4-It.ttf.woff2): Copyright 2014-2021 Adobe (http://www.adobe.com/), with Reserved Font Name 'Source'. All Rights Reserved. Source is a trademark of Adobe in the United diff --git a/rustdoc/latest/FiraSans-Medium.woff b/rustdoc/latest/FiraSans-Medium.woff deleted file mode 100644 index 7d742c5f..00000000 Binary files a/rustdoc/latest/FiraSans-Medium.woff and /dev/null differ diff --git a/rustdoc/latest/FiraSans-Regular.woff b/rustdoc/latest/FiraSans-Regular.woff deleted file mode 100644 index d8e0363f..00000000 Binary files a/rustdoc/latest/FiraSans-Regular.woff and /dev/null differ diff --git a/rustdoc/latest/NanumBarunGothic.ttf.woff b/rustdoc/latest/NanumBarunGothic.ttf.woff deleted file mode 100644 index fb063e8f..00000000 Binary files a/rustdoc/latest/NanumBarunGothic.ttf.woff and /dev/null differ diff --git a/rustdoc/latest/SourceCodePro-It.ttf.woff b/rustdoc/latest/SourceCodePro-It.ttf.woff deleted file mode 100644 index 8d68f2fe..00000000 Binary files a/rustdoc/latest/SourceCodePro-It.ttf.woff and /dev/null differ diff --git a/rustdoc/latest/SourceCodePro-Regular.ttf.woff b/rustdoc/latest/SourceCodePro-Regular.ttf.woff deleted file mode 100644 index 7be076e1..00000000 Binary files a/rustdoc/latest/SourceCodePro-Regular.ttf.woff and /dev/null differ diff --git a/rustdoc/latest/SourceCodePro-Semibold.ttf.woff b/rustdoc/latest/SourceCodePro-Semibold.ttf.woff deleted file mode 100644 index 61bc67b8..00000000 Binary files a/rustdoc/latest/SourceCodePro-Semibold.ttf.woff and /dev/null differ diff --git a/rustdoc/latest/SourceSerif4-Bold.ttf.woff b/rustdoc/latest/SourceSerif4-Bold.ttf.woff deleted file mode 100644 index 8ad41888..00000000 Binary files a/rustdoc/latest/SourceSerif4-Bold.ttf.woff and /dev/null differ diff --git a/rustdoc/latest/SourceSerif4-It.ttf.woff b/rustdoc/latest/SourceSerif4-It.ttf.woff deleted file mode 100644 index 2a34b5c4..00000000 Binary files a/rustdoc/latest/SourceSerif4-It.ttf.woff and /dev/null differ diff --git a/rustdoc/latest/SourceSerif4-Regular.ttf.woff b/rustdoc/latest/SourceSerif4-Regular.ttf.woff deleted file mode 100644 index 45a5521a..00000000 Binary files a/rustdoc/latest/SourceSerif4-Regular.ttf.woff and /dev/null differ diff --git a/rustdoc/latest/halo2/all.html b/rustdoc/latest/halo2/all.html index 09316cf9..94619d97 100644 --- a/rustdoc/latest/halo2/all.html +++ b/rustdoc/latest/halo2/all.html @@ -18,5 +18,5 @@

List of all items

+

Crate halo2

List of all items

\ No newline at end of file diff --git a/rustdoc/latest/halo2/index.html b/rustdoc/latest/halo2/index.html index f4ea9a0d..14b513f1 100644 --- a/rustdoc/latest/halo2/index.html +++ b/rustdoc/latest/halo2/index.html @@ -19,5 +19,5 @@
-

Crate halo2

source · [−]
Expand description

halo2

+

Crate halo2

source · [−]
Expand description

halo2

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/all.html b/rustdoc/latest/halo2_gadgets/all.html index c6f2621a..4a1c323a 100644 --- a/rustdoc/latest/halo2_gadgets/all.html +++ b/rustdoc/latest/halo2_gadgets/all.html @@ -18,5 +18,5 @@

List of all items

Structs

Enums

Traits

Functions

Constants

+

Crate halo2_gadgets

List of all items

Structs

Enums

Traits

Functions

Constants

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.FIXED_BASE_WINDOW_SIZE.html b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.FIXED_BASE_WINDOW_SIZE.html index 526ba0dc..b72c4fd3 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.FIXED_BASE_WINDOW_SIZE.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.FIXED_BASE_WINDOW_SIZE.html @@ -20,5 +20,5 @@

Constant halo2_gadgets::ecc::chip::constants::FIXED_BASE_WINDOW_SIZE

source · [−]
pub const FIXED_BASE_WINDOW_SIZE: usize = 3;
Expand description

Window size for fixed-base scalar multiplication

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.H.html b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.H.html index e31ed635..41b6bf54 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.H.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.H.html @@ -20,5 +20,5 @@

Constant halo2_gadgets::ecc::chip::constants::H

source · [−]
pub const H: usize = 1 << FIXED_BASE_WINDOW_SIZE; // 8usize
Expand description

$2^{FIXED_BASE_WINDOW_SIZE}$

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.NUM_WINDOWS.html b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.NUM_WINDOWS.html index 07ae373d..303a7b44 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.NUM_WINDOWS.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.NUM_WINDOWS.html @@ -20,5 +20,5 @@

Constant halo2_gadgets::ecc::chip::constants::NUM_WINDOWS

source · [−]
pub const NUM_WINDOWS: usize = (pallas::Scalar::NUM_BITS as usize + FIXED_BASE_WINDOW_SIZE - 1) / FIXED_BASE_WINDOW_SIZE; // 85usize
Expand description

Number of windows for a full-width scalar

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.NUM_WINDOWS_SHORT.html b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.NUM_WINDOWS_SHORT.html index 81e9ea09..d26241dc 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.NUM_WINDOWS_SHORT.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/constant.NUM_WINDOWS_SHORT.html @@ -20,5 +20,5 @@

Constant halo2_gadgets::ecc::chip::constants::NUM_WINDOWS_SHORT

source · [−]
pub const NUM_WINDOWS_SHORT: usize = (L_SCALAR_SHORT + FIXED_BASE_WINDOW_SIZE - 1) / FIXED_BASE_WINDOW_SIZE; // 22usize
Expand description

Number of windows for a short signed scalar

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.compute_lagrange_coeffs.html b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.compute_lagrange_coeffs.html index e3c2875c..2fe59f2f 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.compute_lagrange_coeffs.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.compute_lagrange_coeffs.html @@ -21,5 +21,5 @@

In halo2_gadgets::ecc::chip::constants

Function halo2_gadgets::ecc::chip::constants::compute_lagrange_coeffs

source · [−]
pub fn compute_lagrange_coeffs<C: CurveAffine>(
    base: C, 
    num_windows: usize
) -> Vec<[C::Base; 8]>ⓘNotable traits for Vec<u8, A>impl<A> Write for Vec<u8, A> where
    A: Allocator, 
Expand description

For each window, we interpolate the $x$-coordinate. Here, we pre-compute and store the coefficients of the interpolation polynomial.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.find_zs_and_us.html b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.find_zs_and_us.html index 382dc5a4..0b7c4c28 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.find_zs_and_us.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.find_zs_and_us.html @@ -29,5 +29,5 @@ If successful, return a vector of (z: u64, us: [C::Base; H]) for ea bases. The outputs of this function have been stored as constants, and it is not called anywhere in this codebase. However, we keep this function here as a utility for those who wish to use it with different parameters.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.test_lagrange_coeffs.html b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.test_lagrange_coeffs.html index d3138468..17cdb9b7 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.test_lagrange_coeffs.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.test_lagrange_coeffs.html @@ -21,5 +21,5 @@

In halo2_gadgets::ecc::chip::constants

Function halo2_gadgets::ecc::chip::constants::test_lagrange_coeffs

source · [−]
pub fn test_lagrange_coeffs<C: CurveAffine>(base: C, num_windows: usize)
Available on crate feature test-dependencies only.
Expand description

Test that Lagrange interpolation coefficients reproduce the correct x-coordinate for each fixed-base multiple in each window.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.test_zs_and_us.html b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.test_zs_and_us.html index cc65fda3..ba41a006 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.test_zs_and_us.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/fn.test_zs_and_us.html @@ -23,5 +23,5 @@ 1. z + y = u^2, 2. z - y is not a square for the y-coordinate of each fixed-base multiple in each window.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/index.html b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/index.html index bea357d8..d792d5ea 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/constants/index.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/constants/index.html @@ -35,5 +35,5 @@ for each fixed-base multiple in each window.

1. z + y = u^2, 2. z - y is not a square for the y-coordinate of each fixed-base multiple in each window.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/enum.BaseFieldElem.html b/rustdoc/latest/halo2_gadgets/ecc/chip/enum.BaseFieldElem.html index 92142eb0..842d63e2 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/enum.BaseFieldElem.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/enum.BaseFieldElem.html @@ -87,9 +87,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/enum.FullScalar.html b/rustdoc/latest/halo2_gadgets/ecc/chip/enum.FullScalar.html index 5162aac9..69368dad 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/enum.FullScalar.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/enum.FullScalar.html @@ -87,9 +87,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/enum.ShortScalar.html b/rustdoc/latest/halo2_gadgets/ecc/chip/enum.ShortScalar.html index 4a6dd1a3..db7e3d1f 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/enum.ShortScalar.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/enum.ShortScalar.html @@ -87,9 +87,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/index.html b/rustdoc/latest/halo2_gadgets/ecc/chip/index.html index 7e88c3c9..3fd6a134 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/index.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/index.html @@ -53,5 +53,5 @@ multiplication.

Traits

FixedPoint

Returns information about a fixed point that is required by EccChip.

ScalarKind

A trait representing the kind of scalar used with a particular FixedPoint.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccChip.html b/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccChip.html index abc0ec88..face1723 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccChip.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccChip.html @@ -135,9 +135,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccConfig.html b/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccConfig.html index b6f0ff72..b997af7d 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccConfig.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccConfig.html @@ -99,9 +99,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccPoint.html b/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccPoint.html index 53edb216..6521bb02 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccPoint.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccPoint.html @@ -98,9 +98,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccScalarFixed.html b/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccScalarFixed.html index a3810d0a..0525969e 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccScalarFixed.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccScalarFixed.html @@ -94,9 +94,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccScalarFixedShort.html b/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccScalarFixedShort.html index 8ad68b6e..64e315a4 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccScalarFixedShort.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/struct.EccScalarFixedShort.html @@ -100,9 +100,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/struct.NonIdentityEccPoint.html b/rustdoc/latest/halo2_gadgets/ecc/chip/struct.NonIdentityEccPoint.html index 4df117d0..152ec7db 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/struct.NonIdentityEccPoint.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/struct.NonIdentityEccPoint.html @@ -95,9 +95,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/trait.FixedPoint.html b/rustdoc/latest/halo2_gadgets/ecc/chip/trait.FixedPoint.html index bad56ee5..eb3cfab2 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/trait.FixedPoint.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/trait.FixedPoint.html @@ -42,5 +42,5 @@ arrays instead of Vecs.

Returns the $u$ values for this fixed point.

Returns the $z$ value for this fixed point.

Provided Methods

Returns the Lagrange coefficients for this fixed point.

-

Implementors

+

Implementors

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/chip/trait.ScalarKind.html b/rustdoc/latest/halo2_gadgets/ecc/chip/trait.ScalarKind.html index baa60ac8..952a9d70 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/chip/trait.ScalarKind.html +++ b/rustdoc/latest/halo2_gadgets/ecc/chip/trait.ScalarKind.html @@ -24,5 +24,5 @@ }
Expand description

A trait representing the kind of scalar used with a particular FixedPoint.

This trait exists because of limitations around const generics.

Required Associated Constants

The number of windows that this scalar kind requires.

-

Implementors

+

Implementors

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/index.html b/rustdoc/latest/halo2_gadgets/ecc/index.html index a062d0b4..81610645 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/index.html +++ b/rustdoc/latest/halo2_gadgets/ecc/index.html @@ -40,5 +40,5 @@ given curve.

Traits

EccInstructions

The set of circuit instructions required to use the ECC gadgets.

FixedPoints

Defines the fixed points for a given instantiation of the ECC chip.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/struct.FixedPoint.html b/rustdoc/latest/halo2_gadgets/ecc/struct.FixedPoint.html index 936842a1..49fe482b 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/struct.FixedPoint.html +++ b/rustdoc/latest/halo2_gadgets/ecc/struct.FixedPoint.html @@ -94,9 +94,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/struct.FixedPointBaseField.html b/rustdoc/latest/halo2_gadgets/ecc/struct.FixedPointBaseField.html index 231faa6c..7bb9d8a3 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/struct.FixedPointBaseField.html +++ b/rustdoc/latest/halo2_gadgets/ecc/struct.FixedPointBaseField.html @@ -93,9 +93,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/struct.FixedPointShort.html b/rustdoc/latest/halo2_gadgets/ecc/struct.FixedPointShort.html index b1b9a521..9fa2c2aa 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/struct.FixedPointShort.html +++ b/rustdoc/latest/halo2_gadgets/ecc/struct.FixedPointShort.html @@ -93,9 +93,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/struct.NonIdentityPoint.html b/rustdoc/latest/halo2_gadgets/ecc/struct.NonIdentityPoint.html index feb146cb..da948d8b 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/struct.NonIdentityPoint.html +++ b/rustdoc/latest/halo2_gadgets/ecc/struct.NonIdentityPoint.html @@ -102,9 +102,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/struct.Point.html b/rustdoc/latest/halo2_gadgets/ecc/struct.Point.html index 7a8bda53..a05e26c2 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/struct.Point.html +++ b/rustdoc/latest/halo2_gadgets/ecc/struct.Point.html @@ -97,9 +97,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/struct.ScalarFixed.html b/rustdoc/latest/halo2_gadgets/ecc/struct.ScalarFixed.html index 5c27d618..f429158c 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/struct.ScalarFixed.html +++ b/rustdoc/latest/halo2_gadgets/ecc/struct.ScalarFixed.html @@ -85,9 +85,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/struct.ScalarFixedShort.html b/rustdoc/latest/halo2_gadgets/ecc/struct.ScalarFixedShort.html index 22846048..da600ea0 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/struct.ScalarFixedShort.html +++ b/rustdoc/latest/halo2_gadgets/ecc/struct.ScalarFixedShort.html @@ -85,9 +85,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/struct.ScalarVar.html b/rustdoc/latest/halo2_gadgets/ecc/struct.ScalarVar.html index 5927e748..8fdd1d19 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/struct.ScalarVar.html +++ b/rustdoc/latest/halo2_gadgets/ecc/struct.ScalarVar.html @@ -93,9 +93,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/struct.X.html b/rustdoc/latest/halo2_gadgets/ecc/struct.X.html index 3376f290..446a1d44 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/struct.X.html +++ b/rustdoc/latest/halo2_gadgets/ecc/struct.X.html @@ -88,9 +88,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/trait.EccInstructions.html b/rustdoc/latest/halo2_gadgets/ecc/trait.EccInstructions.html index 6bf88bf3..c8bd2eb9 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/trait.EccInstructions.html +++ b/rustdoc/latest/halo2_gadgets/ecc/trait.EccInstructions.html @@ -78,5 +78,5 @@ This returns an error if the point is the identity.

Performs fixed-base scalar multiplication using a base field element as the scalar. In the current implementation, this base field element must be output from another instruction.

-

Implementors

+

Implementors

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/ecc/trait.FixedPoints.html b/rustdoc/latest/halo2_gadgets/ecc/trait.FixedPoints.html index 4355ec3a..38bbfba6 100644 --- a/rustdoc/latest/halo2_gadgets/ecc/trait.FixedPoints.html +++ b/rustdoc/latest/halo2_gadgets/ecc/trait.FixedPoints.html @@ -27,5 +27,5 @@

Required Associated Types

Fixed points that can be used with full-width scalar multiplication.

Fixed points that can be used with short scalar multiplication.

Fixed points that can be multiplied by base field elements.

-

Implementors

+

Implementors

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/index.html b/rustdoc/latest/halo2_gadgets/index.html index 10eb5c8a..a7cfc69c 100644 --- a/rustdoc/latest/halo2_gadgets/index.html +++ b/rustdoc/latest/halo2_gadgets/index.html @@ -26,5 +26,5 @@
sha256

Gadget and chips for the SHA-256 hash function.

sinsemilla

Gadgets for the Sinsemilla hash function.

utilities

Utility gadgets.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/poseidon/enum.PaddedWord.html b/rustdoc/latest/halo2_gadgets/poseidon/enum.PaddedWord.html index dd031f34..b97246b7 100644 --- a/rustdoc/latest/halo2_gadgets/poseidon/enum.PaddedWord.html +++ b/rustdoc/latest/halo2_gadgets/poseidon/enum.PaddedWord.html @@ -19,15 +19,15 @@
-

Enum halo2_gadgets::poseidon::PaddedWord

source · [−]
pub enum PaddedWord<F: Field> {
+    
Enum halo2_gadgets::poseidon::PaddedWord
source · [−]
pub enum PaddedWord<F: Field> {
     Message(AssignedCell<F, F>),
     Padding(F),
 }
Expand description

A word from the padded input to a Poseidon sponge.

Variants

Message(AssignedCell<F, F>)

A message word provided by the prover.

Padding(F)

A padding word, that will be fixed in the circuit parameters.

-

Trait Implementations

Returns a copy of the value. Read more

+

Trait Implementations

Returns a copy of the value. Read more

Performs copy-assignment from source. Read more

-

Formats the value using the given formatter. Read more

+

Formats the value using the given formatter. Read more

Auto Trait Implementations

Blanket Implementations

Gets the TypeId of self. Read more

Immutably borrows from an owned value. Read more

Mutably borrows from an owned value. Read more

@@ -95,9 +95,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/poseidon/index.html b/rustdoc/latest/halo2_gadgets/poseidon/index.html index ead13916..433253e3 100644 --- a/rustdoc/latest/halo2_gadgets/poseidon/index.html +++ b/rustdoc/latest/halo2_gadgets/poseidon/index.html @@ -32,5 +32,5 @@

Traits

PoseidonInstructions

The set of circuit instructions required to use the Poseidon permutation.

PoseidonSpongeInstructions

The set of circuit instructions required to use the Sponge and Hash gadgets.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/poseidon/struct.Hash.html b/rustdoc/latest/halo2_gadgets/poseidon/struct.Hash.html index 0a70a848..14a8deea 100644 --- a/rustdoc/latest/halo2_gadgets/poseidon/struct.Hash.html +++ b/rustdoc/latest/halo2_gadgets/poseidon/struct.Hash.html @@ -19,10 +19,10 @@
-

Struct halo2_gadgets::poseidon::Hash

source · [−]
pub struct Hash<F: FieldExt, PoseidonChip: PoseidonSpongeInstructions<F, S, D, T, RATE>, S: Spec<F, T, RATE>, D: Domain<F, RATE>, const T: usize, const RATE: usize> { /* private fields */ }
Expand description

A Poseidon hash function, built around a sponge.

-

Implementations

Initializes a new hasher.

-

Hashes the given input.

-

Trait Implementations

Formats the value using the given formatter. Read more

+

Struct halo2_gadgets::poseidon::Hash

source · [−]
pub struct Hash<F: FieldExt, PoseidonChip: PoseidonSpongeInstructions<F, S, D, T, RATE>, S: Spec<F, T, RATE>, D: Domain<F, RATE>, const T: usize, const RATE: usize> { /* private fields */ }
Expand description

A Poseidon hash function, built around a sponge.

+

Implementations

Initializes a new hasher.

+

Hashes the given input.

+

Trait Implementations

Formats the value using the given formatter. Read more

Auto Trait Implementations

Blanket Implementations

Gets the TypeId of self. Read more

Immutably borrows from an owned value. Read more

Mutably borrows from an owned value. Read more

@@ -87,9 +87,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/poseidon/struct.Pow5Chip.html b/rustdoc/latest/halo2_gadgets/poseidon/struct.Pow5Chip.html index ba116e52..6e9465ae 100644 --- a/rustdoc/latest/halo2_gadgets/poseidon/struct.Pow5Chip.html +++ b/rustdoc/latest/halo2_gadgets/poseidon/struct.Pow5Chip.html @@ -103,9 +103,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/poseidon/struct.Pow5Config.html b/rustdoc/latest/halo2_gadgets/poseidon/struct.Pow5Config.html index d0c891dd..803e445e 100644 --- a/rustdoc/latest/halo2_gadgets/poseidon/struct.Pow5Config.html +++ b/rustdoc/latest/halo2_gadgets/poseidon/struct.Pow5Config.html @@ -90,9 +90,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/poseidon/struct.Sponge.html b/rustdoc/latest/halo2_gadgets/poseidon/struct.Sponge.html index e4431a83..2251973d 100644 --- a/rustdoc/latest/halo2_gadgets/poseidon/struct.Sponge.html +++ b/rustdoc/latest/halo2_gadgets/poseidon/struct.Sponge.html @@ -19,12 +19,12 @@
-

Struct halo2_gadgets::poseidon::Sponge

source · [−]
pub struct Sponge<F: FieldExt, PoseidonChip: PoseidonSpongeInstructions<F, S, D, T, RATE>, S: Spec<F, T, RATE>, M: SpongeMode, D: Domain<F, RATE>, const T: usize, const RATE: usize> { /* private fields */ }
Expand description

A Poseidon sponge.

-

Implementations

Constructs a new duplex sponge for the given Poseidon specification.

-

Absorbs an element into the sponge.

-

Transitions the sponge into its squeezing state.

-

Squeezes an element from the sponge.

-

Trait Implementations

Formats the value using the given formatter. Read more

+

Struct halo2_gadgets::poseidon::Sponge

source · [−]
pub struct Sponge<F: FieldExt, PoseidonChip: PoseidonSpongeInstructions<F, S, D, T, RATE>, S: Spec<F, T, RATE>, M: SpongeMode, D: Domain<F, RATE>, const T: usize, const RATE: usize> { /* private fields */ }
Expand description

A Poseidon sponge.

+

Implementations

Constructs a new duplex sponge for the given Poseidon specification.

+

Absorbs an element into the sponge.

+

Transitions the sponge into its squeezing state.

+

Squeezes an element from the sponge.

+

Trait Implementations

Formats the value using the given formatter. Read more

Auto Trait Implementations

Blanket Implementations

Gets the TypeId of self. Read more

Immutably borrows from an owned value. Read more

Mutably borrows from an owned value. Read more

@@ -89,9 +89,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/poseidon/struct.StateWord.html b/rustdoc/latest/halo2_gadgets/poseidon/struct.StateWord.html index 20ca17f8..02848525 100644 --- a/rustdoc/latest/halo2_gadgets/poseidon/struct.StateWord.html +++ b/rustdoc/latest/halo2_gadgets/poseidon/struct.StateWord.html @@ -94,9 +94,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/poseidon/struct.Word.html b/rustdoc/latest/halo2_gadgets/poseidon/struct.Word.html index 67eccb7b..8da39ca6 100644 --- a/rustdoc/latest/halo2_gadgets/poseidon/struct.Word.html +++ b/rustdoc/latest/halo2_gadgets/poseidon/struct.Word.html @@ -19,10 +19,10 @@
-

Struct halo2_gadgets::poseidon::Word

source · [−]
pub struct Word<F: FieldExt, PoseidonChip: PoseidonInstructions<F, S, T, RATE>, S: Spec<F, T, RATE>, const T: usize, const RATE: usize> { /* private fields */ }
Expand description

A word over which the Poseidon permutation operates.

-

Implementations

The word contained in this gadget.

-

Construct a Word gadget from the inner word.

-

Trait Implementations

Formats the value using the given formatter. Read more

+

Struct halo2_gadgets::poseidon::Word

source · [−]
pub struct Word<F: FieldExt, PoseidonChip: PoseidonInstructions<F, S, T, RATE>, S: Spec<F, T, RATE>, const T: usize, const RATE: usize> { /* private fields */ }
Expand description

A word over which the Poseidon permutation operates.

+

Implementations

The word contained in this gadget.

+

Construct a Word gadget from the inner word.

+

Trait Implementations

Formats the value using the given formatter. Read more

Auto Trait Implementations

Blanket Implementations

Gets the TypeId of self. Read more

Immutably borrows from an owned value. Read more

Mutably borrows from an owned value. Read more

@@ -87,9 +87,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/poseidon/trait.PoseidonInstructions.html b/rustdoc/latest/halo2_gadgets/poseidon/trait.PoseidonInstructions.html index 11310e11..34d4e364 100644 --- a/rustdoc/latest/halo2_gadgets/poseidon/trait.PoseidonInstructions.html +++ b/rustdoc/latest/halo2_gadgets/poseidon/trait.PoseidonInstructions.html @@ -19,12 +19,12 @@
-

Trait halo2_gadgets::poseidon::PoseidonInstructions

source · [−]
pub trait PoseidonInstructions<F: FieldExt, S: Spec<F, T, RATE>, const T: usize, const RATE: usize>: Chip<F> {
+    
Trait halo2_gadgets::poseidon::PoseidonInstructions
source · [−]
pub trait PoseidonInstructions<F: FieldExt, S: Spec<F, T, RATE>, const T: usize, const RATE: usize>: Chip<F> {
     type Word: Clone + Debug + From<AssignedCell<F, F>> + Into<AssignedCell<F, F>>;
 
     fn permute(
        &self, 
        layouter: &mut impl Layouter<F>, 
        initial_state: &[Self::Word; T]
    ) -> Result<[Self::Word; T], Error>;
 }
Expand description

The set of circuit instructions required to use the Poseidon permutation.

-

Required Associated Types

Variable representing the word over which the Poseidon permutation operates.

-

Required Methods

Applies the Poseidon permutation to the given state.

-

Implementors

+

Required Associated Types

Variable representing the word over which the Poseidon permutation operates.

+

Required Methods

Applies the Poseidon permutation to the given state.

+

Implementors

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/poseidon/trait.PoseidonSpongeInstructions.html b/rustdoc/latest/halo2_gadgets/poseidon/trait.PoseidonSpongeInstructions.html index c1b21d28..5e3338e7 100644 --- a/rustdoc/latest/halo2_gadgets/poseidon/trait.PoseidonSpongeInstructions.html +++ b/rustdoc/latest/halo2_gadgets/poseidon/trait.PoseidonSpongeInstructions.html @@ -19,13 +19,13 @@
-

Trait halo2_gadgets::poseidon::PoseidonSpongeInstructions

source · [−]
pub trait PoseidonSpongeInstructions<F: FieldExt, S: Spec<F, T, RATE>, D: Domain<F, RATE>, const T: usize, const RATE: usize>: PoseidonInstructions<F, S, T, RATE> {
+    
Trait halo2_gadgets::poseidon::PoseidonSpongeInstructions
source · [−]
pub trait PoseidonSpongeInstructions<F: FieldExt, S: Spec<F, T, RATE>, D: Domain<F, RATE>, const T: usize, const RATE: usize>: PoseidonInstructions<F, S, T, RATE> {
     fn initial_state(
        &self, 
        layouter: &mut impl Layouter<F>
    ) -> Result<[Self::Word; T], Error>;
     fn add_input(
        &self, 
        layouter: &mut impl Layouter<F>, 
        initial_state: &[Self::Word; T], 
        input: &Absorbing<PaddedWord<F>, RATE>
    ) -> Result<[Self::Word; T], Error>;
     fn get_output(state: &[Self::Word; T]) -> Squeezing<Self::Word, RATE>;
 }
Expand description

The set of circuit instructions required to use the Sponge and Hash gadgets.

-

Required Methods

Returns the initial empty state for the given domain.

-

Adds the given input to the state.

-

Extracts sponge output from the given state.

-

Implementors

+

Required Methods

Returns the initial empty state for the given domain.

+

Adds the given input to the state.

+

Extracts sponge output from the given state.

+

Implementors

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/index.html b/rustdoc/latest/halo2_gadgets/primitives/index.html index 847134d6..2b020cb1 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/index.html +++ b/rustdoc/latest/halo2_gadgets/primitives/index.html @@ -23,5 +23,5 @@

Modules

poseidon

The Poseidon algebraic hash function.

sinsemilla

The Sinsemilla hash function and commitment scheme.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/poseidon/index.html b/rustdoc/latest/halo2_gadgets/primitives/poseidon/index.html index 20c3db0d..f3a02c1f 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/poseidon/index.html +++ b/rustdoc/latest/halo2_gadgets/primitives/poseidon/index.html @@ -31,5 +31,5 @@ standard number of rounds for 128-bit security “with margin”.

Domain

A domain in which a Poseidon hash function is being used.

Spec

A specification for a Poseidon permutation.

SpongeMode

The state of the Sponge.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.Absorbing.html b/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.Absorbing.html index 31901978..7f80d083 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.Absorbing.html +++ b/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.Absorbing.html @@ -19,9 +19,9 @@
-

Struct halo2_gadgets::primitives::poseidon::Absorbing

source · [−]
pub struct Absorbing<F, const RATE: usize>(_);
Expand description

The absorbing state of the Sponge.

-

Trait Implementations

Formats the value using the given formatter. Read more

-

Auto Trait Implementations

Blanket Implementations

Gets the TypeId of self. Read more

+

Struct halo2_gadgets::primitives::poseidon::Absorbing

source · [−]
pub struct Absorbing<F, const RATE: usize>(_);
Expand description

The absorbing state of the Sponge.

+

Trait Implementations

Formats the value using the given formatter. Read more

+

Auto Trait Implementations

Blanket Implementations

Gets the TypeId of self. Read more

Immutably borrows from an owned value. Read more

Mutably borrows from an owned value. Read more

Converts self into T using Into<T>. Read more

@@ -85,9 +85,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.ConstantLength.html b/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.ConstantLength.html index 950f62c8..b0d5623f 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.ConstantLength.html +++ b/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.ConstantLength.html @@ -19,16 +19,16 @@
-

Struct halo2_gadgets::primitives::poseidon::ConstantLength

source · [−]
pub struct ConstantLength<const L: usize>;
Expand description

A Poseidon hash function used with constant input length.

+

Struct halo2_gadgets::primitives::poseidon::ConstantLength

source · [−]
pub struct ConstantLength<const L: usize>;
Expand description

A Poseidon hash function used with constant input length.

Domain specified in section 4.2 of https://eprint.iacr.org/2019/458.pdf

-

Trait Implementations

Returns a copy of the value. Read more

+

Trait Implementations

Returns a copy of the value. Read more

Performs copy-assignment from source. Read more

-

Formats the value using the given formatter. Read more

-

Iterator that outputs padding field elements.

-

The name of this domain, for debug formatting purposes.

-

The initial capacity element, encoding this domain.

-

Returns the padding to be appended to the input.

-

Auto Trait Implementations

Blanket Implementations

Gets the TypeId of self. Read more

+

Formats the value using the given formatter. Read more

+

Iterator that outputs padding field elements.

+

The name of this domain, for debug formatting purposes.

+

The initial capacity element, encoding this domain.

+

Returns the padding to be appended to the input.

+

Auto Trait Implementations

Blanket Implementations

Gets the TypeId of self. Read more

Immutably borrows from an owned value. Read more

Mutably borrows from an owned value. Read more

Converts self into T using Into<T>. Read more

@@ -95,9 +95,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.Hash.html b/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.Hash.html index b8f288cc..93b00f6e 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.Hash.html +++ b/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.Hash.html @@ -19,10 +19,10 @@
-

Struct halo2_gadgets::primitives::poseidon::Hash

source · [−]
pub struct Hash<F: FieldExt, S: Spec<F, T, RATE>, D: Domain<F, RATE>, const T: usize, const RATE: usize> { /* private fields */ }
Expand description

A Poseidon hash function, built around a sponge.

-

Implementations

Initializes a new hasher.

-

Hashes the given input.

-

Trait Implementations

Formats the value using the given formatter. Read more

+

Struct halo2_gadgets::primitives::poseidon::Hash

source · [−]
pub struct Hash<F: FieldExt, S: Spec<F, T, RATE>, D: Domain<F, RATE>, const T: usize, const RATE: usize> { /* private fields */ }
Expand description

A Poseidon hash function, built around a sponge.

+

Implementations

Initializes a new hasher.

+

Hashes the given input.

+

Trait Implementations

Formats the value using the given formatter. Read more

Auto Trait Implementations

Blanket Implementations

Gets the TypeId of self. Read more

Immutably borrows from an owned value. Read more

Mutably borrows from an owned value. Read more

@@ -87,9 +87,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.P128Pow5T3.html b/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.P128Pow5T3.html index 294c831a..bcc9b5f1 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.P128Pow5T3.html +++ b/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.P128Pow5T3.html @@ -101,9 +101,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.Squeezing.html b/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.Squeezing.html index 133b3a4a..9fdf2623 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.Squeezing.html +++ b/rustdoc/latest/halo2_gadgets/primitives/poseidon/struct.Squeezing.html @@ -19,9 +19,9 @@
-

Struct halo2_gadgets::primitives::poseidon::Squeezing

source · [−]
pub struct Squeezing<F, const RATE: usize>(_);
Expand description

The squeezing state of the Sponge.

-

Trait Implementations

Formats the value using the given formatter. Read more

-

Auto Trait Implementations

Blanket Implementations

Gets the TypeId of self. Read more

+

Struct halo2_gadgets::primitives::poseidon::Squeezing

source · [−]
pub struct Squeezing<F, const RATE: usize>(_);
Expand description

The squeezing state of the Sponge.

+

Trait Implementations

Formats the value using the given formatter. Read more

+

Auto Trait Implementations

Blanket Implementations

Gets the TypeId of self. Read more

Immutably borrows from an owned value. Read more

Mutably borrows from an owned value. Read more

Converts self into T using Into<T>. Read more

@@ -85,9 +85,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/poseidon/trait.Domain.html b/rustdoc/latest/halo2_gadgets/primitives/poseidon/trait.Domain.html index e017f916..0017b104 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/poseidon/trait.Domain.html +++ b/rustdoc/latest/halo2_gadgets/primitives/poseidon/trait.Domain.html @@ -19,16 +19,16 @@
-

Trait halo2_gadgets::primitives::poseidon::Domain

source · [−]
pub trait Domain<F: FieldExt, const RATE: usize> {
+    
Trait halo2_gadgets::primitives::poseidon::Domain
source · [−]
pub trait Domain<F: FieldExt, const RATE: usize> {
     type Padding: IntoIterator<Item = F>;
 
     fn name() -> String;
     fn initial_capacity_element() -> F;
     fn padding(input_len: usize) -> Self::Padding;
 }
Expand description

A domain in which a Poseidon hash function is being used.

-

Required Associated Types

Iterator that outputs padding field elements.

-

Required Methods

The name of this domain, for debug formatting purposes.

-

The initial capacity element, encoding this domain.

-

Returns the padding to be appended to the input.

-

Implementors

+

Required Associated Types

Iterator that outputs padding field elements.

+

Required Methods

The name of this domain, for debug formatting purposes.

+

The initial capacity element, encoding this domain.

+

Returns the padding to be appended to the input.

+

Implementors

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/poseidon/trait.Spec.html b/rustdoc/latest/halo2_gadgets/primitives/poseidon/trait.Spec.html index 50a5af64..ed87d189 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/poseidon/trait.Spec.html +++ b/rustdoc/latest/halo2_gadgets/primitives/poseidon/trait.Spec.html @@ -19,7 +19,7 @@
-

Trait halo2_gadgets::primitives::poseidon::Spec

source · [−]
pub trait Spec<F: FieldExt, const T: usize, const RATE: usize>: Debug {
+    
Trait halo2_gadgets::primitives::poseidon::Spec
source · [−]
pub trait Spec<F: FieldExt, const T: usize, const RATE: usize>: Debug {
     fn full_rounds() -> usize;
     fn partial_rounds() -> usize;
     fn sbox(val: F) -> F;
@@ -27,14 +27,14 @@
 
     fn constants() -> (Vec<[F; T]>, [[F; T]; T], [[F; T]; T]) { ... }
 }
Expand description

A specification for a Poseidon permutation.

-

Required Methods

The number of full rounds for this specification.

+

Required Methods

The number of full rounds for this specification.

This must be an even number.

-

The number of partial rounds for this specification.

-

The S-box for this specification.

-

Side-loaded index of the first correct and secure MDS that will be generated by +

The number of partial rounds for this specification.

+

The S-box for this specification.

+

Side-loaded index of the first correct and secure MDS that will be generated by the reference implementation.

This is used by the default implementation of Spec::constants. If you are hard-coding the constants, you may leave this unimplemented.

-

Provided Methods

Generates (round_constants, mds, mds^-1) corresponding to this specification.

-

Implementors

+

Provided Methods

Generates (round_constants, mds, mds^-1) corresponding to this specification.

+

Implementors

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/poseidon/trait.SpongeMode.html b/rustdoc/latest/halo2_gadgets/primitives/poseidon/trait.SpongeMode.html index 4fc4b0b5..9212dfc5 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/poseidon/trait.SpongeMode.html +++ b/rustdoc/latest/halo2_gadgets/primitives/poseidon/trait.SpongeMode.html @@ -19,6 +19,6 @@
-

Trait halo2_gadgets::primitives::poseidon::SpongeMode

source · [−]
pub trait SpongeMode: SealedSpongeMode { }
Expand description

The state of the Sponge.

-

Implementors

+

Trait halo2_gadgets::primitives::poseidon::SpongeMode

source · [−]
pub trait SpongeMode: SealedSpongeMode { }
Expand description

The state of the Sponge.

+

Implementors

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.C.html b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.C.html index d1d37301..f870696e 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.C.html +++ b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.C.html @@ -21,5 +21,5 @@

In halo2_gadgets::primitives::sinsemilla

Constant halo2_gadgets::primitives::sinsemilla::C

source · [−]
pub const C: usize = 253;
Expand description

The largest integer such that $2^c \leq (r_P - 1) / 2$, where $r_P$ is the order of Pallas.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.INV_TWO_POW_K.html b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.INV_TWO_POW_K.html index 8d2a992b..eae545d5 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.INV_TWO_POW_K.html +++ b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.INV_TWO_POW_K.html @@ -20,5 +20,5 @@

Constant halo2_gadgets::primitives::sinsemilla::INV_TWO_POW_K

source · [−]
pub const INV_TWO_POW_K: [u8; 32];
Expand description

$\frac{1}{2^K}$

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.K.html b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.K.html index 5dd40cfb..587eed80 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.K.html +++ b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.K.html @@ -20,5 +20,5 @@

Constant halo2_gadgets::primitives::sinsemilla::K

source · [−]
pub const K: usize = 10;
Expand description

Number of bits of each message piece in $\mathsf{SinsemillaHashToPoint}$

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.Q_PERSONALIZATION.html b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.Q_PERSONALIZATION.html index f17b8b11..60231abe 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.Q_PERSONALIZATION.html +++ b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.Q_PERSONALIZATION.html @@ -20,5 +20,5 @@

Constant halo2_gadgets::primitives::sinsemilla::Q_PERSONALIZATION

source · [−]
pub const Q_PERSONALIZATION: &str = "z.cash:SinsemillaQ";
Expand description

SWU hash-to-curve personalization for Sinsemilla $Q$ generators.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.SINSEMILLA_S.html b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.SINSEMILLA_S.html index 515290ec..93f61ca4 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.SINSEMILLA_S.html +++ b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.SINSEMILLA_S.html @@ -21,5 +21,5 @@

In halo2_gadgets::primitives::sinsemilla

Constant halo2_gadgets::primitives::sinsemilla::SINSEMILLA_S

source · [−]
pub const SINSEMILLA_S: [(Base, Base); 1024];
Expand description

The precomputed bases for the Sinsemilla hash function.

https://zips.z.cash/protocol/protocol.pdf#concretesinsemillahash

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.S_PERSONALIZATION.html b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.S_PERSONALIZATION.html index e914ba37..eeb5df2f 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.S_PERSONALIZATION.html +++ b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/constant.S_PERSONALIZATION.html @@ -20,5 +20,5 @@

Constant halo2_gadgets::primitives::sinsemilla::S_PERSONALIZATION

source · [−]
pub const S_PERSONALIZATION: &str = "z.cash:SinsemillaS";
Expand description

SWU hash-to-curve personalization for Sinsemilla $S$ generators.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/index.html b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/index.html index 4485b48f..5fd26b26 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/index.html +++ b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/index.html @@ -33,5 +33,5 @@ of Pallas.

Q_PERSONALIZATION

SWU hash-to-curve personalization for Sinsemilla $Q$ generators.

SINSEMILLA_S

The precomputed bases for the Sinsemilla hash function.

S_PERSONALIZATION

SWU hash-to-curve personalization for Sinsemilla $S$ generators.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/struct.CommitDomain.html b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/struct.CommitDomain.html index 06e8c8b0..c4ceb659 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/struct.CommitDomain.html +++ b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/struct.CommitDomain.html @@ -91,9 +91,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/struct.HashDomain.html b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/struct.HashDomain.html index 734ac4d9..969a893d 100644 --- a/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/struct.HashDomain.html +++ b/rustdoc/latest/halo2_gadgets/primitives/sinsemilla/struct.HashDomain.html @@ -97,9 +97,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sha256/constant.BLOCK_SIZE.html b/rustdoc/latest/halo2_gadgets/sha256/constant.BLOCK_SIZE.html index 212dba7f..bae2c7e6 100644 --- a/rustdoc/latest/halo2_gadgets/sha256/constant.BLOCK_SIZE.html +++ b/rustdoc/latest/halo2_gadgets/sha256/constant.BLOCK_SIZE.html @@ -20,5 +20,5 @@

Constant halo2_gadgets::sha256::BLOCK_SIZE

source · [−]
pub const BLOCK_SIZE: usize = 16;
Expand description

The size of a SHA-256 block, in 32-bit words.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sha256/index.html b/rustdoc/latest/halo2_gadgets/sha256/index.html index bfe453cf..6fb65030 100644 --- a/rustdoc/latest/halo2_gadgets/sha256/index.html +++ b/rustdoc/latest/halo2_gadgets/sha256/index.html @@ -31,5 +31,5 @@
BLOCK_SIZE

The size of a SHA-256 block, in 32-bit words.

Traits

Sha256Instructions

The set of circuit instructions required to use the Sha256 gadget.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sha256/struct.BlockWord.html b/rustdoc/latest/halo2_gadgets/sha256/struct.BlockWord.html index eeb7b0a9..7bc2f01c 100644 --- a/rustdoc/latest/halo2_gadgets/sha256/struct.BlockWord.html +++ b/rustdoc/latest/halo2_gadgets/sha256/struct.BlockWord.html @@ -91,9 +91,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sha256/struct.Sha256.html b/rustdoc/latest/halo2_gadgets/sha256/struct.Sha256.html index 8e7fa0b2..4c894f48 100644 --- a/rustdoc/latest/halo2_gadgets/sha256/struct.Sha256.html +++ b/rustdoc/latest/halo2_gadgets/sha256/struct.Sha256.html @@ -91,9 +91,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sha256/struct.Sha256Digest.html b/rustdoc/latest/halo2_gadgets/sha256/struct.Sha256Digest.html index 80c0f321..b1fe1182 100644 --- a/rustdoc/latest/halo2_gadgets/sha256/struct.Sha256Digest.html +++ b/rustdoc/latest/halo2_gadgets/sha256/struct.Sha256Digest.html @@ -85,9 +85,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sha256/struct.Table16Chip.html b/rustdoc/latest/halo2_gadgets/sha256/struct.Table16Chip.html index fd3b67b6..78fb6c7f 100644 --- a/rustdoc/latest/halo2_gadgets/sha256/struct.Table16Chip.html +++ b/rustdoc/latest/halo2_gadgets/sha256/struct.Table16Chip.html @@ -108,9 +108,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sha256/struct.Table16Config.html b/rustdoc/latest/halo2_gadgets/sha256/struct.Table16Config.html index 22226017..7acd3789 100644 --- a/rustdoc/latest/halo2_gadgets/sha256/struct.Table16Config.html +++ b/rustdoc/latest/halo2_gadgets/sha256/struct.Table16Config.html @@ -90,9 +90,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sha256/trait.Sha256Instructions.html b/rustdoc/latest/halo2_gadgets/sha256/trait.Sha256Instructions.html index 8beb6794..07f2e60f 100644 --- a/rustdoc/latest/halo2_gadgets/sha256/trait.Sha256Instructions.html +++ b/rustdoc/latest/halo2_gadgets/sha256/trait.Sha256Instructions.html @@ -36,5 +36,5 @@ function.

Starting from the given initialized state, processes a block of input and returns the final state.

Converts the given state into a message digest.

-

Implementors

+

Implementors

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/chip/index.html b/rustdoc/latest/halo2_gadgets/sinsemilla/chip/index.html index 45facab0..e3162c6c 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/chip/index.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/chip/index.html @@ -19,9 +19,9 @@
-

Module halo2_gadgets::sinsemilla::chip

source · [−]
Expand description

Chip implementations for the Sinsemilla gadgets.

+

Module halo2_gadgets::sinsemilla::chip

source · [−]
Expand description

Chip implementations for the Sinsemilla gadgets.

Structs

SinsemillaChip

A chip that implements 10-bit Sinsemilla using a lookup table and 5 advice columns.

SinsemillaConfig

Configuration for the Sinsemilla hash chip

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/chip/struct.SinsemillaChip.html b/rustdoc/latest/halo2_gadgets/sinsemilla/chip/struct.SinsemillaChip.html index 2226fd50..57c101dc 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/chip/struct.SinsemillaChip.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/chip/struct.SinsemillaChip.html @@ -20,9 +20,9 @@

Struct halo2_gadgets::sinsemilla::chip::SinsemillaChip

source · [−]
pub struct SinsemillaChip<Hash, Commit, Fixed> where
    Hash: HashDomains<Affine>,
    Fixed: FixedPoints<Affine>,
    Commit: CommitDomains<Affine, Fixed, Hash>,  { /* private fields */ }
Expand description

A chip that implements 10-bit Sinsemilla using a lookup table and 5 advice columns.

-

Implementations

Reconstructs this chip from the given config.

+

Implementations

Reconstructs this chip from the given config.

Loads the lookup table required by this chip into the circuit.

-
Side-effects
+
Side-effects

All columns in advices and will be equality-enabled.

Trait Implementations

A type that holds the configuration for this chip, and any other state it may need during circuit synthesis, that can be derived during Circuit::configure. Read more

@@ -37,7 +37,7 @@ synthesis.

This method tests for self and other values to be equal, and is used by ==. Read more

This method tests for !=.

-

A variable in the circuit.

+

A variable in the circuit.

A message composed of Self::MessagePieces.

A piece in a message containing a number of K-bit words. A Self::MessagePiece fits in a single base field element, @@ -51,13 +51,13 @@ that z_next = (z_cur - word_next) /

A type enumerating the fixed points used in CommitDomains.

HashDomains used in this instruction.

CommitDomains used in this instruction.

-

Witness a message piece given a field element. Returns a Self::MessagePiece +

Witness a message piece given a field element. Returns a Self::MessagePiece encoding the given message. Read more

-

Hashes a message to an ECC curve point. +

Hashes a message to an ECC curve point. This returns both the resulting point, as well as the message decomposition in the form of intermediate values in a cumulative sum. Read more

-

Extracts the x-coordinate of the output of a Sinsemilla hash.

+

Extracts the x-coordinate of the output of a Sinsemilla hash.

Auto Trait Implementations

Blanket Implementations

Gets the TypeId of self. Read more

Immutably borrows from an owned value. Read more

Mutably borrows from an owned value. Read more

@@ -125,9 +125,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/chip/struct.SinsemillaConfig.html b/rustdoc/latest/halo2_gadgets/sinsemilla/chip/struct.SinsemillaConfig.html index 62c822b4..a680400d 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/chip/struct.SinsemillaConfig.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/chip/struct.SinsemillaConfig.html @@ -94,9 +94,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/index.html b/rustdoc/latest/halo2_gadgets/sinsemilla/index.html index fdba619f..5647ac0e 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/index.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/index.html @@ -37,5 +37,5 @@ be used.

This trait is bounded on two constant parameters: K, the number of bits in each word accepted by the Sinsemilla hash, and MAX_WORDS, the maximum number of words that a single hash instance can process.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/chip/index.html b/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/chip/index.html index eb327406..dab419c6 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/chip/index.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/chip/index.html @@ -19,9 +19,9 @@
-

Module halo2_gadgets::sinsemilla::merkle::chip

source · [−]
Expand description

Chip implementing a Merkle hash using Sinsemilla as the hash function.

+

Module halo2_gadgets::sinsemilla::merkle::chip

source · [−]
Expand description

Chip implementing a Merkle hash using Sinsemilla as the hash function.

Structs

MerkleChip

Chip implementing MerkleInstructions.

MerkleConfig

Configuration for the MerkleChip implementation.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/chip/struct.MerkleChip.html b/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/chip/struct.MerkleChip.html index 62ca2abd..a5198fe7 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/chip/struct.MerkleChip.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/chip/struct.MerkleChip.html @@ -19,25 +19,25 @@
-

Struct halo2_gadgets::sinsemilla::merkle::chip::MerkleChip

source · [−]
pub struct MerkleChip<Hash, Commit, Fixed> where
    Hash: HashDomains<Affine>,
    Fixed: FixedPoints<Affine>,
    Commit: CommitDomains<Affine, Fixed, Hash>,  { /* private fields */ }
Expand description

Chip implementing MerkleInstructions.

-

Implementations

Configures the MerkleChip.

-

Constructs a MerkleChip given a MerkleConfig.

-

Trait Implementations

A type that holds the configuration for this chip, and any other state it may need +

Struct halo2_gadgets::sinsemilla::merkle::chip::MerkleChip

source · [−]
pub struct MerkleChip<Hash, Commit, Fixed> where
    Hash: HashDomains<Affine>,
    Fixed: FixedPoints<Affine>,
    Commit: CommitDomains<Affine, Fixed, Hash>,  { /* private fields */ }
Expand description

Chip implementing MerkleInstructions.

+

Implementations

Configures the MerkleChip.

+

Constructs a MerkleChip given a MerkleConfig.

+

Trait Implementations

A type that holds the configuration for this chip, and any other state it may need during circuit synthesis, that can be derived during Circuit::configure. Read more

A type that holds any general chip state that needs to be loaded at the start of Circuit::synthesize. This might simply be () for some chips. Read more

-

The chip holds its own configuration.

-

Provides access to general chip state loaded at the beginning of circuit +

The chip holds its own configuration.

+

Provides access to general chip state loaded at the beginning of circuit synthesis. Read more

-

Returns a copy of the value. Read more

+

Returns a copy of the value. Read more

Performs copy-assignment from source. Read more

-

Given an input pair (a,b) and a swap boolean flag, returns +

Given an input pair (a,b) and a swap boolean flag, returns (b,a) if swap is set, else (a,b) if swap is not set. Read more

-

Formats the value using the given formatter. Read more

-

Compute MerkleCRH for a given layer. The hash that computes the root +

Formats the value using the given formatter. Read more

+

Compute MerkleCRH for a given layer. The hash that computes the root is at layer 0, and the hashes that are applied to two leaves are at layer MERKLE_DEPTH - 1 = layer 31. Read more

-

A variable in the circuit.

+

A variable in the circuit.

A message composed of Self::MessagePieces.

A piece in a message containing a number of K-bit words. A Self::MessagePiece fits in a single base field element, @@ -51,14 +51,14 @@ that z_next = (z_cur - word_next) /

A type enumerating the fixed points used in CommitDomains.

HashDomains used in this instruction.

CommitDomains used in this instruction.

-

Witness a message piece given a field element. Returns a Self::MessagePiece +

Witness a message piece given a field element. Returns a Self::MessagePiece encoding the given message. Read more

-

Hashes a message to an ECC curve point. +

Hashes a message to an ECC curve point. This returns both the resulting point, as well as the message decomposition in the form of intermediate values in a cumulative sum. Read more

-

Extracts the x-coordinate of the output of a Sinsemilla hash.

-

Variable in the circuit.

+

Extracts the x-coordinate of the output of a Sinsemilla hash.

+

Variable in the circuit.

Load a variable.

Auto Trait Implementations

Blanket Implementations

Gets the TypeId of self. Read more

Immutably borrows from an owned value. Read more

@@ -127,9 +127,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/chip/struct.MerkleConfig.html b/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/chip/struct.MerkleConfig.html index 422885ff..80c8ef48 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/chip/struct.MerkleConfig.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/chip/struct.MerkleConfig.html @@ -19,10 +19,10 @@
-

Struct halo2_gadgets::sinsemilla::merkle::chip::MerkleConfig

source · [−]
pub struct MerkleConfig<Hash, Commit, Fixed> where
    Hash: HashDomains<Affine>,
    Fixed: FixedPoints<Affine>,
    Commit: CommitDomains<Affine, Fixed, Hash>,  { /* private fields */ }
Expand description

Configuration for the MerkleChip implementation.

-

Trait Implementations

Returns a copy of the value. Read more

+

Struct halo2_gadgets::sinsemilla::merkle::chip::MerkleConfig

source · [−]
pub struct MerkleConfig<Hash, Commit, Fixed> where
    Hash: HashDomains<Affine>,
    Fixed: FixedPoints<Affine>,
    Commit: CommitDomains<Affine, Fixed, Hash>,  { /* private fields */ }
Expand description

Configuration for the MerkleChip implementation.

+

Trait Implementations

Returns a copy of the value. Read more

Performs copy-assignment from source. Read more

-

Formats the value using the given formatter. Read more

+

Formats the value using the given formatter. Read more

Auto Trait Implementations

Blanket Implementations

Gets the TypeId of self. Read more

Immutably borrows from an owned value. Read more

Mutably borrows from an owned value. Read more

@@ -90,9 +90,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/constant.MERKLE_CRH_PERSONALIZATION.html b/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/constant.MERKLE_CRH_PERSONALIZATION.html index 6450a2b7..9c4279f0 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/constant.MERKLE_CRH_PERSONALIZATION.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/constant.MERKLE_CRH_PERSONALIZATION.html @@ -20,5 +20,5 @@

Constant halo2_gadgets::sinsemilla::merkle::MERKLE_CRH_PERSONALIZATION

source · [−]
pub const MERKLE_CRH_PERSONALIZATION: &str = "z.cash:Orchard-MerkleCRH";
Expand description

SWU hash-to-curve personalization for the Merkle CRH generator

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/index.html b/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/index.html index abe16534..7f761d90 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/index.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/index.html @@ -31,5 +31,5 @@ specific position.

MerkleInstructions

Instructions to check the validity of a Merkle path of a given PATH_LENGTH. The hash function used is a Sinsemilla instance with K-bit words. The hash function can process MAX_WORDS words.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/struct.MerklePath.html b/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/struct.MerklePath.html index 36184953..cd5f5f44 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/struct.MerklePath.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/struct.MerklePath.html @@ -93,9 +93,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/trait.MerkleInstructions.html b/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/trait.MerkleInstructions.html index 31482291..02c94b6e 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/trait.MerkleInstructions.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/merkle/trait.MerkleInstructions.html @@ -27,5 +27,5 @@ The hash function can process MAX_WORDS words.

Required Methods

Compute MerkleCRH for a given layer. The hash that computes the root is at layer 0, and the hashes that are applied to two leaves are at layer MERKLE_DEPTH - 1 = layer 31.

-

Implementors

+

Implementors

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/struct.CommitDomain.html b/rustdoc/latest/halo2_gadgets/sinsemilla/struct.CommitDomain.html index 4a44c5f9..74175be2 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/struct.CommitDomain.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/struct.CommitDomain.html @@ -89,9 +89,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/struct.HashDomain.html b/rustdoc/latest/halo2_gadgets/sinsemilla/struct.HashDomain.html index ccfc7fc5..df4dfc40 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/struct.HashDomain.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/struct.HashDomain.html @@ -89,9 +89,9 @@ builds. Read more

Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/struct.Message.html b/rustdoc/latest/halo2_gadgets/sinsemilla/struct.Message.html index df74f536..7d3b578b 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/struct.Message.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/struct.Message.html @@ -92,9 +92,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/struct.MessagePiece.html b/rustdoc/latest/halo2_gadgets/sinsemilla/struct.MessagePiece.html index fc3e29f2..32472184 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/struct.MessagePiece.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/struct.MessagePiece.html @@ -92,9 +92,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/trait.CommitDomains.html b/rustdoc/latest/halo2_gadgets/sinsemilla/trait.CommitDomains.html index 190c8fa5..6e5d8784 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/trait.CommitDomains.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/trait.CommitDomains.html @@ -26,5 +26,5 @@

Required Methods

Returns the fixed point corresponding to the R constant used for randomization in this CommitDomain.

Returns the HashDomain contained in this CommitDomain

-

Implementors

+

Implementors

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/trait.HashDomains.html b/rustdoc/latest/halo2_gadgets/sinsemilla/trait.HashDomains.html index 61ebec45..1fa173fd 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/trait.HashDomains.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/trait.HashDomains.html @@ -23,5 +23,5 @@ fn Q(&self) -> C; }
Expand description

Trait allowing circuit’s Sinsemilla HashDomains to be enumerated.

Required Methods

Returns the Q constant for this domain.

-

Implementors

+

Implementors

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/sinsemilla/trait.SinsemillaInstructions.html b/rustdoc/latest/halo2_gadgets/sinsemilla/trait.SinsemillaInstructions.html index 1716e751..d44b81d4 100644 --- a/rustdoc/latest/halo2_gadgets/sinsemilla/trait.SinsemillaInstructions.html +++ b/rustdoc/latest/halo2_gadgets/sinsemilla/trait.SinsemillaInstructions.html @@ -67,5 +67,5 @@ This returns both the resulting point, as well as the message decomposition in the form of intermediate values in a cumulative sum.

Extracts the x-coordinate of the output of a Sinsemilla hash.

-

Implementors

+

Implementors

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/cond_swap/index.html b/rustdoc/latest/halo2_gadgets/utilities/cond_swap/index.html index 950c2f21..936cf474 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/cond_swap/index.html +++ b/rustdoc/latest/halo2_gadgets/utilities/cond_swap/index.html @@ -25,5 +25,5 @@
CondSwapConfig

Configuration for the CondSwapChip.

Traits

CondSwapInstructions

Instructions for a conditional swap gadget.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/cond_swap/struct.CondSwapChip.html b/rustdoc/latest/halo2_gadgets/utilities/cond_swap/struct.CondSwapChip.html index debdc2de..382ba992 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/cond_swap/struct.CondSwapChip.html +++ b/rustdoc/latest/halo2_gadgets/utilities/cond_swap/struct.CondSwapChip.html @@ -105,9 +105,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/cond_swap/struct.CondSwapConfig.html b/rustdoc/latest/halo2_gadgets/utilities/cond_swap/struct.CondSwapConfig.html index 703260dc..562ddb40 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/cond_swap/struct.CondSwapConfig.html +++ b/rustdoc/latest/halo2_gadgets/utilities/cond_swap/struct.CondSwapConfig.html @@ -90,9 +90,9 @@ builds. Read more

Creates owned data from borrowed data, usually by cloning. Read more

🔬 This is a nightly-only experimental API. (toowned_clone_into)

Uses borrowed data to replace owned data, usually by cloning. Read more

Attempts to convert self into T using TryInto<T>. Read more

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-

The type returned in the event of a conversion error.

+

The type returned in the event of a conversion error.

Performs the conversion.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/cond_swap/trait.CondSwapInstructions.html b/rustdoc/latest/halo2_gadgets/utilities/cond_swap/trait.CondSwapInstructions.html index 9c6fc2d6..6d15f873 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/cond_swap/trait.CondSwapInstructions.html +++ b/rustdoc/latest/halo2_gadgets/utilities/cond_swap/trait.CondSwapInstructions.html @@ -26,5 +26,5 @@ (b,a) if swap is set, else (a,b) if swap is not set.

The second element of the pair is required to be a witnessed value, not a variable that already exists in the circuit.

-

Implementors

+

Implementors

\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/decompose_running_sum/index.html b/rustdoc/latest/halo2_gadgets/utilities/decompose_running_sum/index.html index 225d6b73..6325fd9a 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/decompose_running_sum/index.html +++ b/rustdoc/latest/halo2_gadgets/utilities/decompose_running_sum/index.html @@ -40,5 +40,5 @@ the range check constraint to stay within the degree bound.

Structs

RunningSum

The running sum $[z_0, …, z_W]$. If created in strict mode, $z_W = 0$.

RunningSumConfig

Configuration that provides methods for running sum decomposition.

-
+
\ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/decompose_running_sum/struct.RunningSum.html b/rustdoc/latest/halo2_gadgets/utilities/decompose_running_sum/struct.RunningSum.html index ccb3debe..cebfbb9e 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/decompose_running_sum/struct.RunningSum.html +++ b/rustdoc/latest/halo2_gadgets/utilities/decompose_running_sum/struct.RunningSum.html @@ -61,23 +61,62 @@ as its ‘length’.

v.push(1); assert!(!v.is_empty()); -

Methods from Deref<Target = [T]>

Returns the number of elements in the slice.

+

Methods from Deref<Target = [T]>

🔬 This is a nightly-only experimental API. (slice_flatten)

Takes a &[[T; N]], and flattens it to a &[T].

+
Panics
+

This panics if the length of the resulting slice would overflow a usize.

+

This is only possible when flattening a slice of arrays of zero-sized +types, and thus tends to be irrelevant in practice. If +size_of::<T>() > 0, this will never panic.

Examples
+
#![feature(slice_flatten)]
+
+assert_eq!([[1, 2, 3], [4, 5, 6]].flatten(), &[1, 2, 3, 4, 5, 6]);
+
+assert_eq!(
+    [[1, 2, 3], [4, 5, 6]].flatten(),
+    [[1, 2], [3, 4], [5, 6]].flatten(),
+);
+
+let slice_of_empty_arrays: &[[i32; 0]] = &[[], [], [], [], []];
+assert!(slice_of_empty_arrays.flatten().is_empty());
+
+let empty_slice_of_arrays: &[[u32; 10]] = &[];
+assert!(empty_slice_of_arrays.flatten().is_empty());
+
🔬 This is a nightly-only experimental API. (slice_flatten)

Takes a &mut [[T; N]], and flattens it to a &mut [T].

+
Panics
+

This panics if the length of the resulting slice would overflow a usize.

+

This is only possible when flattening a slice of arrays of zero-sized +types, and thus tends to be irrelevant in practice. If +size_of::<T>() > 0, this will never panic.

+
Examples
+
#![feature(slice_flatten)]
+
+fn add_5_to_all(slice: &mut [i32]) {
+    for i in slice {
+        *i += 5;
+    }
+}
+
+let mut array = [[1, 2, 3], [4, 5, 6], [7, 8, 9]];
+add_5_to_all(array.flatten_mut());
+assert_eq!(array, [[6, 7, 8], [9, 10, 11], [12, 13, 14]]);
+

Returns the number of elements in the slice.

+
Examples
let a = [1, 2, 3];
 assert_eq!(a.len(), 3);

Returns true if the slice has a length of 0.

-
Examples
+
Examples
let a = [1, 2, 3];
 assert!(!a.is_empty());

Returns the first element of the slice, or None if it is empty.

-
Examples
+
Examples
let v = [10, 40, 30];
 assert_eq!(Some(&10), v.first());
 
 let w: &[i32] = &[];
 assert_eq!(None, w.first());

Returns a mutable pointer to the first element of the slice, or None if it is empty.

-
Examples
+
Examples
let x = &mut [0, 1, 2];
 
 if let Some(first) = x.first_mut() {
@@ -85,7 +124,7 @@ as its ‘length’.

 }
 assert_eq!(x, &[5, 1, 2]);

Returns the first and all the rest of the elements of the slice, or None if it is empty.

-
Examples
+
Examples
let x = &[0, 1, 2];
 
 if let Some((first, elements)) = x.split_first() {
@@ -93,7 +132,7 @@ as its ‘length’.

     assert_eq!(elements, &[1, 2]);
 }

Returns the first and all the rest of the elements of the slice, or None if it is empty.

-
Examples
+
Examples
let x = &mut [0, 1, 2];
 
 if let Some((first, elements)) = x.split_first_mut() {
@@ -103,7 +142,7 @@ as its ‘length’.

 }
 assert_eq!(x, &[3, 4, 5]);

Returns the last and all the rest of the elements of the slice, or None if it is empty.

-
Examples
+
Examples
let x = &[0, 1, 2];
 
 if let Some((last, elements)) = x.split_last() {
@@ -111,7 +150,7 @@ as its ‘length’.

     assert_eq!(elements, &[0, 1]);
 }

Returns the last and all the rest of the elements of the slice, or None if it is empty.

-
Examples
+
Examples
let x = &mut [0, 1, 2];
 
 if let Some((last, elements)) = x.split_last_mut() {
@@ -121,14 +160,14 @@ as its ‘length’.

 }
 assert_eq!(x, &[4, 5, 3]);

Returns the last element of the slice, or None if it is empty.

-
Examples
+
Examples
let v = [10, 40, 30];
 assert_eq!(Some(&30), v.last());
 
 let w: &[i32] = &[];
 assert_eq!(None, w.last());

Returns a mutable pointer to the last item in the slice.

-
Examples
+
Examples
let x = &mut [0, 1, 2];
 
 if let Some(last) = x.last_mut() {
@@ -143,7 +182,7 @@ position or None if out of bounds.
 
  • If given a range, returns the subslice corresponding to that range,
 or None if out of bounds.
    • 
 
-
    Examples
    
+
    Examples
    
 
    let v = [10, 40, 30];
 assert_eq!(Some(&40), v.get(1));
 assert_eq!(Some(&[10, 40][..]), v.get(0..2));
@@ -151,7 +190,7 @@ or None if out of bounds.
 assert_eq!(None, v.get(0..4));

    Returns a mutable reference to an element or subslice depending on the type of index (see get) or None if the index is out of bounds.

    -
    Examples
    +
    Examples
    let x = &mut [0, 1, 2];
 
 if let Some(elem) = x.get_mut(1) {
@@ -164,7 +203,7 @@ checking.
    
 
    Safety
    
 
    Calling this method with an out-of-bounds index is undefined behavior
 even if the resulting reference is not used.
    
-
    Examples
    
+
    Examples
    
 
    let x = &[1, 2, 4];
 
 unsafe {
@@ -176,7 +215,7 @@ bounds checking.
    
 
    Safety
    
 
    Calling this method with an out-of-bounds index is undefined behavior
 even if the resulting reference is not used.
    
-
    Examples
    
+
    Examples
    
 
    let x = &mut [1, 2, 4];
 
 unsafe {
@@ -192,7 +231,7 @@ is never written to (except inside an UnsafeCell) using this pointe
 derived from it. If you need to mutate the contents of the slice, use as_mut_ptr.
    
 
    Modifying the container referenced by this slice may cause its buffer
 to be reallocated, which would also make any pointers to it invalid.
    
-
    Examples
    
+
    Examples
    
 
    let x = &[1, 2, 4];
 let x_ptr = x.as_ptr();
 
@@ -206,7 +245,7 @@ to be reallocated, which would also make any pointers to it invalid.
    
 function returns, or else it will end up pointing to garbage.
    
 
    Modifying the container referenced by this slice may cause its buffer
 to be reallocated, which would also make any pointers to it invalid.
    
-
    Examples
    
+
    Examples
    
 
    let x = &mut [1, 2, 4];
 let x_ptr = x.as_mut_ptr();
 
@@ -253,9 +292,9 @@ common in C++.
    
 
  • a - The index of the first element
    • 
 
  • b - The index of the second element
    • 
 
-
    Panics
    
+
    Panics
    
 
    Panics if a or b are out of bounds.
    
-
    Examples
    
+
    Examples
    
 
    let mut v = ["a", "b", "c", "d", "e"];
 v.swap(2, 4);
 assert!(v == ["a", "b", "e", "d", "c"]);
    
@@ -269,7 +308,7 @@ common in C++.
    
 
    Safety
    
 
    Calling this method with an out-of-bounds index is undefined behavior.
 The caller has to ensure that a < self.len() and b < self.len().
    
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_swap_unchecked)]
 
 let mut v = ["a", "b", "c", "d"];
@@ -277,12 +316,12 @@ The caller has to ensure that a < self.len() and b < se
 unsafe { v.swap_unchecked(1, 3) };
 assert!(v == ["a", "d", "c", "b"]);

    Reverses the order of elements in the slice, in place.

    -
    Examples
    +
    Examples
    let mut v = [1, 2, 3];
 v.reverse();
 assert!(v == [3, 2, 1]);

    Returns an iterator over the slice.

    -
    Examples
    +
    Examples
    let x = &[1, 2, 4];
 let mut iterator = x.iter();
 
@@ -291,7 +330,7 @@ The caller has to ensure that a < self.len() and b < se
 assert_eq!(iterator.next(), Some(&4));
 assert_eq!(iterator.next(), None);

    Returns an iterator that allows modifying each value.

    -
    Examples
    +
    Examples
    let x = &mut [1, 2, 4];
 for elem in x.iter_mut() {
     *elem += 2;
@@ -300,9 +339,9 @@ The caller has to ensure that a < self.len() and b < se

    Returns an iterator over all contiguous windows of length size. The windows overlap. If the slice is shorter than size, the iterator returns no values.

    -
    Panics
    +
    Panics

    Panics if size is 0.

    -
    Examples
    +
    Examples
    let slice = ['r', 'u', 's', 't'];
 let mut iter = slice.windows(2);
 assert_eq!(iter.next().unwrap(), &['r', 'u']);
@@ -321,9 +360,9 @@ slice, then the last chunk will not have length chunk_size.
    
 
    See chunks_exact for a variant of this iterator that returns chunks of always exactly
 chunk_size elements, and rchunks for the same iterator but starting at the end of the
 slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let slice = ['l', 'o', 'r', 'e', 'm'];
 let mut iter = slice.chunks(2);
 assert_eq!(iter.next().unwrap(), &['l', 'o']);
@@ -337,9 +376,9 @@ length of the slice, then the last chunk will not have length chunk_sizeSee chunks_exact_mut for a variant of this iterator that returns chunks of always
 exactly chunk_size elements, and rchunks_mut for the same iterator but starting at
 the end of the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
 
@@ -359,9 +398,9 @@ from the remainder function of the iterator.
    
 resulting code better than in the case of chunks.
    
 
    See chunks for a variant of this iterator that also returns the remainder as a smaller
 chunk, and rchunks_exact for the same iterator but starting at the end of the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let slice = ['l', 'o', 'r', 'e', 'm'];
 let mut iter = slice.chunks_exact(2);
 assert_eq!(iter.next().unwrap(), &['l', 'o']);
@@ -378,9 +417,9 @@ resulting code better than in the case of chunks_mut for a variant of this iterator that also returns the remainder as a
 smaller chunk, and rchunks_exact_mut for the same iterator but starting at the end of
 the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
 
@@ -399,7 +438,7 @@ assuming that there’s no remainder.
    
 
  • The slice splits exactly into N-element chunks (aka self.len() % N == 0).
    • 
 
  • N != 0.
    • 
 
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_as_chunks)]
 let slice: &[char] = &['l', 'o', 'r', 'e', 'm', '!'];
 let chunks: &[[char; 1]] =
@@ -417,10 +456,10 @@ assuming that there’s no remainder.
    🔬 This is a nightly-only experimental API. (slice_as_chunks)

    Splits the slice into a slice of N-element arrays, starting at the beginning of the slice, and a remainder slice with length strictly less than N.

    -
    Panics
    +
    Panics

    Panics if N is 0. This check will most probably get changed to a compile time error before this method gets stabilized.

    -
    Examples
    +
    Examples
    #![feature(slice_as_chunks)]
 let slice = ['l', 'o', 'r', 'e', 'm'];
 let (chunks, remainder) = slice.as_chunks();
@@ -429,10 +468,10 @@ error before this method gets stabilized.
    🔬 This is a nightly-only experimental API. (slice_as_chunks)

    Splits the slice into a slice of N-element arrays, starting at the end of the slice, and a remainder slice with length strictly less than N.

    -
    Panics
    +
    Panics

    Panics if N is 0. This check will most probably get changed to a compile time error before this method gets stabilized.

    -
    Examples
    +
    Examples
    #![feature(slice_as_chunks)]
 let slice = ['l', 'o', 'r', 'e', 'm'];
 let (remainder, chunks) = slice.as_rchunks();
@@ -444,10 +483,10 @@ beginning of the slice.
    
 length of the slice, then the last up to N-1 elements will be omitted and can be
 retrieved from the remainder function of the iterator.
    
 
    This method is the const generic equivalent of chunks_exact.
    
-
    Panics
    
+
    Panics
    
 
    Panics if N is 0. This check will most probably get changed to a compile time
 error before this method gets stabilized.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(array_chunks)]
 let slice = ['l', 'o', 'r', 'e', 'm'];
 let mut iter = slice.array_chunks();
@@ -463,7 +502,7 @@ assuming that there’s no remainder.
    
 
  • The slice splits exactly into N-element chunks (aka self.len() % N == 0).
    • 
 
  • N != 0.
    • 
 
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_as_chunks)]
 let slice: &mut [char] = &mut ['l', 'o', 'r', 'e', 'm', '!'];
 let chunks: &mut [[char; 1]] =
@@ -483,10 +522,10 @@ assuming that there’s no remainder.
    🔬 This is a nightly-only experimental API. (slice_as_chunks)

    Splits the slice into a slice of N-element arrays, starting at the beginning of the slice, and a remainder slice with length strictly less than N.

    -
    Panics
    +
    Panics

    Panics if N is 0. This check will most probably get changed to a compile time error before this method gets stabilized.

    -
    Examples
    +
    Examples
    #![feature(slice_as_chunks)]
 let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
@@ -501,10 +540,10 @@ error before this method gets stabilized.
    🔬 This is a nightly-only experimental API. (slice_as_chunks)

    Splits the slice into a slice of N-element arrays, starting at the end of the slice, and a remainder slice with length strictly less than N.

    -
    Panics
    +
    Panics

    Panics if N is 0. This check will most probably get changed to a compile time error before this method gets stabilized.

    -
    Examples
    +
    Examples
    #![feature(slice_as_chunks)]
 let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
@@ -522,10 +561,10 @@ beginning of the slice.
    
 the length of the slice, then the last up to N-1 elements will be omitted and
 can be retrieved from the into_remainder function of the iterator.
    
 
    This method is the const generic equivalent of chunks_exact_mut.
    
-
    Panics
    
+
    Panics
    
 
    Panics if N is 0. This check will most probably get changed to a compile time
 error before this method gets stabilized.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(array_chunks)]
 let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
@@ -539,10 +578,10 @@ error before this method gets stabilized.
    
 starting at the beginning of the slice.
    
 
    This is the const generic equivalent of windows.
    
 
    If N is greater than the size of the slice, it will return no windows.
    
-
    Panics
    
+
    Panics
    
 
    Panics if N is 0. This check will most probably get changed to a compile time
 error before this method gets stabilized.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(array_windows)]
 let slice = [0, 1, 2, 3];
 let mut iter = slice.array_windows();
@@ -557,9 +596,9 @@ slice, then the last chunk will not have length chunk_size.
    
 
    See rchunks_exact for a variant of this iterator that returns chunks of always exactly
 chunk_size elements, and chunks for the same iterator but starting at the beginning
 of the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let slice = ['l', 'o', 'r', 'e', 'm'];
 let mut iter = slice.rchunks(2);
 assert_eq!(iter.next().unwrap(), &['e', 'm']);
@@ -573,9 +612,9 @@ length of the slice, then the last chunk will not have length chunk_sizeSee rchunks_exact_mut for a variant of this iterator that returns chunks of always
 exactly chunk_size elements, and chunks_mut for the same iterator but starting at the
 beginning of the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
 
@@ -596,9 +635,9 @@ resulting code better than in the case of rchunks for a variant of this iterator that also returns the remainder as a smaller
 chunk, and chunks_exact for the same iterator but starting at the beginning of the
 slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let slice = ['l', 'o', 'r', 'e', 'm'];
 let mut iter = slice.rchunks_exact(2);
 assert_eq!(iter.next().unwrap(), &['e', 'm']);
@@ -615,9 +654,9 @@ resulting code better than in the case of rchunks_mut for a variant of this iterator that also returns the remainder as a
 smaller chunk, and chunks_exact_mut for the same iterator but starting at the beginning
 of the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
 
@@ -633,7 +672,7 @@ of elements using the predicate to separate them.
    
 
    The predicate is called on two elements following themselves,
 it means the predicate is called on slice[0] and slice[1]
 then on slice[1] and slice[2] and so on.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_group_by)]
 
 let slice = &[1, 1, 1, 3, 3, 2, 2, 2];
@@ -661,7 +700,7 @@ runs of elements using the predicate to separate them.
    
 
    The predicate is called on two elements following themselves,
 it means the predicate is called on slice[0] and slice[1]
 then on slice[1] and slice[2] and so on.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_group_by)]
 
 let slice = &mut [1, 1, 1, 3, 3, 2, 2, 2];
@@ -688,9 +727,9 @@ then on slice[1] and slice[2] and so on.
    
 
    The first will contain all indices from [0, mid) (excluding
 the index mid itself) and the second will contain all
 indices from [mid, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if mid > len.
    
-
    Examples
    
+
    Examples
    
 
    let v = [1, 2, 3, 4, 5, 6];
 
 {
@@ -714,9 +753,9 @@ indices from [mid, len) (excluding the index len itsel
 
    The first will contain all indices from [0, mid) (excluding
 the index mid itself) and the second will contain all
 indices from [mid, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if mid > len.
    
-
    Examples
    
+
    Examples
    
 
    let mut v = [1, 0, 3, 0, 5, 6];
 let (left, right) = v.split_at_mut(2);
 assert_eq!(left, [1, 0]);
@@ -733,7 +772,7 @@ indices from [mid, len) (excluding the index len itsel
 
    Calling this method with an out-of-bounds index is undefined behavior
 even if the resulting reference is not used. The caller has to ensure that
 0 <= mid <= self.len().
    
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_split_at_unchecked)]
 
 let v = [1, 2, 3, 4, 5, 6];
@@ -764,7 +803,7 @@ indices from [mid, len) (excluding the index len itsel
 
    Calling this method with an out-of-bounds index is undefined behavior
 even if the resulting reference is not used. The caller has to ensure that
 0 <= mid <= self.len().
    
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_split_at_unchecked)]
 
 let mut v = [1, 0, 3, 0, 5, 6];
@@ -781,9 +820,9 @@ even if the resulting reference is not used. The caller has to ensure that
 
    The array will contain all indices from [0, N) (excluding
 the index N itself) and the slice will contain all
 indices from [N, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if N > len.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(split_array)]
 
 let v = &[1, 2, 3, 4, 5, 6][..];
@@ -809,9 +848,9 @@ indices from [N, len) (excluding the index len itself)
 
    The array will contain all indices from [0, N) (excluding
 the index N itself) and the slice will contain all
 indices from [N, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if N > len.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(split_array)]
 
 let mut v = &mut [1, 0, 3, 0, 5, 6][..];
@@ -826,9 +865,9 @@ the end.
    
 
    The slice will contain all indices from [0, len - N) (excluding
 the index len - N itself) and the array will contain all
 indices from [len - N, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if N > len.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(split_array)]
 
 let v = &[1, 2, 3, 4, 5, 6][..];
@@ -855,9 +894,9 @@ index from the end.
    
 
    The slice will contain all indices from [0, len - N) (excluding
 the index N itself) and the array will contain all
 indices from [len - N, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if N > len.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(split_array)]
 
 let mut v = &mut [1, 0, 3, 0, 5, 6][..];
@@ -869,7 +908,7 @@ indices from [len - N, len) (excluding the index len i
 assert_eq!(v, [1, 2, 3, 4, 5, 6]);

    Returns an iterator over subslices separated by elements that match pred. The matched element is not contained in the subslices.

    -
    Examples
    +
    Examples
    let slice = [10, 40, 33, 20];
 let mut iter = slice.split(|num| num % 3 == 0);
 
@@ -899,7 +938,7 @@ present between them:
    
 assert!(iter.next().is_none());

    Returns an iterator over mutable subslices separated by elements that match pred. The matched element is not contained in the subslices.

    -
    Examples
    +
    Examples
    let mut v = [10, 40, 30, 20, 60, 50];
 
 for group in v.split_mut(|num| *num % 3 == 0) {
@@ -909,7 +948,7 @@ match pred. The matched element is not contained in the subslices.<

    Returns an iterator over subslices separated by elements that match pred. The matched element is contained in the end of the previous subslice as a terminator.

    -
    Examples
    +
    Examples
    let slice = [10, 40, 33, 20];
 let mut iter = slice.split_inclusive(|num| num % 3 == 0);
 
@@ -929,7 +968,7 @@ That slice will be the last item returned by the iterator.

    Returns an iterator over mutable subslices separated by elements that match pred. The matched element is contained in the previous subslice as a terminator.

    -
    Examples
    +
    Examples
    let mut v = [10, 40, 30, 20, 60, 50];
 
 for group in v.split_inclusive_mut(|num| *num % 3 == 0) {
@@ -940,7 +979,7 @@ subslice as a terminator.

    Returns an iterator over subslices separated by elements that match pred, starting at the end of the slice and working backwards. The matched element is not contained in the subslices.

    -
    Examples
    +
    Examples
    let slice = [11, 22, 33, 0, 44, 55];
 let mut iter = slice.rsplit(|num| *num == 0);
 
@@ -960,7 +999,7 @@ slice will be the first (or last) item returned by the iterator.

    Returns an iterator over mutable subslices separated by elements that match pred, starting at the end of the slice and working backwards. The matched element is not contained in the subslices.

    -
    Examples
    +
    Examples
    let mut v = [100, 400, 300, 200, 600, 500];
 
 let mut count = 0;
@@ -974,7 +1013,7 @@ backwards. The matched element is not contained in the subslices.
    
 not contained in the subslices.
    
 
    The last element returned, if any, will contain the remainder of the
 slice.
    
-
    Examples
    
+
    Examples
    
 
    Print the slice split once by numbers divisible by 3 (i.e., [10, 40],
 [20, 60, 50]):
    
 
@@ -988,7 +1027,7 @@ slice.
    
 not contained in the subslices.
    
 
    The last element returned, if any, will contain the remainder of the
 slice.
    
-
    Examples
    
+
    Examples
    
 
    let mut v = [10, 40, 30, 20, 60, 50];
 
 for group in v.splitn_mut(2, |num| *num % 3 == 0) {
@@ -1001,7 +1040,7 @@ the slice and works backwards. The matched element is not contained in
 the subslices.
    
 
    The last element returned, if any, will contain the remainder of the
 slice.
    
-
    Examples
    
+
    Examples
    
 
    Print the slice split once, starting from the end, by numbers divisible
 by 3 (i.e., [50], [10, 40, 30, 20]):
    
 
@@ -1016,15 +1055,17 @@ the slice and works backwards. The matched element is not contained in
 the subslices.
    
 
    The last element returned, if any, will contain the remainder of the
 slice.
    
-
    Examples
    
+
    Examples
    
 
    let mut s = [10, 40, 30, 20, 60, 50];
 
 for group in s.rsplitn_mut(2, |num| *num % 3 == 0) {
     group[0] = 1;
 }
 assert_eq!(s, [1, 40, 30, 20, 60, 1]);
    
-

    Returns true if the slice contains an element with the given value.

    -
    Examples
    +

    Returns true if the slice contains an element with the given value.

    +

    This operation is O(n).

    +

    Note that if you have a sorted slice, binary_search may be faster.

    +
    Examples
    let v = [10, 40, 30];
 assert!(v.contains(&30));
 assert!(!v.contains(&50));
    @@ -1035,8 +1076,8 @@ use iter().any:

    let v = [String::from("hello"), String::from("world")]; // slice of `String`
 assert!(v.iter().any(|e| e == "hello")); // search with `&str`
 assert!(!v.iter().any(|e| e == "hi"));
    -

    Returns true if needle is a prefix of the slice.

    -
    Examples
    +

    Returns true if needle is a prefix of the slice.

    +
    Examples
    let v = [10, 40, 30];
 assert!(v.starts_with(&[10]));
 assert!(v.starts_with(&[10, 40]));
@@ -1048,8 +1089,8 @@ use iter().any:
    
 assert!(v.starts_with(&[]));
 let v: &[u8] = &[];
 assert!(v.starts_with(&[]));
    -

    Returns true if needle is a suffix of the slice.

    -
    Examples
    +

    Returns true if needle is a suffix of the slice.

    +
    Examples
    let v = [10, 40, 30];
 assert!(v.ends_with(&[30]));
 assert!(v.ends_with(&[40, 30]));
@@ -1061,11 +1102,11 @@ use iter().any:
    
 assert!(v.ends_with(&[]));
 let v: &[u8] = &[];
 assert!(v.ends_with(&[]));
    -

    Returns a subslice with the prefix removed.

    +

    Returns a subslice with the prefix removed.

    If the slice starts with prefix, returns the subslice after the prefix, wrapped in Some. If prefix is empty, simply returns the original slice.

    If the slice does not start with prefix, returns None.

    -
    Examples
    +
    Examples
    let v = &[10, 40, 30];
 assert_eq!(v.strip_prefix(&[10]), Some(&[40, 30][..]));
 assert_eq!(v.strip_prefix(&[10, 40]), Some(&[30][..]));
@@ -1075,17 +1116,18 @@ If prefix is empty, simply returns the original slice.
    
 let prefix : &str = "he";
 assert_eq!(b"hello".strip_prefix(prefix.as_bytes()),
            Some(b"llo".as_ref()));
    -

    Returns a subslice with the suffix removed.

    +

    Returns a subslice with the suffix removed.

    If the slice ends with suffix, returns the subslice before the suffix, wrapped in Some. If suffix is empty, simply returns the original slice.

    If the slice does not end with suffix, returns None.

    -
    Examples
    +
    Examples
    let v = &[10, 40, 30];
 assert_eq!(v.strip_suffix(&[30]), Some(&[10, 40][..]));
 assert_eq!(v.strip_suffix(&[40, 30]), Some(&[10][..]));
 assert_eq!(v.strip_suffix(&[50]), None);
 assert_eq!(v.strip_suffix(&[50, 30]), None);
    -

    Binary searches this sorted slice for a given element.

    +

    Binary searches this slice for a given element. +This behaves similary to contains if this slice is sorted.

    If the value is found then Result::Ok is returned, containing the index of the matching element. If there are multiple matches, then any one of the matches could be returned. The index is chosen @@ -1094,7 +1136,7 @@ If the value is not found then binary_search_by, binary_search_by_key, and partition_point.

    -
    Examples
    +
    Examples

    Looks up a series of four elements. The first is found, with a uniquely determined position; the second and third are not found; the fourth could match any position in [1, 4].

    @@ -1115,7 +1157,8 @@ sort order, consider using // The above is equivalent to `let idx = s.binary_search(&num).unwrap_or_else(|x| x);` s.insert(idx, num); assert_eq!(s, [0, 1, 1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 42, 55]);
    -

    Binary searches this sorted slice with a comparator function.

    +

    Binary searches this slice with a comparator function. +This behaves similarly to contains if this slice is sorted.

    The comparator function should implement an order consistent with the sort order of the underlying slice, returning an order code that indicates whether its argument is Less, @@ -1128,7 +1171,7 @@ If the value is not found then binary_search, binary_search_by_key, and partition_point.

    -
    Examples
    +
    Examples

    Looks up a series of four elements. The first is found, with a uniquely determined position; the second and third are not found; the fourth could match any position in [1, 4].

    @@ -1144,7 +1187,8 @@ found; the fourth could match any position in [1, 4].

    let seek = 1; let r = s.binary_search_by(|probe| probe.cmp(&seek)); assert!(match r { Ok(1..=4) => true, _ => false, });
    -

    Binary searches this sorted slice with a key extraction function.

    +

    Binary searches this slice with a key extraction function. +This behaves similarly to contains if this slice is sorted.

    Assumes that the slice is sorted by the key, for instance with sort_by_key using the same key extraction function.

    If the value is found then Result::Ok is returned, containing the @@ -1155,7 +1199,7 @@ If the value is not found then binary_search, binary_search_by, and partition_point.

    -
    Examples
    +
    Examples

    Looks up a series of four elements in a slice of pairs sorted by their second elements. The first is found, with a uniquely determined position; the second and third are not found; the @@ -1170,7 +1214,7 @@ fourth could match any position in [1, 4].

    assert_eq!(s.binary_search_by_key(&100, |&(a, b)| b), Err(13)); let r = s.binary_search_by_key(&1, |&(a, b)| b); assert!(match r { Ok(1..=4) => true, _ => false, });
    -

    Sorts the slice, but might not preserve the order of equal elements.

    +

    Sorts the slice, but might not preserve the order of equal elements.

    This sort is unstable (i.e., may reorder equal elements), in-place (i.e., does not allocate), and O(n * log(n)) worst-case.

    Current implementation
    @@ -1181,12 +1225,12 @@ randomization to avoid degenerate cases, but with a fixed seed to always provide deterministic behavior.

    It is typically faster than stable sorting, except in a few special cases, e.g., when the slice consists of several concatenated sorted sequences.

    -
    Examples
    +
    Examples
    let mut v = [-5, 4, 1, -3, 2];
 
 v.sort_unstable();
 assert!(v == [-5, -3, 1, 2, 4]);
    -

    Sorts the slice with a comparator function, but might not preserve the order of equal +

    Sorts the slice with a comparator function, but might not preserve the order of equal elements.

    This sort is unstable (i.e., may reorder equal elements), in-place (i.e., does not allocate), and O(n * log(n)) worst-case.

    @@ -1211,7 +1255,7 @@ randomization to avoid degenerate cases, but with a fixed seed to always provide deterministic behavior.

    It is typically faster than stable sorting, except in a few special cases, e.g., when the slice consists of several concatenated sorted sequences.

    -
    Examples
    +
    Examples
    let mut v = [5, 4, 1, 3, 2];
 v.sort_unstable_by(|a, b| a.cmp(b));
 assert!(v == [1, 2, 3, 4, 5]);
@@ -1219,7 +1263,7 @@ slice consists of several concatenated sorted sequences.
    
 // reverse sorting
 v.sort_unstable_by(|a, b| b.cmp(a));
 assert!(v == [5, 4, 3, 2, 1]);
    -

    Sorts the slice with a key extraction function, but might not preserve the order of equal +

    Sorts the slice with a key extraction function, but might not preserve the order of equal elements.

    This sort is unstable (i.e., may reorder equal elements), in-place (i.e., does not allocate), and O(m * n * log(n)) worst-case, where the key function is @@ -1233,12 +1277,12 @@ deterministic behavior.

    Due to its key calling strategy, sort_unstable_by_key is likely to be slower than sort_by_cached_key in cases where the key function is expensive.

    -
    Examples
    +
    Examples
    let mut v = [-5i32, 4, 1, -3, 2];
 
 v.sort_unstable_by_key(|k| k.abs());
 assert!(v == [1, 2, -3, 4, -5]);
    -

    Reorder the slice such that the element at index is at its final sorted position.

    +

    Reorder the slice such that the element at index is at its final sorted position.

    This reordering has the additional property that any value at position i < index will be less than or equal to any value at a position j > index. Additionally, this reordering is unstable (i.e. any number of equal elements may end up at position index), in-place @@ -1249,9 +1293,9 @@ the one at the given index.

    Current implementation

    The current algorithm is based on the quickselect portion of the same quicksort algorithm used for sort_unstable.

    -
    Panics
    +
    Panics

    Panics when index >= len(), meaning it always panics on empty slices.

    -
    Examples
    +
    Examples
    let mut v = [-5i32, 4, 1, -3, 2];
 
 // Find the median
@@ -1263,7 +1307,7 @@ used for v == [-5, -3, 1, 2, 4] ||
         v == [-3, -5, 1, 4, 2] ||
         v == [-5, -3, 1, 4, 2]);
    -

    Reorder the slice with a comparator function such that the element at index is at its +

    Reorder the slice with a comparator function such that the element at index is at its final sorted position.

    This reordering has the additional property that any value at position i < index will be less than or equal to any value at a position j > index using the comparator function. @@ -1276,9 +1320,9 @@ function.

    Current implementation

    The current algorithm is based on the quickselect portion of the same quicksort algorithm used for sort_unstable.

    -
    Panics
    +
    Panics

    Panics when index >= len(), meaning it always panics on empty slices.

    -
    Examples
    +
    Examples
    let mut v = [-5i32, 4, 1, -3, 2];
 
 // Find the median as if the slice were sorted in descending order.
@@ -1290,7 +1334,7 @@ used for v == [2, 4, 1, -3, -5] ||
         v == [4, 2, 1, -5, -3] ||
         v == [4, 2, 1, -3, -5]);
    -

    Reorder the slice with a key extraction function such that the element at index is at its +

    Reorder the slice with a key extraction function such that the element at index is at its final sorted position.

    This reordering has the additional property that any value at position i < index will be less than or equal to any value at a position j > index using the key extraction function. @@ -1303,9 +1347,9 @@ function.

    Current implementation

    The current algorithm is based on the quickselect portion of the same quicksort algorithm used for sort_unstable.

    -
    Panics
    +
    Panics

    Panics when index >= len(), meaning it always panics on empty slices.

    -
    Examples
    +
    Examples
    let mut v = [-5i32, 4, 1, -3, 2];
 
 // Return the median as if the array were sorted according to absolute value.
@@ -1317,12 +1361,12 @@ used for v == [1, 2, -3, -5, 4] ||
         v == [2, 1, -3, 4, -5] ||
         v == [2, 1, -3, -5, 4]);
    -
    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all consecutive repeated elements to the end of the slice according to the +

    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all consecutive repeated elements to the end of the slice according to the PartialEq trait implementation.

    Returns two slices. The first contains no consecutive repeated elements. The second contains all the duplicates in no specified order.

    If the slice is sorted, the first returned slice contains no duplicates.

    -
    Examples
    +
    Examples
    #![feature(slice_partition_dedup)]
 
 let mut slice = [1, 2, 2, 3, 3, 2, 1, 1];
@@ -1331,7 +1375,7 @@ The second contains all the duplicates in no specified order.
    
 
 assert_eq!(dedup, [1, 2, 3, 2, 1]);
 assert_eq!(duplicates, [2, 3, 1]);
    -
    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all but the first of consecutive elements to the end of the slice satisfying +

    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all but the first of consecutive elements to the end of the slice satisfying a given equality relation.

    Returns two slices. The first contains no consecutive repeated elements. The second contains all the duplicates in no specified order.

    @@ -1340,7 +1384,7 @@ must determine if the elements compare equal. The elements are passed in opposit from their order in the slice, so if same_bucket(a, b) returns true, a is moved at the end of the slice.

    If the slice is sorted, the first returned slice contains no duplicates.

    -
    Examples
    +
    Examples
    #![feature(slice_partition_dedup)]
 
 let mut slice = ["foo", "Foo", "BAZ", "Bar", "bar", "baz", "BAZ"];
@@ -1349,12 +1393,12 @@ at the end of the slice.
    
 
 assert_eq!(dedup, ["foo", "BAZ", "Bar", "baz"]);
 assert_eq!(duplicates, ["bar", "Foo", "BAZ"]);
    -
    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all but the first of consecutive elements to the end of the slice that resolve +

    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all but the first of consecutive elements to the end of the slice that resolve to the same key.

    Returns two slices. The first contains no consecutive repeated elements. The second contains all the duplicates in no specified order.

    If the slice is sorted, the first returned slice contains no duplicates.

    -
    Examples
    +
    Examples
    #![feature(slice_partition_dedup)]
 
 let mut slice = [10, 20, 21, 30, 30, 20, 11, 13];
@@ -1363,17 +1407,17 @@ The second contains all the duplicates in no specified order.
    
 
 assert_eq!(dedup, [10, 20, 30, 20, 11]);
 assert_eq!(duplicates, [21, 30, 13]);
    -

    Rotates the slice in-place such that the first mid elements of the +

    Rotates the slice in-place such that the first mid elements of the slice move to the end while the last self.len() - mid elements move to the front. After calling rotate_left, the element previously at index mid will become the first element in the slice.

    -
    Panics
    +
    Panics

    This function will panic if mid is greater than the length of the slice. Note that mid == self.len() does not panic and is a no-op rotation.

    Complexity

    Takes linear (in self.len()) time.

    -
    Examples
    +
    Examples
    let mut a = ['a', 'b', 'c', 'd', 'e', 'f'];
 a.rotate_left(2);
 assert_eq!(a, ['c', 'd', 'e', 'f', 'a', 'b']);
    @@ -1382,17 +1426,17 @@ rotation.

    let mut a = ['a', 'b', 'c', 'd', 'e', 'f'];
 a[1..5].rotate_left(1);
 assert_eq!(a, ['a', 'c', 'd', 'e', 'b', 'f']);
    -

    Rotates the slice in-place such that the first self.len() - k +

    Rotates the slice in-place such that the first self.len() - k elements of the slice move to the end while the last k elements move to the front. After calling rotate_right, the element previously at index self.len() - k will become the first element in the slice.

    -
    Panics
    +
    Panics

    This function will panic if k is greater than the length of the slice. Note that k == self.len() does not panic and is a no-op rotation.

    Complexity

    Takes linear (in self.len()) time.

    -
    Examples
    +
    Examples
    let mut a = ['a', 'b', 'c', 'd', 'e', 'f'];
 a.rotate_right(2);
 assert_eq!(a, ['e', 'f', 'a', 'b', 'c', 'd']);
    @@ -1401,25 +1445,25 @@ rotation.

    let mut a = ['a', 'b', 'c', 'd', 'e', 'f'];
 a[1..5].rotate_right(1);
 assert_eq!(a, ['a', 'e', 'b', 'c', 'd', 'f']);
    -

    Fills self with elements by cloning value.

    -
    Examples
    +

    Fills self with elements by cloning value.

    +
    Examples
    let mut buf = vec![0; 10];
 buf.fill(1);
 assert_eq!(buf, vec![1; 10]);
    -

    Fills self with elements returned by calling a closure repeatedly.

    +

    Fills self with elements returned by calling a closure repeatedly.

    This method uses a closure to create new values. If you’d rather Clone a given value, use fill. If you want to use the Default trait to generate values, you can pass Default::default as the argument.

    -
    Examples
    +
    Examples
    let mut buf = vec![1; 10];
 buf.fill_with(Default::default);
 assert_eq!(buf, vec![0; 10]);
    -

    Copies the elements from src into self.

    +

    Copies the elements from src into self.

    The length of src must be the same as self.

    -
    Panics
    +
    Panics

    This function will panic if the two slices have different lengths.

    -
    Examples
    +
    Examples

    Cloning two elements from a slice into another:

    let src = [1, 2, 3, 4];
@@ -1451,12 +1495,12 @@ sub-slices from a slice:
    
 }
 
 assert_eq!(slice, [4, 5, 3, 4, 5]);
    -

    Copies all elements from src into self, using a memcpy.

    +

    Copies all elements from src into self, using a memcpy.

    The length of src must be the same as self.

    If T does not implement Copy, use clone_from_slice.

    -
    Panics
    +
    Panics

    This function will panic if the two slices have different lengths.

    -
    Examples
    +
    Examples

    Copying two elements from a slice into another:

    let src = [1, 2, 3, 4];
@@ -1488,16 +1532,16 @@ sub-slices from a slice:
    
 }
 
 assert_eq!(slice, [4, 5, 3, 4, 5]);
    -

    Copies elements from one part of the slice to another part of itself, +

    Copies elements from one part of the slice to another part of itself, using a memmove.

    src is the range within self to copy from. dest is the starting index of the range within self to copy to, which will have the same length as src. The two ranges may overlap. The ends of the two ranges must be less than or equal to self.len().

    -
    Panics
    +
    Panics

    This function will panic if either range exceeds the end of the slice, or if the end of src is before the start.

    -
    Examples
    +
    Examples

    Copying four bytes within a slice:

    let mut bytes = *b"Hello, World!";
@@ -1505,9 +1549,9 @@ or if the end of src is before the start.
    
 bytes.copy_within(1..5, 8);
 
 assert_eq!(&bytes, b"Hello, Wello!");
    -

    Swaps all elements in self with those in other.

    +

    Swaps all elements in self with those in other.

    The length of other must be the same as self.

    -
    Panics
    +
    Panics

    This function will panic if the two slices have different lengths.

    Example

    Swapping two elements across slices:

    @@ -1537,7 +1581,7 @@ mutable sub-slices from a slice:

    } assert_eq!(slice, [4, 5, 3, 1, 2]);
    -

    Transmute the slice to a slice of another type, ensuring alignment of the types is +

    Transmute the slice to a slice of another type, ensuring alignment of the types is maintained.

    This method splits the slice into three distinct slices: prefix, correctly aligned middle slice of a new type, and the suffix slice. The method may make the middle slice the greatest @@ -1549,7 +1593,7 @@ zero-sized and will return the original slice without splitting anything.

    Safety

    This method is essentially a transmute with respect to the elements in the returned middle slice, so all the usual caveats pertaining to transmute::<T, U> also apply here.

    -
    Examples
    +
    Examples

    Basic usage:

    unsafe {
@@ -1559,7 +1603,7 @@ middle slice, so all the usual caveats pertaining to transmute::<T, U&g
     // more_efficient_algorithm_for_aligned_shorts(shorts);
     // less_efficient_algorithm_for_bytes(suffix);
 }
    -

    Transmute the slice to a slice of another type, ensuring alignment of the types is +

    Transmute the slice to a slice of another type, ensuring alignment of the types is maintained.

    This method splits the slice into three distinct slices: prefix, correctly aligned middle slice of a new type, and the suffix slice. The method may make the middle slice the greatest @@ -1571,7 +1615,7 @@ zero-sized and will return the original slice without splitting anything.

    Safety

    This method is essentially a transmute with respect to the elements in the returned middle slice, so all the usual caveats pertaining to transmute::<T, U> also apply here.

    -
    Examples
    +
    Examples

    Basic usage:

    unsafe {
@@ -1581,7 +1625,7 @@ middle slice, so all the usual caveats pertaining to transmute::<T, U&g
     // more_efficient_algorithm_for_aligned_shorts(shorts);
     // less_efficient_algorithm_for_bytes(suffix);
 }
    -
    🔬 This is a nightly-only experimental API. (portable_simd)

    Split a slice into a prefix, a middle of aligned SIMD types, and a suffix.

    +
    🔬 This is a nightly-only experimental API. (portable_simd)

    Split a slice into a prefix, a middle of aligned SIMD types, and a suffix.

    This is a safe wrapper around slice::align_to, so has the same weak postconditions as that method. You’re only assured that self.len() == prefix.len() + middle.len() * LANES + suffix.len().

    @@ -1593,7 +1637,7 @@ postconditions as that method. You’re only assured that

    That said, this is a safe method, so if you’re only writing safe code, then this can at most cause incorrect logic, not unsoundness.

    -
    Panics
    +
    Panics

    This will panic if the size of the SIMD type is different from LANES times that of the scalar.

    At the time of writing, the trait restrictions on Simd<T, LANES> keeps @@ -1601,7 +1645,7 @@ that from ever happening, as only power-of-two numbers of lanes are supported. It’s possible that, in the future, those restrictions might be lifted in a way that would make it possible to see panics from this method for something like LANES == 3.

    -
    Examples
    +
    Examples
    #![feature(portable_simd)]
 
 let short = &[1, 2, 3];
@@ -1628,7 +1672,7 @@ method for something like LANES == 3.
    
 
 let numbers: Vec<f32> = (1..101).map(|x| x as _).collect();
 assert_eq!(basic_simd_sum(&numbers[1..99]), 4949.0);
    -
    🔬 This is a nightly-only experimental API. (portable_simd)

    Split a slice into a prefix, a middle of aligned SIMD types, and a suffix.

    +
    🔬 This is a nightly-only experimental API. (portable_simd)

    Split a slice into a prefix, a middle of aligned SIMD types, and a suffix.

    This is a safe wrapper around slice::align_to_mut, so has the same weak postconditions as that method. You’re only assured that self.len() == prefix.len() + middle.len() * LANES + suffix.len().

    @@ -1641,7 +1685,7 @@ postconditions as that method. You’re only assured that

    That said, this is a safe method, so if you’re only writing safe code, then this can at most cause incorrect logic, not unsoundness.

    This is the mutable version of slice::as_simd; see that for examples.

    -
    Panics
    +
    Panics

    This will panic if the size of the SIMD type is different from LANES times that of the scalar.

    At the time of writing, the trait restrictions on Simd<T, LANES> keeps @@ -1649,13 +1693,13 @@ that from ever happening, as only power-of-two numbers of lanes are supported. It’s possible that, in the future, those restrictions might be lifted in a way that would make it possible to see panics from this method for something like LANES == 3.

    -
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted.

    +
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted.

    That is, for each element a and its following element b, a <= b must hold. If the slice yields exactly zero or one element, true is returned.

    Note that if Self::Item is only PartialOrd, but not Ord, the above definition implies that this function returns false if any two consecutive items are not comparable.

    -
    Examples
    +
    Examples
    #![feature(is_sorted)]
 let empty: [i32; 0] = [];
 
@@ -1664,20 +1708,20 @@ comparable.
    
 assert!([0].is_sorted());
 assert!(empty.is_sorted());
 assert!(![0.0, 1.0, f32::NAN].is_sorted());
    -
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted using the given comparator function.

    +
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted using the given comparator function.

    Instead of using PartialOrd::partial_cmp, this function uses the given compare function to determine the ordering of two elements. Apart from that, it’s equivalent to is_sorted; see its documentation for more information.

    -
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted using the given key extraction function.

    +
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted using the given key extraction function.

    Instead of comparing the slice’s elements directly, this function compares the keys of the elements, as determined by f. Apart from that, it’s equivalent to is_sorted; see its documentation for more information.

    -
    Examples
    +
    Examples
    #![feature(is_sorted)]
 
 assert!(["c", "bb", "aaa"].is_sorted_by_key(|s| s.len()));
 assert!(![-2i32, -1, 0, 3].is_sorted_by_key(|n| n.abs()));
    -

    Returns the index of the partition point according to the given predicate +

    Returns the index of the partition point according to the given predicate (the index of the first element of the second partition).

    The slice is assumed to be partitioned according to the given predicate. This means that all elements for which the predicate returns true are at the start of the slice @@ -1687,7 +1731,7 @@ For example, [7, 15, 3, 5, 4, 12, 6] is a partitioned under the predicate x % 2

    If this slice is not partitioned, the returned result is unspecified and meaningless, as this method performs a kind of binary search.

    See also binary_search, binary_search_by, and binary_search_by_key.

    -
    Examples
    +
    Examples
    let v = [1, 2, 3, 3, 5, 6, 7];
 let i = v.partition_point(|&x| x < 5);
 
@@ -1702,13 +1746,13 @@ sort order:
    
 let idx = s.partition_point(|&x| x < num);
 s.insert(idx, num);
 assert_eq!(s, [0, 1, 1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 42, 55]);
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the subslice corresponding to the given range +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the subslice corresponding to the given range and returns a reference to it.

    Returns None and does not modify the slice if the given range is out of bounds.

    Note that this method only accepts one-sided ranges such as 2.. or ..6, but not 2..6.

    -
    Examples
    +
    Examples

    Taking the first three elements of a slice:

    #![feature(slice_take)]
@@ -1738,13 +1782,13 @@ range is out of bounds.
    
 assert_eq!(None, slice.take(..=4));
 let expected: &[char] = &['a', 'b', 'c', 'd'];
 assert_eq!(Some(expected), slice.take(..4));
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the subslice corresponding to the given range +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the subslice corresponding to the given range and returns a mutable reference to it.

    Returns None and does not modify the slice if the given range is out of bounds.

    Note that this method only accepts one-sided ranges such as 2.. or ..6, but not 2..6.

    -
    Examples
    +
    Examples

    Taking the first three elements of a slice:

    #![feature(slice_take)]
@@ -1774,10 +1818,10 @@ range is out of bounds.
    
 assert_eq!(None, slice.take_mut(..=4));
 let expected: &mut [_] = &mut ['a', 'b', 'c', 'd'];
 assert_eq!(Some(expected), slice.take_mut(..4));
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the first element of the slice and returns a reference +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the first element of the slice and returns a reference to it.

    Returns None if the slice is empty.

    -
    Examples
    +
    Examples
    #![feature(slice_take)]
 
 let mut slice: &[_] = &['a', 'b', 'c'];
@@ -1785,10 +1829,10 @@ to it.
    
 
 assert_eq!(slice, &['b', 'c']);
 assert_eq!(first, &'a');
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the first element of the slice and returns a mutable +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the first element of the slice and returns a mutable reference to it.

    Returns None if the slice is empty.

    -
    Examples
    +
    Examples
    #![feature(slice_take)]
 
 let mut slice: &mut [_] = &mut ['a', 'b', 'c'];
@@ -1797,10 +1841,10 @@ reference to it.
    
 
 assert_eq!(slice, &['b', 'c']);
 assert_eq!(first, &'d');
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the last element of the slice and returns a reference +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the last element of the slice and returns a reference to it.

    Returns None if the slice is empty.

    -
    Examples
    +
    Examples
    #![feature(slice_take)]
 
 let mut slice: &[_] = &['a', 'b', 'c'];
@@ -1808,10 +1852,10 @@ to it.
    
 
 assert_eq!(slice, &['a', 'b']);
 assert_eq!(last, &'c');
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the last element of the slice and returns a mutable +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the last element of the slice and returns a mutable reference to it.

    Returns None if the slice is empty.

    -
    Examples
    +
    Examples
    #![feature(slice_take)]
 
 let mut slice: &mut [_] = &mut ['a', 'b', 'c'];
@@ -1820,45 +1864,6 @@ reference to it.
    
 
 assert_eq!(slice, &['a', 'b']);
 assert_eq!(last, &'d');
    -
    🔬 This is a nightly-only experimental API. (slice_flatten)

    Takes a &[[T; N]], and flattens it to a &[T].

    -
    Panics
    -

    This panics if the length of the resulting slice would overflow a usize.

    -

    This is only possible when flattening a slice of arrays of zero-sized -types, and thus tends to be irrelevant in practice. If -size_of::<T>() > 0, this will never panic.

    -
    Examples
    -
    #![feature(slice_flatten)]
-
-assert_eq!([[1, 2, 3], [4, 5, 6]].flatten(), &[1, 2, 3, 4, 5, 6]);
-
-assert_eq!(
-    [[1, 2, 3], [4, 5, 6]].flatten(),
-    [[1, 2], [3, 4], [5, 6]].flatten(),
-);
-
-let slice_of_empty_arrays: &[[i32; 0]] = &[[], [], [], [], []];
-assert!(slice_of_empty_arrays.flatten().is_empty());
-
-let empty_slice_of_arrays: &[[u32; 10]] = &[];
-assert!(empty_slice_of_arrays.flatten().is_empty());
    -
    🔬 This is a nightly-only experimental API. (slice_flatten)

    Takes a &mut [[T; N]], and flattens it to a &mut [T].

    -
    Panics
    -

    This panics if the length of the resulting slice would overflow a usize.

    -

    This is only possible when flattening a slice of arrays of zero-sized -types, and thus tends to be irrelevant in practice. If -size_of::<T>() > 0, this will never panic.

    -
    Examples
    -
    #![feature(slice_flatten)]
-
-fn add_5_to_all(slice: &mut [i32]) {
-    for i in slice {
-        *i += 5;
-    }
-}
-
-let mut array = [[1, 2, 3], [4, 5, 6], [7, 8, 9]];
-add_5_to_all(array.flatten_mut());
-assert_eq!(array, [[6, 7, 8], [9, 10, 11], [12, 13, 14]]);

    Checks if all bytes in this slice are within the ASCII range.

    Checks that two slices are an ASCII case-insensitive match.

    Same as to_ascii_lowercase(a) == to_ascii_lowercase(b), @@ -2120,9 +2125,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/decompose_running_sum/struct.RunningSumConfig.html b/rustdoc/latest/halo2_gadgets/utilities/decompose_running_sum/struct.RunningSumConfig.html index 6554ff1a..3d908eae 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/decompose_running_sum/struct.RunningSumConfig.html +++ b/rustdoc/latest/halo2_gadgets/utilities/decompose_running_sum/struct.RunningSumConfig.html @@ -104,9 +104,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/fn.bitrange_subset.html b/rustdoc/latest/halo2_gadgets/utilities/fn.bitrange_subset.html index 813d0bcd..99505e23 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/fn.bitrange_subset.html +++ b/rustdoc/latest/halo2_gadgets/utilities/fn.bitrange_subset.html @@ -21,5 +21,5 @@

    In halo2_gadgets::utilities

    Function halo2_gadgets::utilities::bitrange_subset

    source · [−]
    pub fn bitrange_subset<F: PrimeFieldBits>(
        field_elem: &F, 
        bitrange: Range<usize>
    ) -> F
    Expand description

    Takes a specified subsequence of the little-endian bit representation of a field element. The bits are numbered from 0 for the LSB.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/fn.bool_check.html b/rustdoc/latest/halo2_gadgets/utilities/fn.bool_check.html index aba22b9a..92141094 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/fn.bool_check.html +++ b/rustdoc/latest/halo2_gadgets/utilities/fn.bool_check.html @@ -20,5 +20,5 @@

    Function halo2_gadgets::utilities::bool_check

    source · [−]
    pub fn bool_check<F: FieldExt>(value: Expression<F>) -> Expression<F>
    Expand description

    Checks that an expression is either 1 or 0.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/fn.decompose_word.html b/rustdoc/latest/halo2_gadgets/utilities/fn.decompose_word.html index f273131d..f62c45bd 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/fn.decompose_word.html +++ b/rustdoc/latest/halo2_gadgets/utilities/fn.decompose_word.html @@ -25,5 +25,5 @@ is a w-bit value, and scalar = k_0 + k_1 * w + k_n * w^nPanics

    We are returning a Vec<u8> which means the window size is limited to <= 8 bits.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/fn.i2lebsp.html b/rustdoc/latest/halo2_gadgets/utilities/fn.i2lebsp.html index 714dac63..c8905551 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/fn.i2lebsp.html +++ b/rustdoc/latest/halo2_gadgets/utilities/fn.i2lebsp.html @@ -23,5 +23,5 @@

    Panics

    Panics if the expected length of the sequence NUM_BITS exceeds 64.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/fn.lebs2ip.html b/rustdoc/latest/halo2_gadgets/utilities/fn.lebs2ip.html index 8e84a3eb..b9f5206c 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/fn.lebs2ip.html +++ b/rustdoc/latest/halo2_gadgets/utilities/fn.lebs2ip.html @@ -22,5 +22,5 @@

    Function halo2_gadgets::utilities::lebs2ip

    source · [−]
    pub fn lebs2ip<const L: usize>(bits: &[bool; L]) -> u64
    Expand description

    The u64 integer represented by an L-bit little-endian bitstring.

    Panics

    Panics if the bitstring is longer than 64 bits.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/fn.range_check.html b/rustdoc/latest/halo2_gadgets/utilities/fn.range_check.html index 35ba05cd..1b6da3a2 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/fn.range_check.html +++ b/rustdoc/latest/halo2_gadgets/utilities/fn.range_check.html @@ -21,5 +21,5 @@

    In halo2_gadgets::utilities

    Function halo2_gadgets::utilities::range_check

    source · [−]
    pub fn range_check<F: FieldExt>(
        word: Expression<F>, 
        range: usize
    ) -> Expression<F>
    Expand description

    Check that an expression is in the small range [0..range), i.e. 0 ≤ word < range.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/fn.ternary.html b/rustdoc/latest/halo2_gadgets/utilities/fn.ternary.html index 7d01a291..11124019 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/fn.ternary.html +++ b/rustdoc/latest/halo2_gadgets/utilities/fn.ternary.html @@ -21,5 +21,5 @@

    In halo2_gadgets::utilities

    Function halo2_gadgets::utilities::ternary

    source · [−]
    pub fn ternary<F: FieldExt>(
        a: Expression<F>, 
        b: Expression<F>, 
        c: Expression<F>
    ) -> Expression<F>
    Expand description

    If a then b, else c. Returns (a * b) + (1 - a) * c.

    a must be a boolean-constrained expression.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/index.html b/rustdoc/latest/halo2_gadgets/utilities/index.html index 5ffafdcf..15adcef9 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/index.html +++ b/rustdoc/latest/halo2_gadgets/utilities/index.html @@ -43,5 +43,5 @@ is a w-bit value, and scalar = k_0 + k_1 * w + k_n * w^n
    range_check

    Check that an expression is in the small range [0..range), i.e. 0 ≤ word < range.

    ternary

    If a then b, else c. Returns (a * b) + (1 - a) * c.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/lookup_range_check/index.html b/rustdoc/latest/halo2_gadgets/utilities/lookup_range_check/index.html index a3334090..58bcb652 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/lookup_range_check/index.html +++ b/rustdoc/latest/halo2_gadgets/utilities/lookup_range_check/index.html @@ -24,5 +24,5 @@ words.

    Structs

    LookupRangeCheckConfig

    Configuration that provides methods for a lookup range check.

    RunningSum

    The running sum $[z_0, …, z_W]$. If created in strict mode, $z_W = 0$.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/lookup_range_check/struct.LookupRangeCheckConfig.html b/rustdoc/latest/halo2_gadgets/utilities/lookup_range_check/struct.LookupRangeCheckConfig.html index c588218a..bdf54083 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/lookup_range_check/struct.LookupRangeCheckConfig.html +++ b/rustdoc/latest/halo2_gadgets/utilities/lookup_range_check/struct.LookupRangeCheckConfig.html @@ -111,9 +111,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/lookup_range_check/struct.RunningSum.html b/rustdoc/latest/halo2_gadgets/utilities/lookup_range_check/struct.RunningSum.html index 7c109133..ed63d1f5 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/lookup_range_check/struct.RunningSum.html +++ b/rustdoc/latest/halo2_gadgets/utilities/lookup_range_check/struct.RunningSum.html @@ -61,23 +61,62 @@ as its ‘length’.

    v.push(1); assert!(!v.is_empty());     -

    Methods from Deref<Target = [T]>

    Returns the number of elements in the slice.

    +

    Methods from Deref<Target = [T]>

    🔬 This is a nightly-only experimental API. (slice_flatten)

    Takes a &[[T; N]], and flattens it to a &[T].

    +
    Panics
    +

    This panics if the length of the resulting slice would overflow a usize.

    +

    This is only possible when flattening a slice of arrays of zero-sized +types, and thus tends to be irrelevant in practice. If +size_of::<T>() > 0, this will never panic.

    Examples
    +
    #![feature(slice_flatten)]
+
+assert_eq!([[1, 2, 3], [4, 5, 6]].flatten(), &[1, 2, 3, 4, 5, 6]);
+
+assert_eq!(
+    [[1, 2, 3], [4, 5, 6]].flatten(),
+    [[1, 2], [3, 4], [5, 6]].flatten(),
+);
+
+let slice_of_empty_arrays: &[[i32; 0]] = &[[], [], [], [], []];
+assert!(slice_of_empty_arrays.flatten().is_empty());
+
+let empty_slice_of_arrays: &[[u32; 10]] = &[];
+assert!(empty_slice_of_arrays.flatten().is_empty());
    +
    🔬 This is a nightly-only experimental API. (slice_flatten)

    Takes a &mut [[T; N]], and flattens it to a &mut [T].

    +
    Panics
    +

    This panics if the length of the resulting slice would overflow a usize.

    +

    This is only possible when flattening a slice of arrays of zero-sized +types, and thus tends to be irrelevant in practice. If +size_of::<T>() > 0, this will never panic.

    +
    Examples
    +
    #![feature(slice_flatten)]
+
+fn add_5_to_all(slice: &mut [i32]) {
+    for i in slice {
+        *i += 5;
+    }
+}
+
+let mut array = [[1, 2, 3], [4, 5, 6], [7, 8, 9]];
+add_5_to_all(array.flatten_mut());
+assert_eq!(array, [[6, 7, 8], [9, 10, 11], [12, 13, 14]]);
    +

    Returns the number of elements in the slice.

    +
    Examples
    let a = [1, 2, 3];
 assert_eq!(a.len(), 3);

    Returns true if the slice has a length of 0.

    -
    Examples
    +
    Examples
    let a = [1, 2, 3];
 assert!(!a.is_empty());

    Returns the first element of the slice, or None if it is empty.

    -
    Examples
    +
    Examples
    let v = [10, 40, 30];
 assert_eq!(Some(&10), v.first());
 
 let w: &[i32] = &[];
 assert_eq!(None, w.first());

    Returns a mutable pointer to the first element of the slice, or None if it is empty.

    -
    Examples
    +
    Examples
    let x = &mut [0, 1, 2];
 
 if let Some(first) = x.first_mut() {
@@ -85,7 +124,7 @@ as its ‘length’.
    
 }
 assert_eq!(x, &[5, 1, 2]);

    Returns the first and all the rest of the elements of the slice, or None if it is empty.

    -
    Examples
    +
    Examples
    let x = &[0, 1, 2];
 
 if let Some((first, elements)) = x.split_first() {
@@ -93,7 +132,7 @@ as its ‘length’.
    
     assert_eq!(elements, &[1, 2]);
 }

    Returns the first and all the rest of the elements of the slice, or None if it is empty.

    -
    Examples
    +
    Examples
    let x = &mut [0, 1, 2];
 
 if let Some((first, elements)) = x.split_first_mut() {
@@ -103,7 +142,7 @@ as its ‘length’.
    
 }
 assert_eq!(x, &[3, 4, 5]);

    Returns the last and all the rest of the elements of the slice, or None if it is empty.

    -
    Examples
    +
    Examples
    let x = &[0, 1, 2];
 
 if let Some((last, elements)) = x.split_last() {
@@ -111,7 +150,7 @@ as its ‘length’.
    
     assert_eq!(elements, &[0, 1]);
 }

    Returns the last and all the rest of the elements of the slice, or None if it is empty.

    -
    Examples
    +
    Examples
    let x = &mut [0, 1, 2];
 
 if let Some((last, elements)) = x.split_last_mut() {
@@ -121,14 +160,14 @@ as its ‘length’.
    
 }
 assert_eq!(x, &[4, 5, 3]);

    Returns the last element of the slice, or None if it is empty.

    -
    Examples
    +
    Examples
    let v = [10, 40, 30];
 assert_eq!(Some(&30), v.last());
 
 let w: &[i32] = &[];
 assert_eq!(None, w.last());

    Returns a mutable pointer to the last item in the slice.

    -
    Examples
    +
    Examples
    let x = &mut [0, 1, 2];
 
 if let Some(last) = x.last_mut() {
@@ -143,7 +182,7 @@ position or None if out of bounds.
 
  • If given a range, returns the subslice corresponding to that range,
 or None if out of bounds.
    • 
 
-
    Examples
    
+
    Examples
    
 
    let v = [10, 40, 30];
 assert_eq!(Some(&40), v.get(1));
 assert_eq!(Some(&[10, 40][..]), v.get(0..2));
@@ -151,7 +190,7 @@ or None if out of bounds.
 assert_eq!(None, v.get(0..4));

    Returns a mutable reference to an element or subslice depending on the type of index (see get) or None if the index is out of bounds.

    -
    Examples
    +
    Examples
    let x = &mut [0, 1, 2];
 
 if let Some(elem) = x.get_mut(1) {
@@ -164,7 +203,7 @@ checking.
    
 
    Safety
    
 
    Calling this method with an out-of-bounds index is undefined behavior
 even if the resulting reference is not used.
    
-
    Examples
    
+
    Examples
    
 
    let x = &[1, 2, 4];
 
 unsafe {
@@ -176,7 +215,7 @@ bounds checking.
    
 
    Safety
    
 
    Calling this method with an out-of-bounds index is undefined behavior
 even if the resulting reference is not used.
    
-
    Examples
    
+
    Examples
    
 
    let x = &mut [1, 2, 4];
 
 unsafe {
@@ -192,7 +231,7 @@ is never written to (except inside an UnsafeCell) using this pointe
 derived from it. If you need to mutate the contents of the slice, use as_mut_ptr.
    
 
    Modifying the container referenced by this slice may cause its buffer
 to be reallocated, which would also make any pointers to it invalid.
    
-
    Examples
    
+
    Examples
    
 
    let x = &[1, 2, 4];
 let x_ptr = x.as_ptr();
 
@@ -206,7 +245,7 @@ to be reallocated, which would also make any pointers to it invalid.
    
 function returns, or else it will end up pointing to garbage.
    
 
    Modifying the container referenced by this slice may cause its buffer
 to be reallocated, which would also make any pointers to it invalid.
    
-
    Examples
    
+
    Examples
    
 
    let x = &mut [1, 2, 4];
 let x_ptr = x.as_mut_ptr();
 
@@ -253,9 +292,9 @@ common in C++.
    
 
  • a - The index of the first element
    • 
 
  • b - The index of the second element
    • 
 
-
    Panics
    
+
    Panics
    
 
    Panics if a or b are out of bounds.
    
-
    Examples
    
+
    Examples
    
 
    let mut v = ["a", "b", "c", "d", "e"];
 v.swap(2, 4);
 assert!(v == ["a", "b", "e", "d", "c"]);
    
@@ -269,7 +308,7 @@ common in C++.
    
 
    Safety
    
 
    Calling this method with an out-of-bounds index is undefined behavior.
 The caller has to ensure that a < self.len() and b < self.len().
    
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_swap_unchecked)]
 
 let mut v = ["a", "b", "c", "d"];
@@ -277,12 +316,12 @@ The caller has to ensure that a < self.len() and b < se
 unsafe { v.swap_unchecked(1, 3) };
 assert!(v == ["a", "d", "c", "b"]);

    Reverses the order of elements in the slice, in place.

    -
    Examples
    +
    Examples
    let mut v = [1, 2, 3];
 v.reverse();
 assert!(v == [3, 2, 1]);

    Returns an iterator over the slice.

    -
    Examples
    +
    Examples
    let x = &[1, 2, 4];
 let mut iterator = x.iter();
 
@@ -291,7 +330,7 @@ The caller has to ensure that a < self.len() and b < se
 assert_eq!(iterator.next(), Some(&4));
 assert_eq!(iterator.next(), None);

    Returns an iterator that allows modifying each value.

    -
    Examples
    +
    Examples
    let x = &mut [1, 2, 4];
 for elem in x.iter_mut() {
     *elem += 2;
@@ -300,9 +339,9 @@ The caller has to ensure that a < self.len() and b < se

    Returns an iterator over all contiguous windows of length size. The windows overlap. If the slice is shorter than size, the iterator returns no values.

    -
    Panics
    +
    Panics

    Panics if size is 0.

    -
    Examples
    +
    Examples
    let slice = ['r', 'u', 's', 't'];
 let mut iter = slice.windows(2);
 assert_eq!(iter.next().unwrap(), &['r', 'u']);
@@ -321,9 +360,9 @@ slice, then the last chunk will not have length chunk_size.
    
 
    See chunks_exact for a variant of this iterator that returns chunks of always exactly
 chunk_size elements, and rchunks for the same iterator but starting at the end of the
 slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let slice = ['l', 'o', 'r', 'e', 'm'];
 let mut iter = slice.chunks(2);
 assert_eq!(iter.next().unwrap(), &['l', 'o']);
@@ -337,9 +376,9 @@ length of the slice, then the last chunk will not have length chunk_sizeSee chunks_exact_mut for a variant of this iterator that returns chunks of always
 exactly chunk_size elements, and rchunks_mut for the same iterator but starting at
 the end of the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
 
@@ -359,9 +398,9 @@ from the remainder function of the iterator.
    
 resulting code better than in the case of chunks.
    
 
    See chunks for a variant of this iterator that also returns the remainder as a smaller
 chunk, and rchunks_exact for the same iterator but starting at the end of the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let slice = ['l', 'o', 'r', 'e', 'm'];
 let mut iter = slice.chunks_exact(2);
 assert_eq!(iter.next().unwrap(), &['l', 'o']);
@@ -378,9 +417,9 @@ resulting code better than in the case of chunks_mut for a variant of this iterator that also returns the remainder as a
 smaller chunk, and rchunks_exact_mut for the same iterator but starting at the end of
 the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
 
@@ -399,7 +438,7 @@ assuming that there’s no remainder.
    
 
  • The slice splits exactly into N-element chunks (aka self.len() % N == 0).
    • 
 
  • N != 0.
    • 
 
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_as_chunks)]
 let slice: &[char] = &['l', 'o', 'r', 'e', 'm', '!'];
 let chunks: &[[char; 1]] =
@@ -417,10 +456,10 @@ assuming that there’s no remainder.
    🔬 This is a nightly-only experimental API. (slice_as_chunks)

    Splits the slice into a slice of N-element arrays, starting at the beginning of the slice, and a remainder slice with length strictly less than N.

    -
    Panics
    +
    Panics

    Panics if N is 0. This check will most probably get changed to a compile time error before this method gets stabilized.

    -
    Examples
    +
    Examples
    #![feature(slice_as_chunks)]
 let slice = ['l', 'o', 'r', 'e', 'm'];
 let (chunks, remainder) = slice.as_chunks();
@@ -429,10 +468,10 @@ error before this method gets stabilized.
    🔬 This is a nightly-only experimental API. (slice_as_chunks)

    Splits the slice into a slice of N-element arrays, starting at the end of the slice, and a remainder slice with length strictly less than N.

    -
    Panics
    +
    Panics

    Panics if N is 0. This check will most probably get changed to a compile time error before this method gets stabilized.

    -
    Examples
    +
    Examples
    #![feature(slice_as_chunks)]
 let slice = ['l', 'o', 'r', 'e', 'm'];
 let (remainder, chunks) = slice.as_rchunks();
@@ -444,10 +483,10 @@ beginning of the slice.
    
 length of the slice, then the last up to N-1 elements will be omitted and can be
 retrieved from the remainder function of the iterator.
    
 
    This method is the const generic equivalent of chunks_exact.
    
-
    Panics
    
+
    Panics
    
 
    Panics if N is 0. This check will most probably get changed to a compile time
 error before this method gets stabilized.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(array_chunks)]
 let slice = ['l', 'o', 'r', 'e', 'm'];
 let mut iter = slice.array_chunks();
@@ -463,7 +502,7 @@ assuming that there’s no remainder.
    
 
  • The slice splits exactly into N-element chunks (aka self.len() % N == 0).
    • 
 
  • N != 0.
    • 
 
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_as_chunks)]
 let slice: &mut [char] = &mut ['l', 'o', 'r', 'e', 'm', '!'];
 let chunks: &mut [[char; 1]] =
@@ -483,10 +522,10 @@ assuming that there’s no remainder.
    🔬 This is a nightly-only experimental API. (slice_as_chunks)

    Splits the slice into a slice of N-element arrays, starting at the beginning of the slice, and a remainder slice with length strictly less than N.

    -
    Panics
    +
    Panics

    Panics if N is 0. This check will most probably get changed to a compile time error before this method gets stabilized.

    -
    Examples
    +
    Examples
    #![feature(slice_as_chunks)]
 let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
@@ -501,10 +540,10 @@ error before this method gets stabilized.
    🔬 This is a nightly-only experimental API. (slice_as_chunks)

    Splits the slice into a slice of N-element arrays, starting at the end of the slice, and a remainder slice with length strictly less than N.

    -
    Panics
    +
    Panics

    Panics if N is 0. This check will most probably get changed to a compile time error before this method gets stabilized.

    -
    Examples
    +
    Examples
    #![feature(slice_as_chunks)]
 let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
@@ -522,10 +561,10 @@ beginning of the slice.
    
 the length of the slice, then the last up to N-1 elements will be omitted and
 can be retrieved from the into_remainder function of the iterator.
    
 
    This method is the const generic equivalent of chunks_exact_mut.
    
-
    Panics
    
+
    Panics
    
 
    Panics if N is 0. This check will most probably get changed to a compile time
 error before this method gets stabilized.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(array_chunks)]
 let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
@@ -539,10 +578,10 @@ error before this method gets stabilized.
    
 starting at the beginning of the slice.
    
 
    This is the const generic equivalent of windows.
    
 
    If N is greater than the size of the slice, it will return no windows.
    
-
    Panics
    
+
    Panics
    
 
    Panics if N is 0. This check will most probably get changed to a compile time
 error before this method gets stabilized.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(array_windows)]
 let slice = [0, 1, 2, 3];
 let mut iter = slice.array_windows();
@@ -557,9 +596,9 @@ slice, then the last chunk will not have length chunk_size.
    
 
    See rchunks_exact for a variant of this iterator that returns chunks of always exactly
 chunk_size elements, and chunks for the same iterator but starting at the beginning
 of the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let slice = ['l', 'o', 'r', 'e', 'm'];
 let mut iter = slice.rchunks(2);
 assert_eq!(iter.next().unwrap(), &['e', 'm']);
@@ -573,9 +612,9 @@ length of the slice, then the last chunk will not have length chunk_sizeSee rchunks_exact_mut for a variant of this iterator that returns chunks of always
 exactly chunk_size elements, and chunks_mut for the same iterator but starting at the
 beginning of the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
 
@@ -596,9 +635,9 @@ resulting code better than in the case of rchunks for a variant of this iterator that also returns the remainder as a smaller
 chunk, and chunks_exact for the same iterator but starting at the beginning of the
 slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let slice = ['l', 'o', 'r', 'e', 'm'];
 let mut iter = slice.rchunks_exact(2);
 assert_eq!(iter.next().unwrap(), &['e', 'm']);
@@ -615,9 +654,9 @@ resulting code better than in the case of rchunks_mut for a variant of this iterator that also returns the remainder as a
 smaller chunk, and chunks_exact_mut for the same iterator but starting at the beginning
 of the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
 
@@ -633,7 +672,7 @@ of elements using the predicate to separate them.
    
 
    The predicate is called on two elements following themselves,
 it means the predicate is called on slice[0] and slice[1]
 then on slice[1] and slice[2] and so on.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_group_by)]
 
 let slice = &[1, 1, 1, 3, 3, 2, 2, 2];
@@ -661,7 +700,7 @@ runs of elements using the predicate to separate them.
    
 
    The predicate is called on two elements following themselves,
 it means the predicate is called on slice[0] and slice[1]
 then on slice[1] and slice[2] and so on.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_group_by)]
 
 let slice = &mut [1, 1, 1, 3, 3, 2, 2, 2];
@@ -688,9 +727,9 @@ then on slice[1] and slice[2] and so on.
    
 
    The first will contain all indices from [0, mid) (excluding
 the index mid itself) and the second will contain all
 indices from [mid, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if mid > len.
    
-
    Examples
    
+
    Examples
    
 
    let v = [1, 2, 3, 4, 5, 6];
 
 {
@@ -714,9 +753,9 @@ indices from [mid, len) (excluding the index len itsel
 
    The first will contain all indices from [0, mid) (excluding
 the index mid itself) and the second will contain all
 indices from [mid, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if mid > len.
    
-
    Examples
    
+
    Examples
    
 
    let mut v = [1, 0, 3, 0, 5, 6];
 let (left, right) = v.split_at_mut(2);
 assert_eq!(left, [1, 0]);
@@ -733,7 +772,7 @@ indices from [mid, len) (excluding the index len itsel
 
    Calling this method with an out-of-bounds index is undefined behavior
 even if the resulting reference is not used. The caller has to ensure that
 0 <= mid <= self.len().
    
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_split_at_unchecked)]
 
 let v = [1, 2, 3, 4, 5, 6];
@@ -764,7 +803,7 @@ indices from [mid, len) (excluding the index len itsel
 
    Calling this method with an out-of-bounds index is undefined behavior
 even if the resulting reference is not used. The caller has to ensure that
 0 <= mid <= self.len().
    
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_split_at_unchecked)]
 
 let mut v = [1, 0, 3, 0, 5, 6];
@@ -781,9 +820,9 @@ even if the resulting reference is not used. The caller has to ensure that
 
    The array will contain all indices from [0, N) (excluding
 the index N itself) and the slice will contain all
 indices from [N, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if N > len.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(split_array)]
 
 let v = &[1, 2, 3, 4, 5, 6][..];
@@ -809,9 +848,9 @@ indices from [N, len) (excluding the index len itself)
 
    The array will contain all indices from [0, N) (excluding
 the index N itself) and the slice will contain all
 indices from [N, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if N > len.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(split_array)]
 
 let mut v = &mut [1, 0, 3, 0, 5, 6][..];
@@ -826,9 +865,9 @@ the end.
    
 
    The slice will contain all indices from [0, len - N) (excluding
 the index len - N itself) and the array will contain all
 indices from [len - N, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if N > len.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(split_array)]
 
 let v = &[1, 2, 3, 4, 5, 6][..];
@@ -855,9 +894,9 @@ index from the end.
    
 
    The slice will contain all indices from [0, len - N) (excluding
 the index N itself) and the array will contain all
 indices from [len - N, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if N > len.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(split_array)]
 
 let mut v = &mut [1, 0, 3, 0, 5, 6][..];
@@ -869,7 +908,7 @@ indices from [len - N, len) (excluding the index len i
 assert_eq!(v, [1, 2, 3, 4, 5, 6]);

    Returns an iterator over subslices separated by elements that match pred. The matched element is not contained in the subslices.

    -
    Examples
    +
    Examples
    let slice = [10, 40, 33, 20];
 let mut iter = slice.split(|num| num % 3 == 0);
 
@@ -899,7 +938,7 @@ present between them:
    
 assert!(iter.next().is_none());

    Returns an iterator over mutable subslices separated by elements that match pred. The matched element is not contained in the subslices.

    -
    Examples
    +
    Examples
    let mut v = [10, 40, 30, 20, 60, 50];
 
 for group in v.split_mut(|num| *num % 3 == 0) {
@@ -909,7 +948,7 @@ match pred. The matched element is not contained in the subslices.<

    Returns an iterator over subslices separated by elements that match pred. The matched element is contained in the end of the previous subslice as a terminator.

    -
    Examples
    +
    Examples
    let slice = [10, 40, 33, 20];
 let mut iter = slice.split_inclusive(|num| num % 3 == 0);
 
@@ -929,7 +968,7 @@ That slice will be the last item returned by the iterator.

    Returns an iterator over mutable subslices separated by elements that match pred. The matched element is contained in the previous subslice as a terminator.

    -
    Examples
    +
    Examples
    let mut v = [10, 40, 30, 20, 60, 50];
 
 for group in v.split_inclusive_mut(|num| *num % 3 == 0) {
@@ -940,7 +979,7 @@ subslice as a terminator.

    Returns an iterator over subslices separated by elements that match pred, starting at the end of the slice and working backwards. The matched element is not contained in the subslices.

    -
    Examples
    +
    Examples
    let slice = [11, 22, 33, 0, 44, 55];
 let mut iter = slice.rsplit(|num| *num == 0);
 
@@ -960,7 +999,7 @@ slice will be the first (or last) item returned by the iterator.

    Returns an iterator over mutable subslices separated by elements that match pred, starting at the end of the slice and working backwards. The matched element is not contained in the subslices.

    -
    Examples
    +
    Examples
    let mut v = [100, 400, 300, 200, 600, 500];
 
 let mut count = 0;
@@ -974,7 +1013,7 @@ backwards. The matched element is not contained in the subslices.
    
 not contained in the subslices.
    
 
    The last element returned, if any, will contain the remainder of the
 slice.
    
-
    Examples
    
+
    Examples
    
 
    Print the slice split once by numbers divisible by 3 (i.e., [10, 40],
 [20, 60, 50]):
    
 
@@ -988,7 +1027,7 @@ slice.
    
 not contained in the subslices.
    
 
    The last element returned, if any, will contain the remainder of the
 slice.
    
-
    Examples
    
+
    Examples
    
 
    let mut v = [10, 40, 30, 20, 60, 50];
 
 for group in v.splitn_mut(2, |num| *num % 3 == 0) {
@@ -1001,7 +1040,7 @@ the slice and works backwards. The matched element is not contained in
 the subslices.
    
 
    The last element returned, if any, will contain the remainder of the
 slice.
    
-
    Examples
    
+
    Examples
    
 
    Print the slice split once, starting from the end, by numbers divisible
 by 3 (i.e., [50], [10, 40, 30, 20]):
    
 
@@ -1016,15 +1055,17 @@ the slice and works backwards. The matched element is not contained in
 the subslices.
    
 
    The last element returned, if any, will contain the remainder of the
 slice.
    
-
    Examples
    
+
    Examples
    
 
    let mut s = [10, 40, 30, 20, 60, 50];
 
 for group in s.rsplitn_mut(2, |num| *num % 3 == 0) {
     group[0] = 1;
 }
 assert_eq!(s, [1, 40, 30, 20, 60, 1]);
    
-

    Returns true if the slice contains an element with the given value.

    -
    Examples
    +

    Returns true if the slice contains an element with the given value.

    +

    This operation is O(n).

    +

    Note that if you have a sorted slice, binary_search may be faster.

    +
    Examples
    let v = [10, 40, 30];
 assert!(v.contains(&30));
 assert!(!v.contains(&50));
    @@ -1035,8 +1076,8 @@ use iter().any:

    let v = [String::from("hello"), String::from("world")]; // slice of `String`
 assert!(v.iter().any(|e| e == "hello")); // search with `&str`
 assert!(!v.iter().any(|e| e == "hi"));
    -

    Returns true if needle is a prefix of the slice.

    -
    Examples
    +

    Returns true if needle is a prefix of the slice.

    +
    Examples
    let v = [10, 40, 30];
 assert!(v.starts_with(&[10]));
 assert!(v.starts_with(&[10, 40]));
@@ -1048,8 +1089,8 @@ use iter().any:
    
 assert!(v.starts_with(&[]));
 let v: &[u8] = &[];
 assert!(v.starts_with(&[]));
    -

    Returns true if needle is a suffix of the slice.

    -
    Examples
    +

    Returns true if needle is a suffix of the slice.

    +
    Examples
    let v = [10, 40, 30];
 assert!(v.ends_with(&[30]));
 assert!(v.ends_with(&[40, 30]));
@@ -1061,11 +1102,11 @@ use iter().any:
    
 assert!(v.ends_with(&[]));
 let v: &[u8] = &[];
 assert!(v.ends_with(&[]));
    -

    Returns a subslice with the prefix removed.

    +

    Returns a subslice with the prefix removed.

    If the slice starts with prefix, returns the subslice after the prefix, wrapped in Some. If prefix is empty, simply returns the original slice.

    If the slice does not start with prefix, returns None.

    -
    Examples
    +
    Examples
    let v = &[10, 40, 30];
 assert_eq!(v.strip_prefix(&[10]), Some(&[40, 30][..]));
 assert_eq!(v.strip_prefix(&[10, 40]), Some(&[30][..]));
@@ -1075,17 +1116,18 @@ If prefix is empty, simply returns the original slice.
    
 let prefix : &str = "he";
 assert_eq!(b"hello".strip_prefix(prefix.as_bytes()),
            Some(b"llo".as_ref()));
    -

    Returns a subslice with the suffix removed.

    +

    Returns a subslice with the suffix removed.

    If the slice ends with suffix, returns the subslice before the suffix, wrapped in Some. If suffix is empty, simply returns the original slice.

    If the slice does not end with suffix, returns None.

    -
    Examples
    +
    Examples
    let v = &[10, 40, 30];
 assert_eq!(v.strip_suffix(&[30]), Some(&[10, 40][..]));
 assert_eq!(v.strip_suffix(&[40, 30]), Some(&[10][..]));
 assert_eq!(v.strip_suffix(&[50]), None);
 assert_eq!(v.strip_suffix(&[50, 30]), None);
    -

    Binary searches this sorted slice for a given element.

    +

    Binary searches this slice for a given element. +This behaves similary to contains if this slice is sorted.

    If the value is found then Result::Ok is returned, containing the index of the matching element. If there are multiple matches, then any one of the matches could be returned. The index is chosen @@ -1094,7 +1136,7 @@ If the value is not found then binary_search_by, binary_search_by_key, and partition_point.

    -
    Examples
    +
    Examples

    Looks up a series of four elements. The first is found, with a uniquely determined position; the second and third are not found; the fourth could match any position in [1, 4].

    @@ -1115,7 +1157,8 @@ sort order, consider using // The above is equivalent to `let idx = s.binary_search(&num).unwrap_or_else(|x| x);` s.insert(idx, num); assert_eq!(s, [0, 1, 1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 42, 55]);
    -

    Binary searches this sorted slice with a comparator function.

    +

    Binary searches this slice with a comparator function. +This behaves similarly to contains if this slice is sorted.

    The comparator function should implement an order consistent with the sort order of the underlying slice, returning an order code that indicates whether its argument is Less, @@ -1128,7 +1171,7 @@ If the value is not found then binary_search, binary_search_by_key, and partition_point.

    -
    Examples
    +
    Examples

    Looks up a series of four elements. The first is found, with a uniquely determined position; the second and third are not found; the fourth could match any position in [1, 4].

    @@ -1144,7 +1187,8 @@ found; the fourth could match any position in [1, 4].

    let seek = 1; let r = s.binary_search_by(|probe| probe.cmp(&seek)); assert!(match r { Ok(1..=4) => true, _ => false, });
    -

    Binary searches this sorted slice with a key extraction function.

    +

    Binary searches this slice with a key extraction function. +This behaves similarly to contains if this slice is sorted.

    Assumes that the slice is sorted by the key, for instance with sort_by_key using the same key extraction function.

    If the value is found then Result::Ok is returned, containing the @@ -1155,7 +1199,7 @@ If the value is not found then binary_search, binary_search_by, and partition_point.

    -
    Examples
    +
    Examples

    Looks up a series of four elements in a slice of pairs sorted by their second elements. The first is found, with a uniquely determined position; the second and third are not found; the @@ -1170,7 +1214,7 @@ fourth could match any position in [1, 4].

    assert_eq!(s.binary_search_by_key(&100, |&(a, b)| b), Err(13)); let r = s.binary_search_by_key(&1, |&(a, b)| b); assert!(match r { Ok(1..=4) => true, _ => false, });
    -

    Sorts the slice, but might not preserve the order of equal elements.

    +

    Sorts the slice, but might not preserve the order of equal elements.

    This sort is unstable (i.e., may reorder equal elements), in-place (i.e., does not allocate), and O(n * log(n)) worst-case.

    Current implementation
    @@ -1181,12 +1225,12 @@ randomization to avoid degenerate cases, but with a fixed seed to always provide deterministic behavior.

    It is typically faster than stable sorting, except in a few special cases, e.g., when the slice consists of several concatenated sorted sequences.

    -
    Examples
    +
    Examples
    let mut v = [-5, 4, 1, -3, 2];
 
 v.sort_unstable();
 assert!(v == [-5, -3, 1, 2, 4]);
    -

    Sorts the slice with a comparator function, but might not preserve the order of equal +

    Sorts the slice with a comparator function, but might not preserve the order of equal elements.

    This sort is unstable (i.e., may reorder equal elements), in-place (i.e., does not allocate), and O(n * log(n)) worst-case.

    @@ -1211,7 +1255,7 @@ randomization to avoid degenerate cases, but with a fixed seed to always provide deterministic behavior.

    It is typically faster than stable sorting, except in a few special cases, e.g., when the slice consists of several concatenated sorted sequences.

    -
    Examples
    +
    Examples
    let mut v = [5, 4, 1, 3, 2];
 v.sort_unstable_by(|a, b| a.cmp(b));
 assert!(v == [1, 2, 3, 4, 5]);
@@ -1219,7 +1263,7 @@ slice consists of several concatenated sorted sequences.
    
 // reverse sorting
 v.sort_unstable_by(|a, b| b.cmp(a));
 assert!(v == [5, 4, 3, 2, 1]);
    -

    Sorts the slice with a key extraction function, but might not preserve the order of equal +

    Sorts the slice with a key extraction function, but might not preserve the order of equal elements.

    This sort is unstable (i.e., may reorder equal elements), in-place (i.e., does not allocate), and O(m * n * log(n)) worst-case, where the key function is @@ -1233,12 +1277,12 @@ deterministic behavior.

    Due to its key calling strategy, sort_unstable_by_key is likely to be slower than sort_by_cached_key in cases where the key function is expensive.

    -
    Examples
    +
    Examples
    let mut v = [-5i32, 4, 1, -3, 2];
 
 v.sort_unstable_by_key(|k| k.abs());
 assert!(v == [1, 2, -3, 4, -5]);
    -

    Reorder the slice such that the element at index is at its final sorted position.

    +

    Reorder the slice such that the element at index is at its final sorted position.

    This reordering has the additional property that any value at position i < index will be less than or equal to any value at a position j > index. Additionally, this reordering is unstable (i.e. any number of equal elements may end up at position index), in-place @@ -1249,9 +1293,9 @@ the one at the given index.

    Current implementation

    The current algorithm is based on the quickselect portion of the same quicksort algorithm used for sort_unstable.

    -
    Panics
    +
    Panics

    Panics when index >= len(), meaning it always panics on empty slices.

    -
    Examples
    +
    Examples
    let mut v = [-5i32, 4, 1, -3, 2];
 
 // Find the median
@@ -1263,7 +1307,7 @@ used for v == [-5, -3, 1, 2, 4] ||
         v == [-3, -5, 1, 4, 2] ||
         v == [-5, -3, 1, 4, 2]);
    -

    Reorder the slice with a comparator function such that the element at index is at its +

    Reorder the slice with a comparator function such that the element at index is at its final sorted position.

    This reordering has the additional property that any value at position i < index will be less than or equal to any value at a position j > index using the comparator function. @@ -1276,9 +1320,9 @@ function.

    Current implementation

    The current algorithm is based on the quickselect portion of the same quicksort algorithm used for sort_unstable.

    -
    Panics
    +
    Panics

    Panics when index >= len(), meaning it always panics on empty slices.

    -
    Examples
    +
    Examples
    let mut v = [-5i32, 4, 1, -3, 2];
 
 // Find the median as if the slice were sorted in descending order.
@@ -1290,7 +1334,7 @@ used for v == [2, 4, 1, -3, -5] ||
         v == [4, 2, 1, -5, -3] ||
         v == [4, 2, 1, -3, -5]);
    -

    Reorder the slice with a key extraction function such that the element at index is at its +

    Reorder the slice with a key extraction function such that the element at index is at its final sorted position.

    This reordering has the additional property that any value at position i < index will be less than or equal to any value at a position j > index using the key extraction function. @@ -1303,9 +1347,9 @@ function.

    Current implementation

    The current algorithm is based on the quickselect portion of the same quicksort algorithm used for sort_unstable.

    -
    Panics
    +
    Panics

    Panics when index >= len(), meaning it always panics on empty slices.

    -
    Examples
    +
    Examples
    let mut v = [-5i32, 4, 1, -3, 2];
 
 // Return the median as if the array were sorted according to absolute value.
@@ -1317,12 +1361,12 @@ used for v == [1, 2, -3, -5, 4] ||
         v == [2, 1, -3, 4, -5] ||
         v == [2, 1, -3, -5, 4]);
    -
    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all consecutive repeated elements to the end of the slice according to the +

    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all consecutive repeated elements to the end of the slice according to the PartialEq trait implementation.

    Returns two slices. The first contains no consecutive repeated elements. The second contains all the duplicates in no specified order.

    If the slice is sorted, the first returned slice contains no duplicates.

    -
    Examples
    +
    Examples
    #![feature(slice_partition_dedup)]
 
 let mut slice = [1, 2, 2, 3, 3, 2, 1, 1];
@@ -1331,7 +1375,7 @@ The second contains all the duplicates in no specified order.
    
 
 assert_eq!(dedup, [1, 2, 3, 2, 1]);
 assert_eq!(duplicates, [2, 3, 1]);
    -
    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all but the first of consecutive elements to the end of the slice satisfying +

    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all but the first of consecutive elements to the end of the slice satisfying a given equality relation.

    Returns two slices. The first contains no consecutive repeated elements. The second contains all the duplicates in no specified order.

    @@ -1340,7 +1384,7 @@ must determine if the elements compare equal. The elements are passed in opposit from their order in the slice, so if same_bucket(a, b) returns true, a is moved at the end of the slice.

    If the slice is sorted, the first returned slice contains no duplicates.

    -
    Examples
    +
    Examples
    #![feature(slice_partition_dedup)]
 
 let mut slice = ["foo", "Foo", "BAZ", "Bar", "bar", "baz", "BAZ"];
@@ -1349,12 +1393,12 @@ at the end of the slice.
    
 
 assert_eq!(dedup, ["foo", "BAZ", "Bar", "baz"]);
 assert_eq!(duplicates, ["bar", "Foo", "BAZ"]);
    -
    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all but the first of consecutive elements to the end of the slice that resolve +

    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all but the first of consecutive elements to the end of the slice that resolve to the same key.

    Returns two slices. The first contains no consecutive repeated elements. The second contains all the duplicates in no specified order.

    If the slice is sorted, the first returned slice contains no duplicates.

    -
    Examples
    +
    Examples
    #![feature(slice_partition_dedup)]
 
 let mut slice = [10, 20, 21, 30, 30, 20, 11, 13];
@@ -1363,17 +1407,17 @@ The second contains all the duplicates in no specified order.
    
 
 assert_eq!(dedup, [10, 20, 30, 20, 11]);
 assert_eq!(duplicates, [21, 30, 13]);
    -

    Rotates the slice in-place such that the first mid elements of the +

    Rotates the slice in-place such that the first mid elements of the slice move to the end while the last self.len() - mid elements move to the front. After calling rotate_left, the element previously at index mid will become the first element in the slice.

    -
    Panics
    +
    Panics

    This function will panic if mid is greater than the length of the slice. Note that mid == self.len() does not panic and is a no-op rotation.

    Complexity

    Takes linear (in self.len()) time.

    -
    Examples
    +
    Examples
    let mut a = ['a', 'b', 'c', 'd', 'e', 'f'];
 a.rotate_left(2);
 assert_eq!(a, ['c', 'd', 'e', 'f', 'a', 'b']);
    @@ -1382,17 +1426,17 @@ rotation.

    let mut a = ['a', 'b', 'c', 'd', 'e', 'f'];
 a[1..5].rotate_left(1);
 assert_eq!(a, ['a', 'c', 'd', 'e', 'b', 'f']);
    -

    Rotates the slice in-place such that the first self.len() - k +

    Rotates the slice in-place such that the first self.len() - k elements of the slice move to the end while the last k elements move to the front. After calling rotate_right, the element previously at index self.len() - k will become the first element in the slice.

    -
    Panics
    +
    Panics

    This function will panic if k is greater than the length of the slice. Note that k == self.len() does not panic and is a no-op rotation.

    Complexity

    Takes linear (in self.len()) time.

    -
    Examples
    +
    Examples
    let mut a = ['a', 'b', 'c', 'd', 'e', 'f'];
 a.rotate_right(2);
 assert_eq!(a, ['e', 'f', 'a', 'b', 'c', 'd']);
    @@ -1401,25 +1445,25 @@ rotation.

    let mut a = ['a', 'b', 'c', 'd', 'e', 'f'];
 a[1..5].rotate_right(1);
 assert_eq!(a, ['a', 'e', 'b', 'c', 'd', 'f']);
    -

    Fills self with elements by cloning value.

    -
    Examples
    +

    Fills self with elements by cloning value.

    +
    Examples
    let mut buf = vec![0; 10];
 buf.fill(1);
 assert_eq!(buf, vec![1; 10]);
    -

    Fills self with elements returned by calling a closure repeatedly.

    +

    Fills self with elements returned by calling a closure repeatedly.

    This method uses a closure to create new values. If you’d rather Clone a given value, use fill. If you want to use the Default trait to generate values, you can pass Default::default as the argument.

    -
    Examples
    +
    Examples
    let mut buf = vec![1; 10];
 buf.fill_with(Default::default);
 assert_eq!(buf, vec![0; 10]);
    -

    Copies the elements from src into self.

    +

    Copies the elements from src into self.

    The length of src must be the same as self.

    -
    Panics
    +
    Panics

    This function will panic if the two slices have different lengths.

    -
    Examples
    +
    Examples

    Cloning two elements from a slice into another:

    let src = [1, 2, 3, 4];
@@ -1451,12 +1495,12 @@ sub-slices from a slice:
    
 }
 
 assert_eq!(slice, [4, 5, 3, 4, 5]);
    -

    Copies all elements from src into self, using a memcpy.

    +

    Copies all elements from src into self, using a memcpy.

    The length of src must be the same as self.

    If T does not implement Copy, use clone_from_slice.

    -
    Panics
    +
    Panics

    This function will panic if the two slices have different lengths.

    -
    Examples
    +
    Examples

    Copying two elements from a slice into another:

    let src = [1, 2, 3, 4];
@@ -1488,16 +1532,16 @@ sub-slices from a slice:
    
 }
 
 assert_eq!(slice, [4, 5, 3, 4, 5]);
    -

    Copies elements from one part of the slice to another part of itself, +

    Copies elements from one part of the slice to another part of itself, using a memmove.

    src is the range within self to copy from. dest is the starting index of the range within self to copy to, which will have the same length as src. The two ranges may overlap. The ends of the two ranges must be less than or equal to self.len().

    -
    Panics
    +
    Panics

    This function will panic if either range exceeds the end of the slice, or if the end of src is before the start.

    -
    Examples
    +
    Examples

    Copying four bytes within a slice:

    let mut bytes = *b"Hello, World!";
@@ -1505,9 +1549,9 @@ or if the end of src is before the start.
    
 bytes.copy_within(1..5, 8);
 
 assert_eq!(&bytes, b"Hello, Wello!");
    -

    Swaps all elements in self with those in other.

    +

    Swaps all elements in self with those in other.

    The length of other must be the same as self.

    -
    Panics
    +
    Panics

    This function will panic if the two slices have different lengths.

    Example

    Swapping two elements across slices:

    @@ -1537,7 +1581,7 @@ mutable sub-slices from a slice:

    } assert_eq!(slice, [4, 5, 3, 1, 2]);
    -

    Transmute the slice to a slice of another type, ensuring alignment of the types is +

    Transmute the slice to a slice of another type, ensuring alignment of the types is maintained.

    This method splits the slice into three distinct slices: prefix, correctly aligned middle slice of a new type, and the suffix slice. The method may make the middle slice the greatest @@ -1549,7 +1593,7 @@ zero-sized and will return the original slice without splitting anything.

    Safety

    This method is essentially a transmute with respect to the elements in the returned middle slice, so all the usual caveats pertaining to transmute::<T, U> also apply here.

    -
    Examples
    +
    Examples

    Basic usage:

    unsafe {
@@ -1559,7 +1603,7 @@ middle slice, so all the usual caveats pertaining to transmute::<T, U&g
     // more_efficient_algorithm_for_aligned_shorts(shorts);
     // less_efficient_algorithm_for_bytes(suffix);
 }
    -

    Transmute the slice to a slice of another type, ensuring alignment of the types is +

    Transmute the slice to a slice of another type, ensuring alignment of the types is maintained.

    This method splits the slice into three distinct slices: prefix, correctly aligned middle slice of a new type, and the suffix slice. The method may make the middle slice the greatest @@ -1571,7 +1615,7 @@ zero-sized and will return the original slice without splitting anything.

    Safety

    This method is essentially a transmute with respect to the elements in the returned middle slice, so all the usual caveats pertaining to transmute::<T, U> also apply here.

    -
    Examples
    +
    Examples

    Basic usage:

    unsafe {
@@ -1581,7 +1625,7 @@ middle slice, so all the usual caveats pertaining to transmute::<T, U&g
     // more_efficient_algorithm_for_aligned_shorts(shorts);
     // less_efficient_algorithm_for_bytes(suffix);
 }
    -
    🔬 This is a nightly-only experimental API. (portable_simd)

    Split a slice into a prefix, a middle of aligned SIMD types, and a suffix.

    +
    🔬 This is a nightly-only experimental API. (portable_simd)

    Split a slice into a prefix, a middle of aligned SIMD types, and a suffix.

    This is a safe wrapper around slice::align_to, so has the same weak postconditions as that method. You’re only assured that self.len() == prefix.len() + middle.len() * LANES + suffix.len().

    @@ -1593,7 +1637,7 @@ postconditions as that method. You’re only assured that

    That said, this is a safe method, so if you’re only writing safe code, then this can at most cause incorrect logic, not unsoundness.

    -
    Panics
    +
    Panics

    This will panic if the size of the SIMD type is different from LANES times that of the scalar.

    At the time of writing, the trait restrictions on Simd<T, LANES> keeps @@ -1601,7 +1645,7 @@ that from ever happening, as only power-of-two numbers of lanes are supported. It’s possible that, in the future, those restrictions might be lifted in a way that would make it possible to see panics from this method for something like LANES == 3.

    -
    Examples
    +
    Examples
    #![feature(portable_simd)]
 
 let short = &[1, 2, 3];
@@ -1628,7 +1672,7 @@ method for something like LANES == 3.
    
 
 let numbers: Vec<f32> = (1..101).map(|x| x as _).collect();
 assert_eq!(basic_simd_sum(&numbers[1..99]), 4949.0);
    -
    🔬 This is a nightly-only experimental API. (portable_simd)

    Split a slice into a prefix, a middle of aligned SIMD types, and a suffix.

    +
    🔬 This is a nightly-only experimental API. (portable_simd)

    Split a slice into a prefix, a middle of aligned SIMD types, and a suffix.

    This is a safe wrapper around slice::align_to_mut, so has the same weak postconditions as that method. You’re only assured that self.len() == prefix.len() + middle.len() * LANES + suffix.len().

    @@ -1641,7 +1685,7 @@ postconditions as that method. You’re only assured that

    That said, this is a safe method, so if you’re only writing safe code, then this can at most cause incorrect logic, not unsoundness.

    This is the mutable version of slice::as_simd; see that for examples.

    -
    Panics
    +
    Panics

    This will panic if the size of the SIMD type is different from LANES times that of the scalar.

    At the time of writing, the trait restrictions on Simd<T, LANES> keeps @@ -1649,13 +1693,13 @@ that from ever happening, as only power-of-two numbers of lanes are supported. It’s possible that, in the future, those restrictions might be lifted in a way that would make it possible to see panics from this method for something like LANES == 3.

    -
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted.

    +
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted.

    That is, for each element a and its following element b, a <= b must hold. If the slice yields exactly zero or one element, true is returned.

    Note that if Self::Item is only PartialOrd, but not Ord, the above definition implies that this function returns false if any two consecutive items are not comparable.

    -
    Examples
    +
    Examples
    #![feature(is_sorted)]
 let empty: [i32; 0] = [];
 
@@ -1664,20 +1708,20 @@ comparable.
    
 assert!([0].is_sorted());
 assert!(empty.is_sorted());
 assert!(![0.0, 1.0, f32::NAN].is_sorted());
    -
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted using the given comparator function.

    +
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted using the given comparator function.

    Instead of using PartialOrd::partial_cmp, this function uses the given compare function to determine the ordering of two elements. Apart from that, it’s equivalent to is_sorted; see its documentation for more information.

    -
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted using the given key extraction function.

    +
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted using the given key extraction function.

    Instead of comparing the slice’s elements directly, this function compares the keys of the elements, as determined by f. Apart from that, it’s equivalent to is_sorted; see its documentation for more information.

    -
    Examples
    +
    Examples
    #![feature(is_sorted)]
 
 assert!(["c", "bb", "aaa"].is_sorted_by_key(|s| s.len()));
 assert!(![-2i32, -1, 0, 3].is_sorted_by_key(|n| n.abs()));
    -

    Returns the index of the partition point according to the given predicate +

    Returns the index of the partition point according to the given predicate (the index of the first element of the second partition).

    The slice is assumed to be partitioned according to the given predicate. This means that all elements for which the predicate returns true are at the start of the slice @@ -1687,7 +1731,7 @@ For example, [7, 15, 3, 5, 4, 12, 6] is a partitioned under the predicate x % 2

    If this slice is not partitioned, the returned result is unspecified and meaningless, as this method performs a kind of binary search.

    See also binary_search, binary_search_by, and binary_search_by_key.

    -
    Examples
    +
    Examples
    let v = [1, 2, 3, 3, 5, 6, 7];
 let i = v.partition_point(|&x| x < 5);
 
@@ -1702,13 +1746,13 @@ sort order:
    
 let idx = s.partition_point(|&x| x < num);
 s.insert(idx, num);
 assert_eq!(s, [0, 1, 1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 42, 55]);
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the subslice corresponding to the given range +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the subslice corresponding to the given range and returns a reference to it.

    Returns None and does not modify the slice if the given range is out of bounds.

    Note that this method only accepts one-sided ranges such as 2.. or ..6, but not 2..6.

    -
    Examples
    +
    Examples

    Taking the first three elements of a slice:

    #![feature(slice_take)]
@@ -1738,13 +1782,13 @@ range is out of bounds.
    
 assert_eq!(None, slice.take(..=4));
 let expected: &[char] = &['a', 'b', 'c', 'd'];
 assert_eq!(Some(expected), slice.take(..4));
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the subslice corresponding to the given range +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the subslice corresponding to the given range and returns a mutable reference to it.

    Returns None and does not modify the slice if the given range is out of bounds.

    Note that this method only accepts one-sided ranges such as 2.. or ..6, but not 2..6.

    -
    Examples
    +
    Examples

    Taking the first three elements of a slice:

    #![feature(slice_take)]
@@ -1774,10 +1818,10 @@ range is out of bounds.
    
 assert_eq!(None, slice.take_mut(..=4));
 let expected: &mut [_] = &mut ['a', 'b', 'c', 'd'];
 assert_eq!(Some(expected), slice.take_mut(..4));
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the first element of the slice and returns a reference +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the first element of the slice and returns a reference to it.

    Returns None if the slice is empty.

    -
    Examples
    +
    Examples
    #![feature(slice_take)]
 
 let mut slice: &[_] = &['a', 'b', 'c'];
@@ -1785,10 +1829,10 @@ to it.
    
 
 assert_eq!(slice, &['b', 'c']);
 assert_eq!(first, &'a');
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the first element of the slice and returns a mutable +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the first element of the slice and returns a mutable reference to it.

    Returns None if the slice is empty.

    -
    Examples
    +
    Examples
    #![feature(slice_take)]
 
 let mut slice: &mut [_] = &mut ['a', 'b', 'c'];
@@ -1797,10 +1841,10 @@ reference to it.
    
 
 assert_eq!(slice, &['b', 'c']);
 assert_eq!(first, &'d');
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the last element of the slice and returns a reference +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the last element of the slice and returns a reference to it.

    Returns None if the slice is empty.

    -
    Examples
    +
    Examples
    #![feature(slice_take)]
 
 let mut slice: &[_] = &['a', 'b', 'c'];
@@ -1808,10 +1852,10 @@ to it.
    
 
 assert_eq!(slice, &['a', 'b']);
 assert_eq!(last, &'c');
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the last element of the slice and returns a mutable +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the last element of the slice and returns a mutable reference to it.

    Returns None if the slice is empty.

    -
    Examples
    +
    Examples
    #![feature(slice_take)]
 
 let mut slice: &mut [_] = &mut ['a', 'b', 'c'];
@@ -1820,45 +1864,6 @@ reference to it.
    
 
 assert_eq!(slice, &['a', 'b']);
 assert_eq!(last, &'d');
    -
    🔬 This is a nightly-only experimental API. (slice_flatten)

    Takes a &[[T; N]], and flattens it to a &[T].

    -
    Panics
    -

    This panics if the length of the resulting slice would overflow a usize.

    -

    This is only possible when flattening a slice of arrays of zero-sized -types, and thus tends to be irrelevant in practice. If -size_of::<T>() > 0, this will never panic.

    -
    Examples
    -
    #![feature(slice_flatten)]
-
-assert_eq!([[1, 2, 3], [4, 5, 6]].flatten(), &[1, 2, 3, 4, 5, 6]);
-
-assert_eq!(
-    [[1, 2, 3], [4, 5, 6]].flatten(),
-    [[1, 2], [3, 4], [5, 6]].flatten(),
-);
-
-let slice_of_empty_arrays: &[[i32; 0]] = &[[], [], [], [], []];
-assert!(slice_of_empty_arrays.flatten().is_empty());
-
-let empty_slice_of_arrays: &[[u32; 10]] = &[];
-assert!(empty_slice_of_arrays.flatten().is_empty());
    -
    🔬 This is a nightly-only experimental API. (slice_flatten)

    Takes a &mut [[T; N]], and flattens it to a &mut [T].

    -
    Panics
    -

    This panics if the length of the resulting slice would overflow a usize.

    -

    This is only possible when flattening a slice of arrays of zero-sized -types, and thus tends to be irrelevant in practice. If -size_of::<T>() > 0, this will never panic.

    -
    Examples
    -
    #![feature(slice_flatten)]
-
-fn add_5_to_all(slice: &mut [i32]) {
-    for i in slice {
-        *i += 5;
-    }
-}
-
-let mut array = [[1, 2, 3], [4, 5, 6], [7, 8, 9]];
-add_5_to_all(array.flatten_mut());
-assert_eq!(array, [[6, 7, 8], [9, 10, 11], [12, 13, 14]]);

    Checks if all bytes in this slice are within the ASCII range.

    Checks that two slices are an ASCII case-insensitive match.

    Same as to_ascii_lowercase(a) == to_ascii_lowercase(b), @@ -2120,9 +2125,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/trait.UtilitiesInstructions.html b/rustdoc/latest/halo2_gadgets/utilities/trait.UtilitiesInstructions.html index 1cb4e637..b190831b 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/trait.UtilitiesInstructions.html +++ b/rustdoc/latest/halo2_gadgets/utilities/trait.UtilitiesInstructions.html @@ -26,5 +26,5 @@ }
    Expand description

    Trait for utilities used across circuits.

    Required Associated Types

    Variable in the circuit.

    Provided Methods

    Load a variable.

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_gadgets/utilities/trait.Var.html b/rustdoc/latest/halo2_gadgets/utilities/trait.Var.html index b9f64c19..372bcfb2 100644 --- a/rustdoc/latest/halo2_gadgets/utilities/trait.Var.html +++ b/rustdoc/latest/halo2_gadgets/utilities/trait.Var.html @@ -25,5 +25,5 @@ }
    Expand description

    Trait for a variable in the circuit.

    Required Methods

    The cell at which this variable was allocated.

    The value allocated to this variable.

    -

    Implementations on Foreign Types

    Implementors

    +

    Implementations on Foreign Types

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/all.html b/rustdoc/latest/halo2_proofs/all.html index f1717dd3..09097323 100644 --- a/rustdoc/latest/halo2_proofs/all.html +++ b/rustdoc/latest/halo2_proofs/all.html @@ -18,5 +18,5 @@

    List of all items

    Structs

    Enums

    Traits

    Functions

    +

    Crate halo2_proofs

    List of all items

    Structs

    Enums

    Traits

    Functions

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/fn.best_fft.html b/rustdoc/latest/halo2_proofs/arithmetic/fn.best_fft.html index ca63c04e..99f8556f 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/fn.best_fft.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/fn.best_fft.html @@ -28,5 +28,5 @@ distinct powers of $\omega$. This transformation is invertible by providing $\omega^{-1}$ in place of $\omega$ and dividing each resulting field element by $n$.

    This will use multithreading if beneficial.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/fn.best_multiexp.html b/rustdoc/latest/halo2_proofs/arithmetic/fn.best_multiexp.html index 8eca0ed7..b7bece52 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/fn.best_multiexp.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/fn.best_multiexp.html @@ -22,5 +22,5 @@

    Function halo2_proofs::arithmetic::best_multiexp

    source · [−]
    pub fn best_multiexp<C: CurveAffine>(
        coeffs: &[C::Scalar], 
        bases: &[C]
    ) -> C::Curve
    Expand description

    Performs a multi-exponentiation operation.

    This function will panic if coeffs and bases have a different length.

    This will use multithreading if beneficial.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/fn.compute_inner_product.html b/rustdoc/latest/halo2_proofs/arithmetic/fn.compute_inner_product.html index ac8174b1..4625d9fb 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/fn.compute_inner_product.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/fn.compute_inner_product.html @@ -21,5 +21,5 @@

    In halo2_proofs::arithmetic

    Function halo2_proofs::arithmetic::compute_inner_product

    source · [−]
    pub fn compute_inner_product<F: Field>(a: &[F], b: &[F]) -> F
    Expand description

    This computes the inner product of two vectors a and b.

    This function will panic if the two vectors are not the same size.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/fn.eval_polynomial.html b/rustdoc/latest/halo2_proofs/arithmetic/fn.eval_polynomial.html index 33c7d551..34bf58c5 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/fn.eval_polynomial.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/fn.eval_polynomial.html @@ -20,5 +20,5 @@

    Function halo2_proofs::arithmetic::eval_polynomial

    source · [−]
    pub fn eval_polynomial<F: Field>(poly: &[F], point: F) -> F
    Expand description

    This evaluates a provided polynomial (in coefficient form) at point.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/fn.kate_division.html b/rustdoc/latest/halo2_proofs/arithmetic/fn.kate_division.html index 9694d547..dc5898a3 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/fn.kate_division.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/fn.kate_division.html @@ -21,5 +21,5 @@

    In halo2_proofs::arithmetic

    Function halo2_proofs::arithmetic::kate_division

    source · [−]
    pub fn kate_division<'a, F: Field, I: IntoIterator<Item = &'a F>>(
        a: I, 
        b: F
    ) -> Vec<F>ⓘNotable traits for Vec<u8, A>impl<A> Write for Vec<u8, A> where
        A: Allocator,      where
        I::IntoIter: DoubleEndedIterator + ExactSizeIterator, 
    Expand description

    Divides polynomial a in X by X - b with no remainder.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/fn.lagrange_interpolate.html b/rustdoc/latest/halo2_proofs/arithmetic/fn.lagrange_interpolate.html index 54e00aaf..e9713844 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/fn.lagrange_interpolate.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/fn.lagrange_interpolate.html @@ -22,5 +22,5 @@

    Function halo2_proofs::arithmetic::lagrange_interpolate

    source · [−]
    pub fn lagrange_interpolate<F: FieldExt>(points: &[F], evals: &[F]) -> Vec<F>ⓘNotable traits for Vec<u8, A>impl<A> Write for Vec<u8, A> where
        A: Allocator, 
    Expand description

    Returns coefficients of an n - 1 degree polynomial given a set of n points and their evaluations. This function will panic if two values in points are the same.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/fn.parallelize.html b/rustdoc/latest/halo2_proofs/arithmetic/fn.parallelize.html index bdad9f83..d573ab25 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/fn.parallelize.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/fn.parallelize.html @@ -21,5 +21,5 @@

    In halo2_proofs::arithmetic

    Function halo2_proofs::arithmetic::parallelize

    source · [−]
    pub fn parallelize<T: Send, F: Fn(&mut [T], usize) + Send + Sync + Clone>(
        v: &mut [T], 
        f: F
    )
    Expand description

    This simple utility function will parallelize an operation that is to be performed over a mutable slice.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/fn.small_multiexp.html b/rustdoc/latest/halo2_proofs/arithmetic/fn.small_multiexp.html index ccd14826..fc6e8c34 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/fn.small_multiexp.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/fn.small_multiexp.html @@ -21,5 +21,5 @@

    In halo2_proofs::arithmetic

    Function halo2_proofs::arithmetic::small_multiexp

    source · [−]
    pub fn small_multiexp<C: CurveAffine>(
        coeffs: &[C::Scalar], 
        bases: &[C]
    ) -> C::Curve
    Expand description

    Performs a small multi-exponentiation operation. Uses the double-and-add algorithm with doublings shared across points.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/index.html b/rustdoc/latest/halo2_proofs/arithmetic/index.html index f4505288..1b7fb704 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/index.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/index.html @@ -59,5 +59,5 @@ are the same.

    performed over a mutable slice.

    small_multiexp

    Performs a small multi-exponentiation operation. Uses the double-and-add algorithm with doublings shared across points.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/struct.Coordinates.html b/rustdoc/latest/halo2_proofs/arithmetic/struct.Coordinates.html index 9c1c9dde..cb55bfac 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/struct.Coordinates.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/struct.Coordinates.html @@ -104,9 +104,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/struct.SqrtTables.html b/rustdoc/latest/halo2_proofs/arithmetic/struct.SqrtTables.html index 6dde8712..9db6e71e 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/struct.SqrtTables.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/struct.SqrtTables.html @@ -96,9 +96,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/trait.CurveAffine.html b/rustdoc/latest/halo2_proofs/arithmetic/trait.CurveAffine.html index 248fb63c..1d4d9435 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/trait.CurveAffine.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/trait.CurveAffine.html @@ -43,5 +43,5 @@ curve.

    always be true unless an “unchecked” API was used.

    Returns the curve constant $a$.

    Returns the curve constant $b$.

    -

    Implementations on Foreign Types

    Implementors

    +

    Implementations on Foreign Types

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/trait.CurveExt.html b/rustdoc/latest/halo2_proofs/arithmetic/trait.CurveExt.html index f8ccaf4b..0a3e945c 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/trait.CurveExt.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/trait.CurveExt.html @@ -68,5 +68,5 @@ if the coordinates are not on the curve.

    by an element of multiplicative order 3.

    Apply the curve endomorphism by multiplying the x-coordinate by an element of multiplicative order 3.

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/trait.Field.html b/rustdoc/latest/halo2_proofs/arithmetic/trait.Field.html index 2b2d2e9d..c8ec75c1 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/trait.Field.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/trait.Field.html @@ -59,5 +59,5 @@ failing if the element is zero.

    Computes the square root of this element, if it exists.

    Computes the multiplicative inverse of this element, failing if the element is zero.

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/trait.FieldExt.html b/rustdoc/latest/halo2_proofs/arithmetic/trait.FieldExt.html index 718ad46b..004e0dbe 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/trait.FieldExt.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/trait.FieldExt.html @@ -45,15 +45,15 @@ byte representation of an integer.

    canonically.

    Provided Methods

    Exponentiates self by by, where by is a little-endian order integer exponent.

    -

    Implementations on Foreign Types

    Converts a 512-bit little endian integer into -a Fp by reducing by the modulus.

    -

    Converts a 512-bit little endian integer into +

    Implementations on Foreign Types

    Converts a 512-bit little endian integer into a Fq by reducing by the modulus.

    -

    Implementors

    +

    Converts a 512-bit little endian integer into +a Fp by reducing by the modulus.

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/trait.Group.html b/rustdoc/latest/halo2_proofs/arithmetic/trait.Group.html index 97aa9cc0..8caf2a56 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/trait.Group.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/trait.Group.html @@ -35,5 +35,5 @@ associated scalar field of size $p$.

    Adds rhs to this group element.

    Subtracts rhs from this group element.

    Scales this group element by a scalar.

    -

    Implementations on Foreign Types

    Implementors

    +

    Implementations on Foreign Types

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/arithmetic/trait.SqrtRatio.html b/rustdoc/latest/halo2_proofs/arithmetic/trait.SqrtRatio.html index 2d4a3685..c6c40c79 100644 --- a/rustdoc/latest/halo2_proofs/arithmetic/trait.SqrtRatio.html +++ b/rustdoc/latest/halo2_proofs/arithmetic/trait.SqrtRatio.html @@ -52,5 +52,5 @@ generator being fixed; it may be changed in future crate versions to simplify th implementation of the SSWU hash-to-curve algorithm.

    The choice of root from sqrt is unspecified.

    Equivalent to Self::sqrt_ratio(self, one()).

    -

    Implementations on Foreign Types

    Implementors

    +

    Implementations on Foreign Types

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/floor_planner/index.html b/rustdoc/latest/halo2_proofs/circuit/floor_planner/index.html index 111f303e..ad8d552f 100644 --- a/rustdoc/latest/halo2_proofs/circuit/floor_planner/index.html +++ b/rustdoc/latest/halo2_proofs/circuit/floor_planner/index.html @@ -23,5 +23,5 @@

    Structs

    V1

    The version 1 FloorPlanner provided by halo2.

    V1Pass

    A single pass of the V1 layouter.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/floor_planner/struct.V1.html b/rustdoc/latest/halo2_proofs/circuit/floor_planner/struct.V1.html index 9d53f8bc..bef230d5 100644 --- a/rustdoc/latest/halo2_proofs/circuit/floor_planner/struct.V1.html +++ b/rustdoc/latest/halo2_proofs/circuit/floor_planner/struct.V1.html @@ -94,9 +94,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/floor_planner/struct.V1Pass.html b/rustdoc/latest/halo2_proofs/circuit/floor_planner/struct.V1Pass.html index e71fe7af..46f15152 100644 --- a/rustdoc/latest/halo2_proofs/circuit/floor_planner/struct.V1Pass.html +++ b/rustdoc/latest/halo2_proofs/circuit/floor_planner/struct.V1Pass.html @@ -95,9 +95,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/index.html b/rustdoc/latest/halo2_proofs/circuit/index.html index 4ff3eeac..2f1acdf4 100644 --- a/rustdoc/latest/halo2_proofs/circuit/index.html +++ b/rustdoc/latest/halo2_proofs/circuit/index.html @@ -37,5 +37,5 @@ context) and, when dropped, pops out of the namespace context.

    Chip

    A chip implements a set of instructions that can be used by gadgets.

    Layouter

    A layout strategy within a circuit. The layouter is chip-agnostic and applies its strategy to the context and config it is given.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/layouter/enum.RegionColumn.html b/rustdoc/latest/halo2_proofs/circuit/layouter/enum.RegionColumn.html index 27690387..ee541247 100644 --- a/rustdoc/latest/halo2_proofs/circuit/layouter/enum.RegionColumn.html +++ b/rustdoc/latest/halo2_proofs/circuit/layouter/enum.RegionColumn.html @@ -114,9 +114,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/layouter/index.html b/rustdoc/latest/halo2_proofs/circuit/layouter/index.html index 821b991e..0e02a63b 100644 --- a/rustdoc/latest/halo2_proofs/circuit/layouter/index.html +++ b/rustdoc/latest/halo2_proofs/circuit/layouter/index.html @@ -29,5 +29,5 @@ as well as selectors that are not concrete columns at this stage.

    Traits

    RegionLayouter

    Helper trait for implementing a custom Layouter.

    TableLayouter

    Helper trait for implementing a custom Layouter.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/layouter/struct.RegionShape.html b/rustdoc/latest/halo2_proofs/circuit/layouter/struct.RegionShape.html index 94f7775b..ff2fee81 100644 --- a/rustdoc/latest/halo2_proofs/circuit/layouter/struct.RegionShape.html +++ b/rustdoc/latest/halo2_proofs/circuit/layouter/struct.RegionShape.html @@ -103,9 +103,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/layouter/trait.RegionLayouter.html b/rustdoc/latest/halo2_proofs/circuit/layouter/trait.RegionLayouter.html index 5bfc9153..44b529f3 100644 --- a/rustdoc/latest/halo2_proofs/circuit/layouter/trait.RegionLayouter.html +++ b/rustdoc/latest/halo2_proofs/circuit/layouter/trait.RegionLayouter.html @@ -66,5 +66,5 @@ configured via ConstraintSystem::enable_constant.

    Constraint two cells to have the same value.

    Returns an error if either of the cells is not within the given permutation.

    Trait Implementations

    Converts to this type from the input type.

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/layouter/trait.TableLayouter.html b/rustdoc/latest/halo2_proofs/circuit/layouter/trait.TableLayouter.html index c5977d37..510e2faf 100644 --- a/rustdoc/latest/halo2_proofs/circuit/layouter/trait.TableLayouter.html +++ b/rustdoc/latest/halo2_proofs/circuit/layouter/trait.TableLayouter.html @@ -26,5 +26,5 @@

    Required Methods

    Assigns a fixed value to a table cell.

    Returns an error if the table cell has already been assigned to.

    Trait Implementations

    Converts to this type from the input type.

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/struct.AssignedCell.html b/rustdoc/latest/halo2_proofs/circuit/struct.AssignedCell.html index 707aff55..8beca75b 100644 --- a/rustdoc/latest/halo2_proofs/circuit/struct.AssignedCell.html +++ b/rustdoc/latest/halo2_proofs/circuit/struct.AssignedCell.html @@ -99,9 +99,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/struct.Cell.html b/rustdoc/latest/halo2_proofs/circuit/struct.Cell.html index 45f5373a..1d0d2a65 100644 --- a/rustdoc/latest/halo2_proofs/circuit/struct.Cell.html +++ b/rustdoc/latest/halo2_proofs/circuit/struct.Cell.html @@ -90,9 +90,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/struct.NamespacedLayouter.html b/rustdoc/latest/halo2_proofs/circuit/struct.NamespacedLayouter.html index e65b06ab..f17dddde 100644 --- a/rustdoc/latest/halo2_proofs/circuit/struct.NamespacedLayouter.html +++ b/rustdoc/latest/halo2_proofs/circuit/struct.NamespacedLayouter.html @@ -97,9 +97,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/struct.Region.html b/rustdoc/latest/halo2_proofs/circuit/struct.Region.html index c8e185ec..87ba62dd 100644 --- a/rustdoc/latest/halo2_proofs/circuit/struct.Region.html +++ b/rustdoc/latest/halo2_proofs/circuit/struct.Region.html @@ -109,9 +109,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/struct.RegionIndex.html b/rustdoc/latest/halo2_proofs/circuit/struct.RegionIndex.html index 6ac35d3a..629b75b9 100644 --- a/rustdoc/latest/halo2_proofs/circuit/struct.RegionIndex.html +++ b/rustdoc/latest/halo2_proofs/circuit/struct.RegionIndex.html @@ -93,9 +93,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/struct.RegionStart.html b/rustdoc/latest/halo2_proofs/circuit/struct.RegionStart.html index 33f5c6cc..4725af95 100644 --- a/rustdoc/latest/halo2_proofs/circuit/struct.RegionStart.html +++ b/rustdoc/latest/halo2_proofs/circuit/struct.RegionStart.html @@ -96,9 +96,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/struct.SimpleFloorPlanner.html b/rustdoc/latest/halo2_proofs/circuit/struct.SimpleFloorPlanner.html index f1100264..a7360403 100644 --- a/rustdoc/latest/halo2_proofs/circuit/struct.SimpleFloorPlanner.html +++ b/rustdoc/latest/halo2_proofs/circuit/struct.SimpleFloorPlanner.html @@ -89,9 +89,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/struct.Table.html b/rustdoc/latest/halo2_proofs/circuit/struct.Table.html index 0b852d7c..0012d549 100644 --- a/rustdoc/latest/halo2_proofs/circuit/struct.Table.html +++ b/rustdoc/latest/halo2_proofs/circuit/struct.Table.html @@ -89,9 +89,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/trait.Chip.html b/rustdoc/latest/halo2_proofs/circuit/trait.Chip.html index cc5f1ec8..5a053393 100644 --- a/rustdoc/latest/halo2_proofs/circuit/trait.Chip.html +++ b/rustdoc/latest/halo2_proofs/circuit/trait.Chip.html @@ -39,5 +39,5 @@ during circuit synthesis, that can be derived during

    Provides access to general chip state loaded at the beginning of circuit synthesis.

    Panics if called before Chip::load.

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/circuit/trait.Layouter.html b/rustdoc/latest/halo2_proofs/circuit/trait.Layouter.html index cd6ea6ba..7ce4dfdf 100644 --- a/rustdoc/latest/halo2_proofs/circuit/trait.Layouter.html +++ b/rustdoc/latest/halo2_proofs/circuit/trait.Layouter.html @@ -60,5 +60,5 @@ absolute position.

    Exits out of the existing namespace.

    Not intended for downstream consumption; use Layouter::namespace instead.

    Provided Methods

    Enters into a namespace.

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/cost/index.html b/rustdoc/latest/halo2_proofs/dev/cost/index.html index 17772c45..b382b98c 100644 --- a/rustdoc/latest/halo2_proofs/dev/cost/index.html +++ b/rustdoc/latest/halo2_proofs/dev/cost/index.html @@ -24,5 +24,5 @@
    CircuitCost

    Measures a circuit to determine its costs, and explain what contributes to them.

    MarginalProofSize

    The marginal size of a Halo 2 proof, broken down into its contributing factors.

    ProofSize

    The size of a Halo 2 proof, broken down into its contributing factors.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/cost/struct.CircuitCost.html b/rustdoc/latest/halo2_proofs/dev/cost/struct.CircuitCost.html index 4f67d2db..fea48b55 100644 --- a/rustdoc/latest/halo2_proofs/dev/cost/struct.CircuitCost.html +++ b/rustdoc/latest/halo2_proofs/dev/cost/struct.CircuitCost.html @@ -89,9 +89,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/cost/struct.MarginalProofSize.html b/rustdoc/latest/halo2_proofs/dev/cost/struct.MarginalProofSize.html index 0ec617d2..c27037ec 100644 --- a/rustdoc/latest/halo2_proofs/dev/cost/struct.MarginalProofSize.html +++ b/rustdoc/latest/halo2_proofs/dev/cost/struct.MarginalProofSize.html @@ -86,9 +86,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/cost/struct.ProofSize.html b/rustdoc/latest/halo2_proofs/dev/cost/struct.ProofSize.html index 43d3b896..c83d1e95 100644 --- a/rustdoc/latest/halo2_proofs/dev/cost/struct.ProofSize.html +++ b/rustdoc/latest/halo2_proofs/dev/cost/struct.ProofSize.html @@ -86,9 +86,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/enum.FailureLocation.html b/rustdoc/latest/halo2_proofs/dev/enum.FailureLocation.html index 55075a2f..e0cc6514 100644 --- a/rustdoc/latest/halo2_proofs/dev/enum.FailureLocation.html +++ b/rustdoc/latest/halo2_proofs/dev/enum.FailureLocation.html @@ -104,9 +104,9 @@ builds. Read more

    builds. Read more

    Converts the given value to a String. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/enum.VerifyFailure.html b/rustdoc/latest/halo2_proofs/dev/enum.VerifyFailure.html index a5c8dd68..c5f60469 100644 --- a/rustdoc/latest/halo2_proofs/dev/enum.VerifyFailure.html +++ b/rustdoc/latest/halo2_proofs/dev/enum.VerifyFailure.html @@ -150,9 +150,9 @@ builds. Read more

    builds. Read more

    Converts the given value to a String. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/fn.circuit_dot_graph.html b/rustdoc/latest/halo2_proofs/dev/fn.circuit_dot_graph.html index 11eb70b4..20417267 100644 --- a/rustdoc/latest/halo2_proofs/dev/fn.circuit_dot_graph.html +++ b/rustdoc/latest/halo2_proofs/dev/fn.circuit_dot_graph.html @@ -22,5 +22,5 @@

    Function halo2_proofs::dev::circuit_dot_graph

    source · [−]
    pub fn circuit_dot_graph<F: Field, ConcreteCircuit: Circuit<F>>(
        circuit: &ConcreteCircuit
    ) -> String
    Expand description

    Builds a dot graph string representing the given circuit.

    The graph is built from calls to Layouter::namespace both within the circuit, and inside the gadgets and chips that it uses.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/index.html b/rustdoc/latest/halo2_proofs/dev/index.html index dbcab9aa..291d2661 100644 --- a/rustdoc/latest/halo2_proofs/dev/index.html +++ b/rustdoc/latest/halo2_proofs/dev/index.html @@ -33,5 +33,5 @@
    VerifyFailure

    The reasons why a particular circuit is not satisfied.

    Functions

    circuit_dot_graph

    Builds a dot graph string representing the given circuit.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/metadata/index.html b/rustdoc/latest/halo2_proofs/dev/metadata/index.html index 2b359b82..d96d585e 100644 --- a/rustdoc/latest/halo2_proofs/dev/metadata/index.html +++ b/rustdoc/latest/halo2_proofs/dev/metadata/index.html @@ -27,5 +27,5 @@
    Region

    Metadata about an assigned region within a circuit.

    VirtualCell

    A “virtual cell” is a PLONK cell that has been queried at a particular relative offset within a custom gate.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/metadata/struct.Column.html b/rustdoc/latest/halo2_proofs/dev/metadata/struct.Column.html index 94ce2362..2826f356 100644 --- a/rustdoc/latest/halo2_proofs/dev/metadata/struct.Column.html +++ b/rustdoc/latest/halo2_proofs/dev/metadata/struct.Column.html @@ -108,9 +108,9 @@ builds. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Converts the given value to a String. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/metadata/struct.Constraint.html b/rustdoc/latest/halo2_proofs/dev/metadata/struct.Constraint.html index d5220bb5..bcb4c738 100644 --- a/rustdoc/latest/halo2_proofs/dev/metadata/struct.Constraint.html +++ b/rustdoc/latest/halo2_proofs/dev/metadata/struct.Constraint.html @@ -91,9 +91,9 @@ builds. Read more

    builds. Read more

    Converts the given value to a String. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/metadata/struct.Gate.html b/rustdoc/latest/halo2_proofs/dev/metadata/struct.Gate.html index 1af47542..a0b5e621 100644 --- a/rustdoc/latest/halo2_proofs/dev/metadata/struct.Gate.html +++ b/rustdoc/latest/halo2_proofs/dev/metadata/struct.Gate.html @@ -91,9 +91,9 @@ builds. Read more

    builds. Read more

    Converts the given value to a String. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/metadata/struct.Region.html b/rustdoc/latest/halo2_proofs/dev/metadata/struct.Region.html index c0e01fc6..9fd668fb 100644 --- a/rustdoc/latest/halo2_proofs/dev/metadata/struct.Region.html +++ b/rustdoc/latest/halo2_proofs/dev/metadata/struct.Region.html @@ -97,9 +97,9 @@ builds. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Converts the given value to a String. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/metadata/struct.VirtualCell.html b/rustdoc/latest/halo2_proofs/dev/metadata/struct.VirtualCell.html index 906cd57f..69972e25 100644 --- a/rustdoc/latest/halo2_proofs/dev/metadata/struct.VirtualCell.html +++ b/rustdoc/latest/halo2_proofs/dev/metadata/struct.VirtualCell.html @@ -104,9 +104,9 @@ builds. Read more

    builds. Read more

    Converts the given value to a String. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/struct.CircuitGates.html b/rustdoc/latest/halo2_proofs/dev/struct.CircuitGates.html index 392a1be7..888acade 100644 --- a/rustdoc/latest/halo2_proofs/dev/struct.CircuitGates.html +++ b/rustdoc/latest/halo2_proofs/dev/struct.CircuitGates.html @@ -151,9 +151,9 @@ builds. Read more

    builds. Read more

    Converts the given value to a String. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/struct.CircuitLayout.html b/rustdoc/latest/halo2_proofs/dev/struct.CircuitLayout.html index 349357b1..654d03a0 100644 --- a/rustdoc/latest/halo2_proofs/dev/struct.CircuitLayout.html +++ b/rustdoc/latest/halo2_proofs/dev/struct.CircuitLayout.html @@ -112,9 +112,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/dev/struct.MockProver.html b/rustdoc/latest/halo2_proofs/dev/struct.MockProver.html index 8dc35f4a..d44613e9 100644 --- a/rustdoc/latest/halo2_proofs/dev/struct.MockProver.html +++ b/rustdoc/latest/halo2_proofs/dev/struct.MockProver.html @@ -214,9 +214,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/index.html b/rustdoc/latest/halo2_proofs/index.html index 7303ef7b..2e7c749b 100644 --- a/rustdoc/latest/halo2_proofs/index.html +++ b/rustdoc/latest/halo2_proofs/index.html @@ -33,5 +33,5 @@ various forms, including computing commitments to them and provably opening the committed polynomials at arbitrary points.

    transcript

    This module contains utilities and traits for dealing with Fiat-Shamir transcripts.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/enum.Any.html b/rustdoc/latest/halo2_proofs/plonk/enum.Any.html index dd668757..1787b3da 100644 --- a/rustdoc/latest/halo2_proofs/plonk/enum.Any.html +++ b/rustdoc/latest/halo2_proofs/plonk/enum.Any.html @@ -116,9 +116,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/enum.Assigned.html b/rustdoc/latest/halo2_proofs/plonk/enum.Assigned.html index 944e2201..20711976 100644 --- a/rustdoc/latest/halo2_proofs/plonk/enum.Assigned.html +++ b/rustdoc/latest/halo2_proofs/plonk/enum.Assigned.html @@ -142,9 +142,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/enum.Error.html b/rustdoc/latest/halo2_proofs/plonk/enum.Error.html index 5f007ca9..a2c803cb 100644 --- a/rustdoc/latest/halo2_proofs/plonk/enum.Error.html +++ b/rustdoc/latest/halo2_proofs/plonk/enum.Error.html @@ -119,9 +119,9 @@ builds. Read more

    builds. Read more

    Converts the given value to a String. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/enum.Expression.html b/rustdoc/latest/halo2_proofs/plonk/enum.Expression.html index 829a209e..f13c5be3 100644 --- a/rustdoc/latest/halo2_proofs/plonk/enum.Expression.html +++ b/rustdoc/latest/halo2_proofs/plonk/enum.Expression.html @@ -146,9 +146,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/fn.create_proof.html b/rustdoc/latest/halo2_proofs/plonk/fn.create_proof.html index 2d72cdd6..114b283a 100644 --- a/rustdoc/latest/halo2_proofs/plonk/fn.create_proof.html +++ b/rustdoc/latest/halo2_proofs/plonk/fn.create_proof.html @@ -23,5 +23,5 @@ parameters params and the proving key ProvingKey that was generated previously for the same circuit. The provided instances are zero-padded internally.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/fn.keygen_pk.html b/rustdoc/latest/halo2_proofs/plonk/fn.keygen_pk.html index 569441e3..edde3992 100644 --- a/rustdoc/latest/halo2_proofs/plonk/fn.keygen_pk.html +++ b/rustdoc/latest/halo2_proofs/plonk/fn.keygen_pk.html @@ -20,5 +20,5 @@

    Function halo2_proofs::plonk::keygen_pk

    source · [−]
    pub fn keygen_pk<C, ConcreteCircuit>(
        params: &Params<C>, 
        vk: VerifyingKey<C>, 
        circuit: &ConcreteCircuit
    ) -> Result<ProvingKey<C>, Error> where
        C: CurveAffine,
        ConcreteCircuit: Circuit<C::Scalar>, 
    Expand description

    Generate a ProvingKey from a VerifyingKey and an instance of Circuit.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/fn.keygen_vk.html b/rustdoc/latest/halo2_proofs/plonk/fn.keygen_vk.html index 3d122029..5d0e4093 100644 --- a/rustdoc/latest/halo2_proofs/plonk/fn.keygen_vk.html +++ b/rustdoc/latest/halo2_proofs/plonk/fn.keygen_vk.html @@ -20,5 +20,5 @@

    Function halo2_proofs::plonk::keygen_vk

    source · [−]
    pub fn keygen_vk<C, ConcreteCircuit>(
        params: &Params<C>, 
        circuit: &ConcreteCircuit
    ) -> Result<VerifyingKey<C>, Error> where
        C: CurveAffine,
        ConcreteCircuit: Circuit<C::Scalar>, 
    Expand description

    Generate a VerifyingKey from an instance of Circuit.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/fn.verify_proof.html b/rustdoc/latest/halo2_proofs/plonk/fn.verify_proof.html index 4dcca513..99d3981e 100644 --- a/rustdoc/latest/halo2_proofs/plonk/fn.verify_proof.html +++ b/rustdoc/latest/halo2_proofs/plonk/fn.verify_proof.html @@ -20,5 +20,5 @@

    Function halo2_proofs::plonk::verify_proof

    source · [−]
    pub fn verify_proof<'params, C: CurveAffine, E: EncodedChallenge<C>, T: TranscriptRead<C, E>, V: VerificationStrategy<'params, C>>(
        params: &'params Params<C>, 
        vk: &VerifyingKey<C>, 
        strategy: V, 
        instances: &[&[&[C::Scalar]]], 
        transcript: &mut T
    ) -> Result<V::Output, Error>
    Expand description

    Returns a boolean indicating whether or not the proof is valid

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/index.html b/rustdoc/latest/halo2_proofs/plonk/index.html index 9dde0d1d..6837af47 100644 --- a/rustdoc/latest/halo2_proofs/plonk/index.html +++ b/rustdoc/latest/halo2_proofs/plonk/index.html @@ -66,5 +66,5 @@ are zero-padded internally.

    keygen_pk

    Generate a ProvingKey from a VerifyingKey and an instance of Circuit.

    keygen_vk

    Generate a VerifyingKey from an instance of Circuit.

    verify_proof

    Returns a boolean indicating whether or not the proof is valid

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.Advice.html b/rustdoc/latest/halo2_proofs/plonk/struct.Advice.html index 929a7c1a..1fb38396 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.Advice.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.Advice.html @@ -96,9 +96,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.BatchVerifier.html b/rustdoc/latest/halo2_proofs/plonk/struct.BatchVerifier.html index 7cce1275..ffef938f 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.BatchVerifier.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.BatchVerifier.html @@ -92,9 +92,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.Column.html b/rustdoc/latest/halo2_proofs/plonk/struct.Column.html index 70b74cc7..791d0c29 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.Column.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.Column.html @@ -118,9 +118,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.Constraint.html b/rustdoc/latest/halo2_proofs/plonk/struct.Constraint.html index 8b8f1b34..e250ec84 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.Constraint.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.Constraint.html @@ -88,9 +88,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.ConstraintSystem.html b/rustdoc/latest/halo2_proofs/plonk/struct.ConstraintSystem.html index 105f705f..a5acf02d 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.ConstraintSystem.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.ConstraintSystem.html @@ -125,9 +125,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.Constraints.html b/rustdoc/latest/halo2_proofs/plonk/struct.Constraints.html index 10a70497..abc3c454 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.Constraints.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.Constraints.html @@ -119,9 +119,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.Fixed.html b/rustdoc/latest/halo2_proofs/plonk/struct.Fixed.html index 292580e0..97aad25e 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.Fixed.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.Fixed.html @@ -96,9 +96,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.Instance.html b/rustdoc/latest/halo2_proofs/plonk/struct.Instance.html index 2755a4e7..b27160ca 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.Instance.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.Instance.html @@ -96,9 +96,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.PinnedConstraintSystem.html b/rustdoc/latest/halo2_proofs/plonk/struct.PinnedConstraintSystem.html index a37e1fc6..bacd33a8 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.PinnedConstraintSystem.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.PinnedConstraintSystem.html @@ -85,9 +85,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.PinnedVerificationKey.html b/rustdoc/latest/halo2_proofs/plonk/struct.PinnedVerificationKey.html index 32b4e12b..7df4ce26 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.PinnedVerificationKey.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.PinnedVerificationKey.html @@ -86,9 +86,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.ProvingKey.html b/rustdoc/latest/halo2_proofs/plonk/struct.ProvingKey.html index 9839d6bf..83d1b55a 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.ProvingKey.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.ProvingKey.html @@ -87,9 +87,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.Selector.html b/rustdoc/latest/halo2_proofs/plonk/struct.Selector.html index dd721a66..431f69a6 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.Selector.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.Selector.html @@ -136,9 +136,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.SingleVerifier.html b/rustdoc/latest/halo2_proofs/plonk/struct.SingleVerifier.html index fec33779..ceed6880 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.SingleVerifier.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.SingleVerifier.html @@ -89,9 +89,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.TableColumn.html b/rustdoc/latest/halo2_proofs/plonk/struct.TableColumn.html index a6680397..782b4de7 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.TableColumn.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.TableColumn.html @@ -100,9 +100,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.VerifyingKey.html b/rustdoc/latest/halo2_proofs/plonk/struct.VerifyingKey.html index d6c1132c..2d1a1dac 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.VerifyingKey.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.VerifyingKey.html @@ -90,9 +90,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/struct.VirtualCells.html b/rustdoc/latest/halo2_proofs/plonk/struct.VirtualCells.html index e950b2e2..a6033aa6 100644 --- a/rustdoc/latest/halo2_proofs/plonk/struct.VirtualCells.html +++ b/rustdoc/latest/halo2_proofs/plonk/struct.VirtualCells.html @@ -91,9 +91,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/trait.Assignment.html b/rustdoc/latest/halo2_proofs/plonk/trait.Assignment.html index 747dae22..89c10b00 100644 --- a/rustdoc/latest/halo2_proofs/plonk/trait.Assignment.html +++ b/rustdoc/latest/halo2_proofs/plonk/trait.Assignment.html @@ -49,5 +49,5 @@ for a constraint system.

    Not intended for downstream consumption; use Layouter::namespace instead.

    Exits out of the existing namespace.

    Not intended for downstream consumption; use Layouter::namespace instead.

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/trait.Circuit.html b/rustdoc/latest/halo2_proofs/plonk/trait.Circuit.html index 7fde9dbe..b414193f 100644 --- a/rustdoc/latest/halo2_proofs/plonk/trait.Circuit.html +++ b/rustdoc/latest/halo2_proofs/plonk/trait.Circuit.html @@ -39,5 +39,5 @@ arrangement, column arrangement, etc.

    Given the provided cs, synthesize the circuit. The concrete type of the caller will be different depending on the context, and they may or may not expect to have a witness present.

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/trait.ColumnType.html b/rustdoc/latest/halo2_proofs/plonk/trait.ColumnType.html index e9bb1e31..e3bf2875 100644 --- a/rustdoc/latest/halo2_proofs/plonk/trait.ColumnType.html +++ b/rustdoc/latest/halo2_proofs/plonk/trait.ColumnType.html @@ -20,5 +20,5 @@

    Trait halo2_proofs::plonk::ColumnType

    source · [−]
    pub trait ColumnType: 'static + Sized + Copy + Debug + PartialEq + Eq + Into<Any> { }
    Expand description

    A column type

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/trait.FloorPlanner.html b/rustdoc/latest/halo2_proofs/plonk/trait.FloorPlanner.html index bf38bcea..dbf84986 100644 --- a/rustdoc/latest/halo2_proofs/plonk/trait.FloorPlanner.html +++ b/rustdoc/latest/halo2_proofs/plonk/trait.FloorPlanner.html @@ -34,5 +34,5 @@ global constant values. These columns will all have been equality-enabled.

    calls to Circuit::default().synthesize(config, &mut layouter).
  • Call circuit.synthesize(config, &mut layouter) exactly once.
    • -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/plonk/trait.VerificationStrategy.html b/rustdoc/latest/halo2_proofs/plonk/trait.VerificationStrategy.html index fbb9f3d9..b95378bb 100644 --- a/rustdoc/latest/halo2_proofs/plonk/trait.VerificationStrategy.html +++ b/rustdoc/latest/halo2_proofs/plonk/trait.VerificationStrategy.html @@ -27,5 +27,5 @@

    Required Associated Types

    The output type of this verification strategy after processing a proof.

    Required Methods

    Obtains an MSM from the verifier strategy and yields back the strategy’s output.

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/commitment/fn.create_proof.html b/rustdoc/latest/halo2_proofs/poly/commitment/fn.create_proof.html index 884f63bc..483ae36e 100644 --- a/rustdoc/latest/halo2_proofs/poly/commitment/fn.create_proof.html +++ b/rustdoc/latest/halo2_proofs/poly/commitment/fn.create_proof.html @@ -30,5 +30,5 @@ already seen the common inputs: the polynomial commitment P, the claimed opening v, and the point x. It’s probably also nice for the transcript to have seen the elliptic curve description and the URS, if you want to be rigorous.

    -
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    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/commitment/fn.verify_proof.html b/rustdoc/latest/halo2_proofs/poly/commitment/fn.verify_proof.html index 90ccf704..b37d9677 100644 --- a/rustdoc/latest/halo2_proofs/poly/commitment/fn.verify_proof.html +++ b/rustdoc/latest/halo2_proofs/poly/commitment/fn.verify_proof.html @@ -22,5 +22,5 @@

    Function halo2_proofs::poly::commitment::verify_proof

    source · [−]
    pub fn verify_proof<'a, C: CurveAffine, E: EncodedChallenge<C>, T: TranscriptRead<C, E>>(
        params: &'a Params<C>, 
        msm: MSM<'a, C>, 
        transcript: &mut T, 
        x: C::Scalar, 
        v: C::Scalar
    ) -> Result<Guard<'a, C, E>, Error>
    Expand description

    Checks to see if the proof represented within transcript is valid, and a point x that the polynomial commitment P opens purportedly to the value v. The provided msm should evaluate to the commitment P being opened.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/commitment/index.html b/rustdoc/latest/halo2_proofs/poly/commitment/index.html index 9470231e..aa07f4e4 100644 --- a/rustdoc/latest/halo2_proofs/poly/commitment/index.html +++ b/rustdoc/latest/halo2_proofs/poly/commitment/index.html @@ -35,5 +35,5 @@ evaluated at.

    verify_proof

    Checks to see if the proof represented within transcript is valid, and a point x that the polynomial commitment P opens purportedly to the value v. The provided msm should evaluate to the commitment P being opened.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/commitment/struct.Accumulator.html b/rustdoc/latest/halo2_proofs/poly/commitment/struct.Accumulator.html index 43b20503..2765cd32 100644 --- a/rustdoc/latest/halo2_proofs/poly/commitment/struct.Accumulator.html +++ b/rustdoc/latest/halo2_proofs/poly/commitment/struct.Accumulator.html @@ -96,9 +96,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/commitment/struct.Blind.html b/rustdoc/latest/halo2_proofs/poly/commitment/struct.Blind.html index e7788620..5e92c3d2 100644 --- a/rustdoc/latest/halo2_proofs/poly/commitment/struct.Blind.html +++ b/rustdoc/latest/halo2_proofs/poly/commitment/struct.Blind.html @@ -102,9 +102,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/commitment/struct.Guard.html b/rustdoc/latest/halo2_proofs/poly/commitment/struct.Guard.html index b305f79c..04fbcec6 100644 --- a/rustdoc/latest/halo2_proofs/poly/commitment/struct.Guard.html +++ b/rustdoc/latest/halo2_proofs/poly/commitment/struct.Guard.html @@ -95,9 +95,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/commitment/struct.MSM.html b/rustdoc/latest/halo2_proofs/poly/commitment/struct.MSM.html index ceed4510..4c2d52c3 100644 --- a/rustdoc/latest/halo2_proofs/poly/commitment/struct.MSM.html +++ b/rustdoc/latest/halo2_proofs/poly/commitment/struct.MSM.html @@ -100,9 +100,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/commitment/struct.Params.html b/rustdoc/latest/halo2_proofs/poly/commitment/struct.Params.html index c641b490..7783fb1d 100644 --- a/rustdoc/latest/halo2_proofs/poly/commitment/struct.Params.html +++ b/rustdoc/latest/halo2_proofs/poly/commitment/struct.Params.html @@ -98,9 +98,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/enum.Error.html b/rustdoc/latest/halo2_proofs/poly/enum.Error.html index 2c6725b5..20bf231d 100644 --- a/rustdoc/latest/halo2_proofs/poly/enum.Error.html +++ b/rustdoc/latest/halo2_proofs/poly/enum.Error.html @@ -90,9 +90,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/index.html b/rustdoc/latest/halo2_proofs/poly/index.html index 146a547b..3b4860d9 100644 --- a/rustdoc/latest/halo2_proofs/poly/index.html +++ b/rustdoc/latest/halo2_proofs/poly/index.html @@ -45,5 +45,5 @@ rotations. Zero represents no rotation.

    Error

    This is an error that could occur during proving or circuit synthesis.

    Traits

    Basis

    The basis over which a polynomial is described.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/multiopen/fn.create_proof.html b/rustdoc/latest/halo2_proofs/poly/multiopen/fn.create_proof.html index 935ebed4..7c18b987 100644 --- a/rustdoc/latest/halo2_proofs/poly/multiopen/fn.create_proof.html +++ b/rustdoc/latest/halo2_proofs/poly/multiopen/fn.create_proof.html @@ -20,5 +20,5 @@

    Function halo2_proofs::poly::multiopen::create_proof

    source · [−]
    pub fn create_proof<'a, I, C: CurveAffine, E: EncodedChallenge<C>, R: RngCore, T: TranscriptWrite<C, E>>(
        params: &Params<C>, 
        rng: R, 
        transcript: &mut T, 
        queries: I
    ) -> Result<()> where
        I: IntoIterator<Item = ProverQuery<'a, C>> + Clone, 
    Expand description

    Create a multi-opening proof

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/multiopen/fn.verify_proof.html b/rustdoc/latest/halo2_proofs/poly/multiopen/fn.verify_proof.html index d306bffa..9e5b15dc 100644 --- a/rustdoc/latest/halo2_proofs/poly/multiopen/fn.verify_proof.html +++ b/rustdoc/latest/halo2_proofs/poly/multiopen/fn.verify_proof.html @@ -20,5 +20,5 @@

    Function halo2_proofs::poly::multiopen::verify_proof

    source · [−]
    pub fn verify_proof<'r, 'params: 'r, I, C: CurveAffine, E: EncodedChallenge<C>, T: TranscriptRead<C, E>>(
        params: &'params Params<C>, 
        transcript: &mut T, 
        queries: I, 
        msm: MSM<'params, C>
    ) -> Result<Guard<'params, C, E>, Error> where
        I: IntoIterator<Item = VerifierQuery<'r, 'params, C>> + Clone, 
    Expand description

    Verify a multi-opening proof

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/multiopen/index.html b/rustdoc/latest/halo2_proofs/poly/multiopen/index.html index 41c34b23..dd52c69e 100644 --- a/rustdoc/latest/halo2_proofs/poly/multiopen/index.html +++ b/rustdoc/latest/halo2_proofs/poly/multiopen/index.html @@ -27,5 +27,5 @@ scheme described in the Halo pap

    Functions

    create_proof

    Create a multi-opening proof

    verify_proof

    Verify a multi-opening proof

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/multiopen/struct.ProverQuery.html b/rustdoc/latest/halo2_proofs/poly/multiopen/struct.ProverQuery.html index 702cad1a..71344877 100644 --- a/rustdoc/latest/halo2_proofs/poly/multiopen/struct.ProverQuery.html +++ b/rustdoc/latest/halo2_proofs/poly/multiopen/struct.ProverQuery.html @@ -97,9 +97,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/multiopen/struct.VerifierQuery.html b/rustdoc/latest/halo2_proofs/poly/multiopen/struct.VerifierQuery.html index 5112784c..491ad5f9 100644 --- a/rustdoc/latest/halo2_proofs/poly/multiopen/struct.VerifierQuery.html +++ b/rustdoc/latest/halo2_proofs/poly/multiopen/struct.VerifierQuery.html @@ -92,9 +92,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/struct.Coeff.html b/rustdoc/latest/halo2_proofs/poly/struct.Coeff.html index 5c0f836e..7497b842 100644 --- a/rustdoc/latest/halo2_proofs/poly/struct.Coeff.html +++ b/rustdoc/latest/halo2_proofs/poly/struct.Coeff.html @@ -90,9 +90,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/struct.EvaluationDomain.html b/rustdoc/latest/halo2_proofs/poly/struct.EvaluationDomain.html index 2e731e87..54de82b7 100644 --- a/rustdoc/latest/halo2_proofs/poly/struct.EvaluationDomain.html +++ b/rustdoc/latest/halo2_proofs/poly/struct.EvaluationDomain.html @@ -148,9 +148,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/struct.ExtendedLagrangeCoeff.html b/rustdoc/latest/halo2_proofs/poly/struct.ExtendedLagrangeCoeff.html index fe6ab82c..cfd54f99 100644 --- a/rustdoc/latest/halo2_proofs/poly/struct.ExtendedLagrangeCoeff.html +++ b/rustdoc/latest/halo2_proofs/poly/struct.ExtendedLagrangeCoeff.html @@ -91,9 +91,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/struct.LagrangeCoeff.html b/rustdoc/latest/halo2_proofs/poly/struct.LagrangeCoeff.html index 3515dfb7..f8bc24e0 100644 --- a/rustdoc/latest/halo2_proofs/poly/struct.LagrangeCoeff.html +++ b/rustdoc/latest/halo2_proofs/poly/struct.LagrangeCoeff.html @@ -90,9 +90,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/struct.PinnedEvaluationDomain.html b/rustdoc/latest/halo2_proofs/poly/struct.PinnedEvaluationDomain.html index 8a24a980..fd79ccd0 100644 --- a/rustdoc/latest/halo2_proofs/poly/struct.PinnedEvaluationDomain.html +++ b/rustdoc/latest/halo2_proofs/poly/struct.PinnedEvaluationDomain.html @@ -85,9 +85,9 @@ builds. Read more

    Calls .tap_deref_mut() only in debug builds, and is erased in release builds. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/struct.Polynomial.html b/rustdoc/latest/halo2_proofs/poly/struct.Polynomial.html index e0262130..4811572c 100644 --- a/rustdoc/latest/halo2_proofs/poly/struct.Polynomial.html +++ b/rustdoc/latest/halo2_proofs/poly/struct.Polynomial.html @@ -28,23 +28,62 @@ evaluation form depending on the basis B.

    Gets the size of this polynomial in terms of the number of coefficients used to describe it.

    Rotates the values in a Lagrange basis polynomial by Rotation

    -

    Methods from Deref<Target = [F]>

    Returns the number of elements in the slice.

    +

    Methods from Deref<Target = [F]>

    🔬 This is a nightly-only experimental API. (slice_flatten)

    Takes a &[[T; N]], and flattens it to a &[T].

    +
    Panics
    +

    This panics if the length of the resulting slice would overflow a usize.

    +

    This is only possible when flattening a slice of arrays of zero-sized +types, and thus tends to be irrelevant in practice. If +size_of::<T>() > 0, this will never panic.

    Examples
    +
    #![feature(slice_flatten)]
+
+assert_eq!([[1, 2, 3], [4, 5, 6]].flatten(), &[1, 2, 3, 4, 5, 6]);
+
+assert_eq!(
+    [[1, 2, 3], [4, 5, 6]].flatten(),
+    [[1, 2], [3, 4], [5, 6]].flatten(),
+);
+
+let slice_of_empty_arrays: &[[i32; 0]] = &[[], [], [], [], []];
+assert!(slice_of_empty_arrays.flatten().is_empty());
+
+let empty_slice_of_arrays: &[[u32; 10]] = &[];
+assert!(empty_slice_of_arrays.flatten().is_empty());
    +
    🔬 This is a nightly-only experimental API. (slice_flatten)

    Takes a &mut [[T; N]], and flattens it to a &mut [T].

    +
    Panics
    +

    This panics if the length of the resulting slice would overflow a usize.

    +

    This is only possible when flattening a slice of arrays of zero-sized +types, and thus tends to be irrelevant in practice. If +size_of::<T>() > 0, this will never panic.

    +
    Examples
    +
    #![feature(slice_flatten)]
+
+fn add_5_to_all(slice: &mut [i32]) {
+    for i in slice {
+        *i += 5;
+    }
+}
+
+let mut array = [[1, 2, 3], [4, 5, 6], [7, 8, 9]];
+add_5_to_all(array.flatten_mut());
+assert_eq!(array, [[6, 7, 8], [9, 10, 11], [12, 13, 14]]);
    +

    Returns the number of elements in the slice.

    +
    Examples
    let a = [1, 2, 3];
 assert_eq!(a.len(), 3);

    Returns true if the slice has a length of 0.

    -
    Examples
    +
    Examples
    let a = [1, 2, 3];
 assert!(!a.is_empty());

    Returns the first element of the slice, or None if it is empty.

    -
    Examples
    +
    Examples
    let v = [10, 40, 30];
 assert_eq!(Some(&10), v.first());
 
 let w: &[i32] = &[];
 assert_eq!(None, w.first());

    Returns a mutable pointer to the first element of the slice, or None if it is empty.

    -
    Examples
    +
    Examples
    let x = &mut [0, 1, 2];
 
 if let Some(first) = x.first_mut() {
@@ -52,7 +91,7 @@ coefficients used to describe it.
    
 }
 assert_eq!(x, &[5, 1, 2]);

    Returns the first and all the rest of the elements of the slice, or None if it is empty.

    -
    Examples
    +
    Examples
    let x = &[0, 1, 2];
 
 if let Some((first, elements)) = x.split_first() {
@@ -60,7 +99,7 @@ coefficients used to describe it.
    
     assert_eq!(elements, &[1, 2]);
 }

    Returns the first and all the rest of the elements of the slice, or None if it is empty.

    -
    Examples
    +
    Examples
    let x = &mut [0, 1, 2];
 
 if let Some((first, elements)) = x.split_first_mut() {
@@ -70,7 +109,7 @@ coefficients used to describe it.
    
 }
 assert_eq!(x, &[3, 4, 5]);

    Returns the last and all the rest of the elements of the slice, or None if it is empty.

    -
    Examples
    +
    Examples
    let x = &[0, 1, 2];
 
 if let Some((last, elements)) = x.split_last() {
@@ -78,7 +117,7 @@ coefficients used to describe it.
    
     assert_eq!(elements, &[0, 1]);
 }

    Returns the last and all the rest of the elements of the slice, or None if it is empty.

    -
    Examples
    +
    Examples
    let x = &mut [0, 1, 2];
 
 if let Some((last, elements)) = x.split_last_mut() {
@@ -88,14 +127,14 @@ coefficients used to describe it.
    
 }
 assert_eq!(x, &[4, 5, 3]);

    Returns the last element of the slice, or None if it is empty.

    -
    Examples
    +
    Examples
    let v = [10, 40, 30];
 assert_eq!(Some(&30), v.last());
 
 let w: &[i32] = &[];
 assert_eq!(None, w.last());

    Returns a mutable pointer to the last item in the slice.

    -
    Examples
    +
    Examples
    let x = &mut [0, 1, 2];
 
 if let Some(last) = x.last_mut() {
@@ -110,7 +149,7 @@ position or None if out of bounds.
 
  • If given a range, returns the subslice corresponding to that range,
 or None if out of bounds.
    • 
 
-
    Examples
    
+
    Examples
    
 
    let v = [10, 40, 30];
 assert_eq!(Some(&40), v.get(1));
 assert_eq!(Some(&[10, 40][..]), v.get(0..2));
@@ -118,7 +157,7 @@ or None if out of bounds.
 assert_eq!(None, v.get(0..4));

    Returns a mutable reference to an element or subslice depending on the type of index (see get) or None if the index is out of bounds.

    -
    Examples
    +
    Examples
    let x = &mut [0, 1, 2];
 
 if let Some(elem) = x.get_mut(1) {
@@ -131,7 +170,7 @@ checking.
    
 
    Safety
    
 
    Calling this method with an out-of-bounds index is undefined behavior
 even if the resulting reference is not used.
    
-
    Examples
    
+
    Examples
    
 
    let x = &[1, 2, 4];
 
 unsafe {
@@ -143,7 +182,7 @@ bounds checking.
    
 
    Safety
    
 
    Calling this method with an out-of-bounds index is undefined behavior
 even if the resulting reference is not used.
    
-
    Examples
    
+
    Examples
    
 
    let x = &mut [1, 2, 4];
 
 unsafe {
@@ -159,7 +198,7 @@ is never written to (except inside an UnsafeCell) using this pointe
 derived from it. If you need to mutate the contents of the slice, use as_mut_ptr.
    
 
    Modifying the container referenced by this slice may cause its buffer
 to be reallocated, which would also make any pointers to it invalid.
    
-
    Examples
    
+
    Examples
    
 
    let x = &[1, 2, 4];
 let x_ptr = x.as_ptr();
 
@@ -173,7 +212,7 @@ to be reallocated, which would also make any pointers to it invalid.
    
 function returns, or else it will end up pointing to garbage.
    
 
    Modifying the container referenced by this slice may cause its buffer
 to be reallocated, which would also make any pointers to it invalid.
    
-
    Examples
    
+
    Examples
    
 
    let x = &mut [1, 2, 4];
 let x_ptr = x.as_mut_ptr();
 
@@ -220,9 +259,9 @@ common in C++.
    
 
  • a - The index of the first element
    • 
 
  • b - The index of the second element
    • 
 
-
    Panics
    
+
    Panics
    
 
    Panics if a or b are out of bounds.
    
-
    Examples
    
+
    Examples
    
 
    let mut v = ["a", "b", "c", "d", "e"];
 v.swap(2, 4);
 assert!(v == ["a", "b", "e", "d", "c"]);
    
@@ -236,7 +275,7 @@ common in C++.
    
 
    Safety
    
 
    Calling this method with an out-of-bounds index is undefined behavior.
 The caller has to ensure that a < self.len() and b < self.len().
    
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_swap_unchecked)]
 
 let mut v = ["a", "b", "c", "d"];
@@ -244,12 +283,12 @@ The caller has to ensure that a < self.len() and b < se
 unsafe { v.swap_unchecked(1, 3) };
 assert!(v == ["a", "d", "c", "b"]);

    Reverses the order of elements in the slice, in place.

    -
    Examples
    +
    Examples
    let mut v = [1, 2, 3];
 v.reverse();
 assert!(v == [3, 2, 1]);

    Returns an iterator over the slice.

    -
    Examples
    +
    Examples
    let x = &[1, 2, 4];
 let mut iterator = x.iter();
 
@@ -258,7 +297,7 @@ The caller has to ensure that a < self.len() and b < se
 assert_eq!(iterator.next(), Some(&4));
 assert_eq!(iterator.next(), None);

    Returns an iterator that allows modifying each value.

    -
    Examples
    +
    Examples
    let x = &mut [1, 2, 4];
 for elem in x.iter_mut() {
     *elem += 2;
@@ -267,9 +306,9 @@ The caller has to ensure that a < self.len() and b < se

    Returns an iterator over all contiguous windows of length size. The windows overlap. If the slice is shorter than size, the iterator returns no values.

    -
    Panics
    +
    Panics

    Panics if size is 0.

    -
    Examples
    +
    Examples
    let slice = ['r', 'u', 's', 't'];
 let mut iter = slice.windows(2);
 assert_eq!(iter.next().unwrap(), &['r', 'u']);
@@ -288,9 +327,9 @@ slice, then the last chunk will not have length chunk_size.
    
 
    See chunks_exact for a variant of this iterator that returns chunks of always exactly
 chunk_size elements, and rchunks for the same iterator but starting at the end of the
 slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let slice = ['l', 'o', 'r', 'e', 'm'];
 let mut iter = slice.chunks(2);
 assert_eq!(iter.next().unwrap(), &['l', 'o']);
@@ -304,9 +343,9 @@ length of the slice, then the last chunk will not have length chunk_sizeSee chunks_exact_mut for a variant of this iterator that returns chunks of always
 exactly chunk_size elements, and rchunks_mut for the same iterator but starting at
 the end of the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
 
@@ -326,9 +365,9 @@ from the remainder function of the iterator.
    
 resulting code better than in the case of chunks.
    
 
    See chunks for a variant of this iterator that also returns the remainder as a smaller
 chunk, and rchunks_exact for the same iterator but starting at the end of the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let slice = ['l', 'o', 'r', 'e', 'm'];
 let mut iter = slice.chunks_exact(2);
 assert_eq!(iter.next().unwrap(), &['l', 'o']);
@@ -345,9 +384,9 @@ resulting code better than in the case of chunks_mut for a variant of this iterator that also returns the remainder as a
 smaller chunk, and rchunks_exact_mut for the same iterator but starting at the end of
 the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
 
@@ -366,7 +405,7 @@ assuming that there’s no remainder.
    
 
  • The slice splits exactly into N-element chunks (aka self.len() % N == 0).
    • 
 
  • N != 0.
    • 
 
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_as_chunks)]
 let slice: &[char] = &['l', 'o', 'r', 'e', 'm', '!'];
 let chunks: &[[char; 1]] =
@@ -384,10 +423,10 @@ assuming that there’s no remainder.
    🔬 This is a nightly-only experimental API. (slice_as_chunks)

    Splits the slice into a slice of N-element arrays, starting at the beginning of the slice, and a remainder slice with length strictly less than N.

    -
    Panics
    +
    Panics

    Panics if N is 0. This check will most probably get changed to a compile time error before this method gets stabilized.

    -
    Examples
    +
    Examples
    #![feature(slice_as_chunks)]
 let slice = ['l', 'o', 'r', 'e', 'm'];
 let (chunks, remainder) = slice.as_chunks();
@@ -396,10 +435,10 @@ error before this method gets stabilized.
    🔬 This is a nightly-only experimental API. (slice_as_chunks)

    Splits the slice into a slice of N-element arrays, starting at the end of the slice, and a remainder slice with length strictly less than N.

    -
    Panics
    +
    Panics

    Panics if N is 0. This check will most probably get changed to a compile time error before this method gets stabilized.

    -
    Examples
    +
    Examples
    #![feature(slice_as_chunks)]
 let slice = ['l', 'o', 'r', 'e', 'm'];
 let (remainder, chunks) = slice.as_rchunks();
@@ -411,10 +450,10 @@ beginning of the slice.
    
 length of the slice, then the last up to N-1 elements will be omitted and can be
 retrieved from the remainder function of the iterator.
    
 
    This method is the const generic equivalent of chunks_exact.
    
-
    Panics
    
+
    Panics
    
 
    Panics if N is 0. This check will most probably get changed to a compile time
 error before this method gets stabilized.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(array_chunks)]
 let slice = ['l', 'o', 'r', 'e', 'm'];
 let mut iter = slice.array_chunks();
@@ -430,7 +469,7 @@ assuming that there’s no remainder.
    
 
  • The slice splits exactly into N-element chunks (aka self.len() % N == 0).
    • 
 
  • N != 0.
    • 
 
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_as_chunks)]
 let slice: &mut [char] = &mut ['l', 'o', 'r', 'e', 'm', '!'];
 let chunks: &mut [[char; 1]] =
@@ -450,10 +489,10 @@ assuming that there’s no remainder.
    🔬 This is a nightly-only experimental API. (slice_as_chunks)

    Splits the slice into a slice of N-element arrays, starting at the beginning of the slice, and a remainder slice with length strictly less than N.

    -
    Panics
    +
    Panics

    Panics if N is 0. This check will most probably get changed to a compile time error before this method gets stabilized.

    -
    Examples
    +
    Examples
    #![feature(slice_as_chunks)]
 let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
@@ -468,10 +507,10 @@ error before this method gets stabilized.
    🔬 This is a nightly-only experimental API. (slice_as_chunks)

    Splits the slice into a slice of N-element arrays, starting at the end of the slice, and a remainder slice with length strictly less than N.

    -
    Panics
    +
    Panics

    Panics if N is 0. This check will most probably get changed to a compile time error before this method gets stabilized.

    -
    Examples
    +
    Examples
    #![feature(slice_as_chunks)]
 let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
@@ -489,10 +528,10 @@ beginning of the slice.
    
 the length of the slice, then the last up to N-1 elements will be omitted and
 can be retrieved from the into_remainder function of the iterator.
    
 
    This method is the const generic equivalent of chunks_exact_mut.
    
-
    Panics
    
+
    Panics
    
 
    Panics if N is 0. This check will most probably get changed to a compile time
 error before this method gets stabilized.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(array_chunks)]
 let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
@@ -506,10 +545,10 @@ error before this method gets stabilized.
    
 starting at the beginning of the slice.
    
 
    This is the const generic equivalent of windows.
    
 
    If N is greater than the size of the slice, it will return no windows.
    
-
    Panics
    
+
    Panics
    
 
    Panics if N is 0. This check will most probably get changed to a compile time
 error before this method gets stabilized.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(array_windows)]
 let slice = [0, 1, 2, 3];
 let mut iter = slice.array_windows();
@@ -524,9 +563,9 @@ slice, then the last chunk will not have length chunk_size.
    
 
    See rchunks_exact for a variant of this iterator that returns chunks of always exactly
 chunk_size elements, and chunks for the same iterator but starting at the beginning
 of the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let slice = ['l', 'o', 'r', 'e', 'm'];
 let mut iter = slice.rchunks(2);
 assert_eq!(iter.next().unwrap(), &['e', 'm']);
@@ -540,9 +579,9 @@ length of the slice, then the last chunk will not have length chunk_sizeSee rchunks_exact_mut for a variant of this iterator that returns chunks of always
 exactly chunk_size elements, and chunks_mut for the same iterator but starting at the
 beginning of the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
 
@@ -563,9 +602,9 @@ resulting code better than in the case of rchunks for a variant of this iterator that also returns the remainder as a smaller
 chunk, and chunks_exact for the same iterator but starting at the beginning of the
 slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let slice = ['l', 'o', 'r', 'e', 'm'];
 let mut iter = slice.rchunks_exact(2);
 assert_eq!(iter.next().unwrap(), &['e', 'm']);
@@ -582,9 +621,9 @@ resulting code better than in the case of rchunks_mut for a variant of this iterator that also returns the remainder as a
 smaller chunk, and chunks_exact_mut for the same iterator but starting at the beginning
 of the slice.
    
-
    Panics
    
+
    Panics
    
 
    Panics if chunk_size is 0.
    
-
    Examples
    
+
    Examples
    
 
    let v = &mut [0, 0, 0, 0, 0];
 let mut count = 1;
 
@@ -600,7 +639,7 @@ of elements using the predicate to separate them.
    
 
    The predicate is called on two elements following themselves,
 it means the predicate is called on slice[0] and slice[1]
 then on slice[1] and slice[2] and so on.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_group_by)]
 
 let slice = &[1, 1, 1, 3, 3, 2, 2, 2];
@@ -628,7 +667,7 @@ runs of elements using the predicate to separate them.
    
 
    The predicate is called on two elements following themselves,
 it means the predicate is called on slice[0] and slice[1]
 then on slice[1] and slice[2] and so on.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_group_by)]
 
 let slice = &mut [1, 1, 1, 3, 3, 2, 2, 2];
@@ -655,9 +694,9 @@ then on slice[1] and slice[2] and so on.
    
 
    The first will contain all indices from [0, mid) (excluding
 the index mid itself) and the second will contain all
 indices from [mid, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if mid > len.
    
-
    Examples
    
+
    Examples
    
 
    let v = [1, 2, 3, 4, 5, 6];
 
 {
@@ -681,9 +720,9 @@ indices from [mid, len) (excluding the index len itsel
 
    The first will contain all indices from [0, mid) (excluding
 the index mid itself) and the second will contain all
 indices from [mid, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if mid > len.
    
-
    Examples
    
+
    Examples
    
 
    let mut v = [1, 0, 3, 0, 5, 6];
 let (left, right) = v.split_at_mut(2);
 assert_eq!(left, [1, 0]);
@@ -700,7 +739,7 @@ indices from [mid, len) (excluding the index len itsel
 
    Calling this method with an out-of-bounds index is undefined behavior
 even if the resulting reference is not used. The caller has to ensure that
 0 <= mid <= self.len().
    
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_split_at_unchecked)]
 
 let v = [1, 2, 3, 4, 5, 6];
@@ -731,7 +770,7 @@ indices from [mid, len) (excluding the index len itsel
 
    Calling this method with an out-of-bounds index is undefined behavior
 even if the resulting reference is not used. The caller has to ensure that
 0 <= mid <= self.len().
    
-
    Examples
    
+
    Examples
    
 
    #![feature(slice_split_at_unchecked)]
 
 let mut v = [1, 0, 3, 0, 5, 6];
@@ -748,9 +787,9 @@ even if the resulting reference is not used. The caller has to ensure that
 
    The array will contain all indices from [0, N) (excluding
 the index N itself) and the slice will contain all
 indices from [N, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if N > len.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(split_array)]
 
 let v = &[1, 2, 3, 4, 5, 6][..];
@@ -776,9 +815,9 @@ indices from [N, len) (excluding the index len itself)
 
    The array will contain all indices from [0, N) (excluding
 the index N itself) and the slice will contain all
 indices from [N, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if N > len.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(split_array)]
 
 let mut v = &mut [1, 0, 3, 0, 5, 6][..];
@@ -793,9 +832,9 @@ the end.
    
 
    The slice will contain all indices from [0, len - N) (excluding
 the index len - N itself) and the array will contain all
 indices from [len - N, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if N > len.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(split_array)]
 
 let v = &[1, 2, 3, 4, 5, 6][..];
@@ -822,9 +861,9 @@ index from the end.
    
 
    The slice will contain all indices from [0, len - N) (excluding
 the index N itself) and the array will contain all
 indices from [len - N, len) (excluding the index len itself).
    
-
    Panics
    
+
    Panics
    
 
    Panics if N > len.
    
-
    Examples
    
+
    Examples
    
 
    #![feature(split_array)]
 
 let mut v = &mut [1, 0, 3, 0, 5, 6][..];
@@ -836,7 +875,7 @@ indices from [len - N, len) (excluding the index len i
 assert_eq!(v, [1, 2, 3, 4, 5, 6]);

    Returns an iterator over subslices separated by elements that match pred. The matched element is not contained in the subslices.

    -
    Examples
    +
    Examples
    let slice = [10, 40, 33, 20];
 let mut iter = slice.split(|num| num % 3 == 0);
 
@@ -866,7 +905,7 @@ present between them:
    
 assert!(iter.next().is_none());

    Returns an iterator over mutable subslices separated by elements that match pred. The matched element is not contained in the subslices.

    -
    Examples
    +
    Examples
    let mut v = [10, 40, 30, 20, 60, 50];
 
 for group in v.split_mut(|num| *num % 3 == 0) {
@@ -876,7 +915,7 @@ match pred. The matched element is not contained in the subslices.<

    Returns an iterator over subslices separated by elements that match pred. The matched element is contained in the end of the previous subslice as a terminator.

    -
    Examples
    +
    Examples
    let slice = [10, 40, 33, 20];
 let mut iter = slice.split_inclusive(|num| num % 3 == 0);
 
@@ -896,7 +935,7 @@ That slice will be the last item returned by the iterator.

    Returns an iterator over mutable subslices separated by elements that match pred. The matched element is contained in the previous subslice as a terminator.

    -
    Examples
    +
    Examples
    let mut v = [10, 40, 30, 20, 60, 50];
 
 for group in v.split_inclusive_mut(|num| *num % 3 == 0) {
@@ -907,7 +946,7 @@ subslice as a terminator.

    Returns an iterator over subslices separated by elements that match pred, starting at the end of the slice and working backwards. The matched element is not contained in the subslices.

    -
    Examples
    +
    Examples
    let slice = [11, 22, 33, 0, 44, 55];
 let mut iter = slice.rsplit(|num| *num == 0);
 
@@ -927,7 +966,7 @@ slice will be the first (or last) item returned by the iterator.

    Returns an iterator over mutable subslices separated by elements that match pred, starting at the end of the slice and working backwards. The matched element is not contained in the subslices.

    -
    Examples
    +
    Examples
    let mut v = [100, 400, 300, 200, 600, 500];
 
 let mut count = 0;
@@ -941,7 +980,7 @@ backwards. The matched element is not contained in the subslices.
    
 not contained in the subslices.
    
 
    The last element returned, if any, will contain the remainder of the
 slice.
    
-
    Examples
    
+
    Examples
    
 
    Print the slice split once by numbers divisible by 3 (i.e., [10, 40],
 [20, 60, 50]):
    
 
@@ -955,7 +994,7 @@ slice.
    
 not contained in the subslices.
    
 
    The last element returned, if any, will contain the remainder of the
 slice.
    
-
    Examples
    
+
    Examples
    
 
    let mut v = [10, 40, 30, 20, 60, 50];
 
 for group in v.splitn_mut(2, |num| *num % 3 == 0) {
@@ -968,7 +1007,7 @@ the slice and works backwards. The matched element is not contained in
 the subslices.
    
 
    The last element returned, if any, will contain the remainder of the
 slice.
    
-
    Examples
    
+
    Examples
    
 
    Print the slice split once, starting from the end, by numbers divisible
 by 3 (i.e., [50], [10, 40, 30, 20]):
    
 
@@ -983,15 +1022,17 @@ the slice and works backwards. The matched element is not contained in
 the subslices.
    
 
    The last element returned, if any, will contain the remainder of the
 slice.
    
-
    Examples
    
+
    Examples
    
 
    let mut s = [10, 40, 30, 20, 60, 50];
 
 for group in s.rsplitn_mut(2, |num| *num % 3 == 0) {
     group[0] = 1;
 }
 assert_eq!(s, [1, 40, 30, 20, 60, 1]);
    
-

    Returns true if the slice contains an element with the given value.

    -
    Examples
    +

    Returns true if the slice contains an element with the given value.

    +

    This operation is O(n).

    +

    Note that if you have a sorted slice, binary_search may be faster.

    +
    Examples
    let v = [10, 40, 30];
 assert!(v.contains(&30));
 assert!(!v.contains(&50));
    @@ -1002,8 +1043,8 @@ use iter().any:

    let v = [String::from("hello"), String::from("world")]; // slice of `String`
 assert!(v.iter().any(|e| e == "hello")); // search with `&str`
 assert!(!v.iter().any(|e| e == "hi"));
    -

    Returns true if needle is a prefix of the slice.

    -
    Examples
    +

    Returns true if needle is a prefix of the slice.

    +
    Examples
    let v = [10, 40, 30];
 assert!(v.starts_with(&[10]));
 assert!(v.starts_with(&[10, 40]));
@@ -1015,8 +1056,8 @@ use iter().any:
    
 assert!(v.starts_with(&[]));
 let v: &[u8] = &[];
 assert!(v.starts_with(&[]));
    -

    Returns true if needle is a suffix of the slice.

    -
    Examples
    +

    Returns true if needle is a suffix of the slice.

    +
    Examples
    let v = [10, 40, 30];
 assert!(v.ends_with(&[30]));
 assert!(v.ends_with(&[40, 30]));
@@ -1028,11 +1069,11 @@ use iter().any:
    
 assert!(v.ends_with(&[]));
 let v: &[u8] = &[];
 assert!(v.ends_with(&[]));
    -

    Returns a subslice with the prefix removed.

    +

    Returns a subslice with the prefix removed.

    If the slice starts with prefix, returns the subslice after the prefix, wrapped in Some. If prefix is empty, simply returns the original slice.

    If the slice does not start with prefix, returns None.

    -
    Examples
    +
    Examples
    let v = &[10, 40, 30];
 assert_eq!(v.strip_prefix(&[10]), Some(&[40, 30][..]));
 assert_eq!(v.strip_prefix(&[10, 40]), Some(&[30][..]));
@@ -1042,17 +1083,18 @@ If prefix is empty, simply returns the original slice.
    
 let prefix : &str = "he";
 assert_eq!(b"hello".strip_prefix(prefix.as_bytes()),
            Some(b"llo".as_ref()));
    -

    Returns a subslice with the suffix removed.

    +

    Returns a subslice with the suffix removed.

    If the slice ends with suffix, returns the subslice before the suffix, wrapped in Some. If suffix is empty, simply returns the original slice.

    If the slice does not end with suffix, returns None.

    -
    Examples
    +
    Examples
    let v = &[10, 40, 30];
 assert_eq!(v.strip_suffix(&[30]), Some(&[10, 40][..]));
 assert_eq!(v.strip_suffix(&[40, 30]), Some(&[10][..]));
 assert_eq!(v.strip_suffix(&[50]), None);
 assert_eq!(v.strip_suffix(&[50, 30]), None);
    -

    Binary searches this sorted slice for a given element.

    +

    Binary searches this slice for a given element. +This behaves similary to contains if this slice is sorted.

    If the value is found then Result::Ok is returned, containing the index of the matching element. If there are multiple matches, then any one of the matches could be returned. The index is chosen @@ -1061,7 +1103,7 @@ If the value is not found then binary_search_by, binary_search_by_key, and partition_point.

    -
    Examples
    +
    Examples

    Looks up a series of four elements. The first is found, with a uniquely determined position; the second and third are not found; the fourth could match any position in [1, 4].

    @@ -1082,7 +1124,8 @@ sort order, consider using // The above is equivalent to `let idx = s.binary_search(&num).unwrap_or_else(|x| x);` s.insert(idx, num); assert_eq!(s, [0, 1, 1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 42, 55]);
    -

    Binary searches this sorted slice with a comparator function.

    +

    Binary searches this slice with a comparator function. +This behaves similarly to contains if this slice is sorted.

    The comparator function should implement an order consistent with the sort order of the underlying slice, returning an order code that indicates whether its argument is Less, @@ -1095,7 +1138,7 @@ If the value is not found then binary_search, binary_search_by_key, and partition_point.

    -
    Examples
    +
    Examples

    Looks up a series of four elements. The first is found, with a uniquely determined position; the second and third are not found; the fourth could match any position in [1, 4].

    @@ -1111,7 +1154,8 @@ found; the fourth could match any position in [1, 4].

    let seek = 1; let r = s.binary_search_by(|probe| probe.cmp(&seek)); assert!(match r { Ok(1..=4) => true, _ => false, });
    -

    Binary searches this sorted slice with a key extraction function.

    +

    Binary searches this slice with a key extraction function. +This behaves similarly to contains if this slice is sorted.

    Assumes that the slice is sorted by the key, for instance with sort_by_key using the same key extraction function.

    If the value is found then Result::Ok is returned, containing the @@ -1122,7 +1166,7 @@ If the value is not found then binary_search, binary_search_by, and partition_point.

    -
    Examples
    +
    Examples

    Looks up a series of four elements in a slice of pairs sorted by their second elements. The first is found, with a uniquely determined position; the second and third are not found; the @@ -1137,7 +1181,7 @@ fourth could match any position in [1, 4].

    assert_eq!(s.binary_search_by_key(&100, |&(a, b)| b), Err(13)); let r = s.binary_search_by_key(&1, |&(a, b)| b); assert!(match r { Ok(1..=4) => true, _ => false, });
    -

    Sorts the slice, but might not preserve the order of equal elements.

    +

    Sorts the slice, but might not preserve the order of equal elements.

    This sort is unstable (i.e., may reorder equal elements), in-place (i.e., does not allocate), and O(n * log(n)) worst-case.

    Current implementation
    @@ -1148,12 +1192,12 @@ randomization to avoid degenerate cases, but with a fixed seed to always provide deterministic behavior.

    It is typically faster than stable sorting, except in a few special cases, e.g., when the slice consists of several concatenated sorted sequences.

    -
    Examples
    +
    Examples
    let mut v = [-5, 4, 1, -3, 2];
 
 v.sort_unstable();
 assert!(v == [-5, -3, 1, 2, 4]);
    -

    Sorts the slice with a comparator function, but might not preserve the order of equal +

    Sorts the slice with a comparator function, but might not preserve the order of equal elements.

    This sort is unstable (i.e., may reorder equal elements), in-place (i.e., does not allocate), and O(n * log(n)) worst-case.

    @@ -1178,7 +1222,7 @@ randomization to avoid degenerate cases, but with a fixed seed to always provide deterministic behavior.

    It is typically faster than stable sorting, except in a few special cases, e.g., when the slice consists of several concatenated sorted sequences.

    -
    Examples
    +
    Examples
    let mut v = [5, 4, 1, 3, 2];
 v.sort_unstable_by(|a, b| a.cmp(b));
 assert!(v == [1, 2, 3, 4, 5]);
@@ -1186,7 +1230,7 @@ slice consists of several concatenated sorted sequences.
    
 // reverse sorting
 v.sort_unstable_by(|a, b| b.cmp(a));
 assert!(v == [5, 4, 3, 2, 1]);
    -

    Sorts the slice with a key extraction function, but might not preserve the order of equal +

    Sorts the slice with a key extraction function, but might not preserve the order of equal elements.

    This sort is unstable (i.e., may reorder equal elements), in-place (i.e., does not allocate), and O(m * n * log(n)) worst-case, where the key function is @@ -1200,12 +1244,12 @@ deterministic behavior.

    Due to its key calling strategy, sort_unstable_by_key is likely to be slower than sort_by_cached_key in cases where the key function is expensive.

    -
    Examples
    +
    Examples
    let mut v = [-5i32, 4, 1, -3, 2];
 
 v.sort_unstable_by_key(|k| k.abs());
 assert!(v == [1, 2, -3, 4, -5]);
    -

    Reorder the slice such that the element at index is at its final sorted position.

    +

    Reorder the slice such that the element at index is at its final sorted position.

    This reordering has the additional property that any value at position i < index will be less than or equal to any value at a position j > index. Additionally, this reordering is unstable (i.e. any number of equal elements may end up at position index), in-place @@ -1216,9 +1260,9 @@ the one at the given index.

    Current implementation

    The current algorithm is based on the quickselect portion of the same quicksort algorithm used for sort_unstable.

    -
    Panics
    +
    Panics

    Panics when index >= len(), meaning it always panics on empty slices.

    -
    Examples
    +
    Examples
    let mut v = [-5i32, 4, 1, -3, 2];
 
 // Find the median
@@ -1230,7 +1274,7 @@ used for v == [-5, -3, 1, 2, 4] ||
         v == [-3, -5, 1, 4, 2] ||
         v == [-5, -3, 1, 4, 2]);
    -

    Reorder the slice with a comparator function such that the element at index is at its +

    Reorder the slice with a comparator function such that the element at index is at its final sorted position.

    This reordering has the additional property that any value at position i < index will be less than or equal to any value at a position j > index using the comparator function. @@ -1243,9 +1287,9 @@ function.

    Current implementation

    The current algorithm is based on the quickselect portion of the same quicksort algorithm used for sort_unstable.

    -
    Panics
    +
    Panics

    Panics when index >= len(), meaning it always panics on empty slices.

    -
    Examples
    +
    Examples
    let mut v = [-5i32, 4, 1, -3, 2];
 
 // Find the median as if the slice were sorted in descending order.
@@ -1257,7 +1301,7 @@ used for v == [2, 4, 1, -3, -5] ||
         v == [4, 2, 1, -5, -3] ||
         v == [4, 2, 1, -3, -5]);
    -

    Reorder the slice with a key extraction function such that the element at index is at its +

    Reorder the slice with a key extraction function such that the element at index is at its final sorted position.

    This reordering has the additional property that any value at position i < index will be less than or equal to any value at a position j > index using the key extraction function. @@ -1270,9 +1314,9 @@ function.

    Current implementation

    The current algorithm is based on the quickselect portion of the same quicksort algorithm used for sort_unstable.

    -
    Panics
    +
    Panics

    Panics when index >= len(), meaning it always panics on empty slices.

    -
    Examples
    +
    Examples
    let mut v = [-5i32, 4, 1, -3, 2];
 
 // Return the median as if the array were sorted according to absolute value.
@@ -1284,12 +1328,12 @@ used for v == [1, 2, -3, -5, 4] ||
         v == [2, 1, -3, 4, -5] ||
         v == [2, 1, -3, -5, 4]);
    -
    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all consecutive repeated elements to the end of the slice according to the +

    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all consecutive repeated elements to the end of the slice according to the PartialEq trait implementation.

    Returns two slices. The first contains no consecutive repeated elements. The second contains all the duplicates in no specified order.

    If the slice is sorted, the first returned slice contains no duplicates.

    -
    Examples
    +
    Examples
    #![feature(slice_partition_dedup)]
 
 let mut slice = [1, 2, 2, 3, 3, 2, 1, 1];
@@ -1298,7 +1342,7 @@ The second contains all the duplicates in no specified order.
    
 
 assert_eq!(dedup, [1, 2, 3, 2, 1]);
 assert_eq!(duplicates, [2, 3, 1]);
    -
    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all but the first of consecutive elements to the end of the slice satisfying +

    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all but the first of consecutive elements to the end of the slice satisfying a given equality relation.

    Returns two slices. The first contains no consecutive repeated elements. The second contains all the duplicates in no specified order.

    @@ -1307,7 +1351,7 @@ must determine if the elements compare equal. The elements are passed in opposit from their order in the slice, so if same_bucket(a, b) returns true, a is moved at the end of the slice.

    If the slice is sorted, the first returned slice contains no duplicates.

    -
    Examples
    +
    Examples
    #![feature(slice_partition_dedup)]
 
 let mut slice = ["foo", "Foo", "BAZ", "Bar", "bar", "baz", "BAZ"];
@@ -1316,12 +1360,12 @@ at the end of the slice.
    
 
 assert_eq!(dedup, ["foo", "BAZ", "Bar", "baz"]);
 assert_eq!(duplicates, ["bar", "Foo", "BAZ"]);
    -
    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all but the first of consecutive elements to the end of the slice that resolve +

    🔬 This is a nightly-only experimental API. (slice_partition_dedup)

    Moves all but the first of consecutive elements to the end of the slice that resolve to the same key.

    Returns two slices. The first contains no consecutive repeated elements. The second contains all the duplicates in no specified order.

    If the slice is sorted, the first returned slice contains no duplicates.

    -
    Examples
    +
    Examples
    #![feature(slice_partition_dedup)]
 
 let mut slice = [10, 20, 21, 30, 30, 20, 11, 13];
@@ -1330,17 +1374,17 @@ The second contains all the duplicates in no specified order.
    
 
 assert_eq!(dedup, [10, 20, 30, 20, 11]);
 assert_eq!(duplicates, [21, 30, 13]);
    -

    Rotates the slice in-place such that the first mid elements of the +

    Rotates the slice in-place such that the first mid elements of the slice move to the end while the last self.len() - mid elements move to the front. After calling rotate_left, the element previously at index mid will become the first element in the slice.

    -
    Panics
    +
    Panics

    This function will panic if mid is greater than the length of the slice. Note that mid == self.len() does not panic and is a no-op rotation.

    Complexity

    Takes linear (in self.len()) time.

    -
    Examples
    +
    Examples
    let mut a = ['a', 'b', 'c', 'd', 'e', 'f'];
 a.rotate_left(2);
 assert_eq!(a, ['c', 'd', 'e', 'f', 'a', 'b']);
    @@ -1349,17 +1393,17 @@ rotation.

    let mut a = ['a', 'b', 'c', 'd', 'e', 'f'];
 a[1..5].rotate_left(1);
 assert_eq!(a, ['a', 'c', 'd', 'e', 'b', 'f']);
    -

    Rotates the slice in-place such that the first self.len() - k +

    Rotates the slice in-place such that the first self.len() - k elements of the slice move to the end while the last k elements move to the front. After calling rotate_right, the element previously at index self.len() - k will become the first element in the slice.

    -
    Panics
    +
    Panics

    This function will panic if k is greater than the length of the slice. Note that k == self.len() does not panic and is a no-op rotation.

    Complexity

    Takes linear (in self.len()) time.

    -
    Examples
    +
    Examples
    let mut a = ['a', 'b', 'c', 'd', 'e', 'f'];
 a.rotate_right(2);
 assert_eq!(a, ['e', 'f', 'a', 'b', 'c', 'd']);
    @@ -1368,25 +1412,25 @@ rotation.

    let mut a = ['a', 'b', 'c', 'd', 'e', 'f'];
 a[1..5].rotate_right(1);
 assert_eq!(a, ['a', 'e', 'b', 'c', 'd', 'f']);
    -

    Fills self with elements by cloning value.

    -
    Examples
    +

    Fills self with elements by cloning value.

    +
    Examples
    let mut buf = vec![0; 10];
 buf.fill(1);
 assert_eq!(buf, vec![1; 10]);
    -

    Fills self with elements returned by calling a closure repeatedly.

    +

    Fills self with elements returned by calling a closure repeatedly.

    This method uses a closure to create new values. If you’d rather Clone a given value, use fill. If you want to use the Default trait to generate values, you can pass Default::default as the argument.

    -
    Examples
    +
    Examples
    let mut buf = vec![1; 10];
 buf.fill_with(Default::default);
 assert_eq!(buf, vec![0; 10]);
    -

    Copies the elements from src into self.

    +

    Copies the elements from src into self.

    The length of src must be the same as self.

    -
    Panics
    +
    Panics

    This function will panic if the two slices have different lengths.

    -
    Examples
    +
    Examples

    Cloning two elements from a slice into another:

    let src = [1, 2, 3, 4];
@@ -1418,12 +1462,12 @@ sub-slices from a slice:
    
 }
 
 assert_eq!(slice, [4, 5, 3, 4, 5]);
    -

    Copies all elements from src into self, using a memcpy.

    +

    Copies all elements from src into self, using a memcpy.

    The length of src must be the same as self.

    If T does not implement Copy, use clone_from_slice.

    -
    Panics
    +
    Panics

    This function will panic if the two slices have different lengths.

    -
    Examples
    +
    Examples

    Copying two elements from a slice into another:

    let src = [1, 2, 3, 4];
@@ -1455,16 +1499,16 @@ sub-slices from a slice:
    
 }
 
 assert_eq!(slice, [4, 5, 3, 4, 5]);
    -

    Copies elements from one part of the slice to another part of itself, +

    Copies elements from one part of the slice to another part of itself, using a memmove.

    src is the range within self to copy from. dest is the starting index of the range within self to copy to, which will have the same length as src. The two ranges may overlap. The ends of the two ranges must be less than or equal to self.len().

    -
    Panics
    +
    Panics

    This function will panic if either range exceeds the end of the slice, or if the end of src is before the start.

    -
    Examples
    +
    Examples

    Copying four bytes within a slice:

    let mut bytes = *b"Hello, World!";
@@ -1472,9 +1516,9 @@ or if the end of src is before the start.
    
 bytes.copy_within(1..5, 8);
 
 assert_eq!(&bytes, b"Hello, Wello!");
    -

    Swaps all elements in self with those in other.

    +

    Swaps all elements in self with those in other.

    The length of other must be the same as self.

    -
    Panics
    +
    Panics

    This function will panic if the two slices have different lengths.

    Example

    Swapping two elements across slices:

    @@ -1504,7 +1548,7 @@ mutable sub-slices from a slice:

    } assert_eq!(slice, [4, 5, 3, 1, 2]);
    -

    Transmute the slice to a slice of another type, ensuring alignment of the types is +

    Transmute the slice to a slice of another type, ensuring alignment of the types is maintained.

    This method splits the slice into three distinct slices: prefix, correctly aligned middle slice of a new type, and the suffix slice. The method may make the middle slice the greatest @@ -1516,7 +1560,7 @@ zero-sized and will return the original slice without splitting anything.

    Safety

    This method is essentially a transmute with respect to the elements in the returned middle slice, so all the usual caveats pertaining to transmute::<T, U> also apply here.

    -
    Examples
    +
    Examples

    Basic usage:

    unsafe {
@@ -1526,7 +1570,7 @@ middle slice, so all the usual caveats pertaining to transmute::<T, U&g
     // more_efficient_algorithm_for_aligned_shorts(shorts);
     // less_efficient_algorithm_for_bytes(suffix);
 }
    -

    Transmute the slice to a slice of another type, ensuring alignment of the types is +

    Transmute the slice to a slice of another type, ensuring alignment of the types is maintained.

    This method splits the slice into three distinct slices: prefix, correctly aligned middle slice of a new type, and the suffix slice. The method may make the middle slice the greatest @@ -1538,7 +1582,7 @@ zero-sized and will return the original slice without splitting anything.

    Safety

    This method is essentially a transmute with respect to the elements in the returned middle slice, so all the usual caveats pertaining to transmute::<T, U> also apply here.

    -
    Examples
    +
    Examples

    Basic usage:

    unsafe {
@@ -1548,7 +1592,7 @@ middle slice, so all the usual caveats pertaining to transmute::<T, U&g
     // more_efficient_algorithm_for_aligned_shorts(shorts);
     // less_efficient_algorithm_for_bytes(suffix);
 }
    -
    🔬 This is a nightly-only experimental API. (portable_simd)

    Split a slice into a prefix, a middle of aligned SIMD types, and a suffix.

    +
    🔬 This is a nightly-only experimental API. (portable_simd)

    Split a slice into a prefix, a middle of aligned SIMD types, and a suffix.

    This is a safe wrapper around slice::align_to, so has the same weak postconditions as that method. You’re only assured that self.len() == prefix.len() + middle.len() * LANES + suffix.len().

    @@ -1560,7 +1604,7 @@ postconditions as that method. You’re only assured that

    That said, this is a safe method, so if you’re only writing safe code, then this can at most cause incorrect logic, not unsoundness.

    -
    Panics
    +
    Panics

    This will panic if the size of the SIMD type is different from LANES times that of the scalar.

    At the time of writing, the trait restrictions on Simd<T, LANES> keeps @@ -1568,7 +1612,7 @@ that from ever happening, as only power-of-two numbers of lanes are supported. It’s possible that, in the future, those restrictions might be lifted in a way that would make it possible to see panics from this method for something like LANES == 3.

    -
    Examples
    +
    Examples
    #![feature(portable_simd)]
 
 let short = &[1, 2, 3];
@@ -1595,7 +1639,7 @@ method for something like LANES == 3.
    
 
 let numbers: Vec<f32> = (1..101).map(|x| x as _).collect();
 assert_eq!(basic_simd_sum(&numbers[1..99]), 4949.0);
    -
    🔬 This is a nightly-only experimental API. (portable_simd)

    Split a slice into a prefix, a middle of aligned SIMD types, and a suffix.

    +
    🔬 This is a nightly-only experimental API. (portable_simd)

    Split a slice into a prefix, a middle of aligned SIMD types, and a suffix.

    This is a safe wrapper around slice::align_to_mut, so has the same weak postconditions as that method. You’re only assured that self.len() == prefix.len() + middle.len() * LANES + suffix.len().

    @@ -1608,7 +1652,7 @@ postconditions as that method. You’re only assured that

    That said, this is a safe method, so if you’re only writing safe code, then this can at most cause incorrect logic, not unsoundness.

    This is the mutable version of slice::as_simd; see that for examples.

    -
    Panics
    +
    Panics

    This will panic if the size of the SIMD type is different from LANES times that of the scalar.

    At the time of writing, the trait restrictions on Simd<T, LANES> keeps @@ -1616,13 +1660,13 @@ that from ever happening, as only power-of-two numbers of lanes are supported. It’s possible that, in the future, those restrictions might be lifted in a way that would make it possible to see panics from this method for something like LANES == 3.

    -
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted.

    +
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted.

    That is, for each element a and its following element b, a <= b must hold. If the slice yields exactly zero or one element, true is returned.

    Note that if Self::Item is only PartialOrd, but not Ord, the above definition implies that this function returns false if any two consecutive items are not comparable.

    -
    Examples
    +
    Examples
    #![feature(is_sorted)]
 let empty: [i32; 0] = [];
 
@@ -1631,20 +1675,20 @@ comparable.
    
 assert!([0].is_sorted());
 assert!(empty.is_sorted());
 assert!(![0.0, 1.0, f32::NAN].is_sorted());
    -
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted using the given comparator function.

    +
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted using the given comparator function.

    Instead of using PartialOrd::partial_cmp, this function uses the given compare function to determine the ordering of two elements. Apart from that, it’s equivalent to is_sorted; see its documentation for more information.

    -
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted using the given key extraction function.

    +
    🔬 This is a nightly-only experimental API. (is_sorted)

    Checks if the elements of this slice are sorted using the given key extraction function.

    Instead of comparing the slice’s elements directly, this function compares the keys of the elements, as determined by f. Apart from that, it’s equivalent to is_sorted; see its documentation for more information.

    -
    Examples
    +
    Examples
    #![feature(is_sorted)]
 
 assert!(["c", "bb", "aaa"].is_sorted_by_key(|s| s.len()));
 assert!(![-2i32, -1, 0, 3].is_sorted_by_key(|n| n.abs()));
    -

    Returns the index of the partition point according to the given predicate +

    Returns the index of the partition point according to the given predicate (the index of the first element of the second partition).

    The slice is assumed to be partitioned according to the given predicate. This means that all elements for which the predicate returns true are at the start of the slice @@ -1654,7 +1698,7 @@ For example, [7, 15, 3, 5, 4, 12, 6] is a partitioned under the predicate x % 2

    If this slice is not partitioned, the returned result is unspecified and meaningless, as this method performs a kind of binary search.

    See also binary_search, binary_search_by, and binary_search_by_key.

    -
    Examples
    +
    Examples
    let v = [1, 2, 3, 3, 5, 6, 7];
 let i = v.partition_point(|&x| x < 5);
 
@@ -1669,13 +1713,13 @@ sort order:
    
 let idx = s.partition_point(|&x| x < num);
 s.insert(idx, num);
 assert_eq!(s, [0, 1, 1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 42, 55]);
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the subslice corresponding to the given range +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the subslice corresponding to the given range and returns a reference to it.

    Returns None and does not modify the slice if the given range is out of bounds.

    Note that this method only accepts one-sided ranges such as 2.. or ..6, but not 2..6.

    -
    Examples
    +
    Examples

    Taking the first three elements of a slice:

    #![feature(slice_take)]
@@ -1705,13 +1749,13 @@ range is out of bounds.
    
 assert_eq!(None, slice.take(..=4));
 let expected: &[char] = &['a', 'b', 'c', 'd'];
 assert_eq!(Some(expected), slice.take(..4));
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the subslice corresponding to the given range +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the subslice corresponding to the given range and returns a mutable reference to it.

    Returns None and does not modify the slice if the given range is out of bounds.

    Note that this method only accepts one-sided ranges such as 2.. or ..6, but not 2..6.

    -
    Examples
    +
    Examples

    Taking the first three elements of a slice:

    #![feature(slice_take)]
@@ -1741,10 +1785,10 @@ range is out of bounds.
    
 assert_eq!(None, slice.take_mut(..=4));
 let expected: &mut [_] = &mut ['a', 'b', 'c', 'd'];
 assert_eq!(Some(expected), slice.take_mut(..4));
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the first element of the slice and returns a reference +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the first element of the slice and returns a reference to it.

    Returns None if the slice is empty.

    -
    Examples
    +
    Examples
    #![feature(slice_take)]
 
 let mut slice: &[_] = &['a', 'b', 'c'];
@@ -1752,10 +1796,10 @@ to it.
    
 
 assert_eq!(slice, &['b', 'c']);
 assert_eq!(first, &'a');
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the first element of the slice and returns a mutable +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the first element of the slice and returns a mutable reference to it.

    Returns None if the slice is empty.

    -
    Examples
    +
    Examples
    #![feature(slice_take)]
 
 let mut slice: &mut [_] = &mut ['a', 'b', 'c'];
@@ -1764,10 +1808,10 @@ reference to it.
    
 
 assert_eq!(slice, &['b', 'c']);
 assert_eq!(first, &'d');
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the last element of the slice and returns a reference +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the last element of the slice and returns a reference to it.

    Returns None if the slice is empty.

    -
    Examples
    +
    Examples
    #![feature(slice_take)]
 
 let mut slice: &[_] = &['a', 'b', 'c'];
@@ -1775,10 +1819,10 @@ to it.
    
 
 assert_eq!(slice, &['a', 'b']);
 assert_eq!(last, &'c');
    -
    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the last element of the slice and returns a mutable +

    🔬 This is a nightly-only experimental API. (slice_take)

    Removes the last element of the slice and returns a mutable reference to it.

    Returns None if the slice is empty.

    -
    Examples
    +
    Examples
    #![feature(slice_take)]
 
 let mut slice: &mut [_] = &mut ['a', 'b', 'c'];
@@ -1787,45 +1831,6 @@ reference to it.
    
 
 assert_eq!(slice, &['a', 'b']);
 assert_eq!(last, &'d');
    -
    🔬 This is a nightly-only experimental API. (slice_flatten)

    Takes a &[[T; N]], and flattens it to a &[T].

    -
    Panics
    -

    This panics if the length of the resulting slice would overflow a usize.

    -

    This is only possible when flattening a slice of arrays of zero-sized -types, and thus tends to be irrelevant in practice. If -size_of::<T>() > 0, this will never panic.

    -
    Examples
    -
    #![feature(slice_flatten)]
-
-assert_eq!([[1, 2, 3], [4, 5, 6]].flatten(), &[1, 2, 3, 4, 5, 6]);
-
-assert_eq!(
-    [[1, 2, 3], [4, 5, 6]].flatten(),
-    [[1, 2], [3, 4], [5, 6]].flatten(),
-);
-
-let slice_of_empty_arrays: &[[i32; 0]] = &[[], [], [], [], []];
-assert!(slice_of_empty_arrays.flatten().is_empty());
-
-let empty_slice_of_arrays: &[[u32; 10]] = &[];
-assert!(empty_slice_of_arrays.flatten().is_empty());
    -
    🔬 This is a nightly-only experimental API. (slice_flatten)

    Takes a &mut [[T; N]], and flattens it to a &mut [T].

    -
    Panics
    -

    This panics if the length of the resulting slice would overflow a usize.

    -

    This is only possible when flattening a slice of arrays of zero-sized -types, and thus tends to be irrelevant in practice. If -size_of::<T>() > 0, this will never panic.

    -
    Examples
    -
    #![feature(slice_flatten)]
-
-fn add_5_to_all(slice: &mut [i32]) {
-    for i in slice {
-        *i += 5;
-    }
-}
-
-let mut array = [[1, 2, 3], [4, 5, 6], [7, 8, 9]];
-add_5_to_all(array.flatten_mut());
-assert_eq!(array, [[6, 7, 8], [9, 10, 11], [12, 13, 14]]);

    Checks if all bytes in this slice are within the ASCII range.

    Checks that two slices are an ASCII case-insensitive match.

    Same as to_ascii_lowercase(a) == to_ascii_lowercase(b), @@ -2106,9 +2111,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/struct.Rotation.html b/rustdoc/latest/halo2_proofs/poly/struct.Rotation.html index 71823e04..f6aedd5c 100644 --- a/rustdoc/latest/halo2_proofs/poly/struct.Rotation.html +++ b/rustdoc/latest/halo2_proofs/poly/struct.Rotation.html @@ -98,9 +98,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

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    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/poly/trait.Basis.html b/rustdoc/latest/halo2_proofs/poly/trait.Basis.html index faa679a5..5ddc3e42 100644 --- a/rustdoc/latest/halo2_proofs/poly/trait.Basis.html +++ b/rustdoc/latest/halo2_proofs/poly/trait.Basis.html @@ -20,5 +20,5 @@

    Trait halo2_proofs::poly::Basis

    source · [−]
    pub trait Basis: Copy + Debug + Send + Sync { }
    Expand description

    The basis over which a polynomial is described.

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/transcript/index.html b/rustdoc/latest/halo2_proofs/transcript/index.html index 71d29786..d0544b2d 100644 --- a/rustdoc/latest/halo2_proofs/transcript/index.html +++ b/rustdoc/latest/halo2_proofs/transcript/index.html @@ -35,5 +35,5 @@ the real C::Scalar that the challenge encoding represents. input stream of data from the prover to the verifier.

    TranscriptWrite

    Transcript view from the perspective of a prover that has access to an output stream of messages from the prover to the verifier.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/transcript/struct.Blake2bRead.html b/rustdoc/latest/halo2_proofs/transcript/struct.Blake2bRead.html index 9f79384e..3eec469e 100644 --- a/rustdoc/latest/halo2_proofs/transcript/struct.Blake2bRead.html +++ b/rustdoc/latest/halo2_proofs/transcript/struct.Blake2bRead.html @@ -99,9 +99,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/transcript/struct.Blake2bWrite.html b/rustdoc/latest/halo2_proofs/transcript/struct.Blake2bWrite.html index d967392e..c0ee0966 100644 --- a/rustdoc/latest/halo2_proofs/transcript/struct.Blake2bWrite.html +++ b/rustdoc/latest/halo2_proofs/transcript/struct.Blake2bWrite.html @@ -100,9 +100,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/transcript/struct.Challenge255.html b/rustdoc/latest/halo2_proofs/transcript/struct.Challenge255.html index 89aed855..f40e1f25 100644 --- a/rustdoc/latest/halo2_proofs/transcript/struct.Challenge255.html +++ b/rustdoc/latest/halo2_proofs/transcript/struct.Challenge255.html @@ -114,9 +114,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/transcript/struct.ChallengeScalar.html b/rustdoc/latest/halo2_proofs/transcript/struct.ChallengeScalar.html index 259de362..27fec6f2 100644 --- a/rustdoc/latest/halo2_proofs/transcript/struct.ChallengeScalar.html +++ b/rustdoc/latest/halo2_proofs/transcript/struct.ChallengeScalar.html @@ -96,9 +96,9 @@ builds. Read more

    Creates owned data from borrowed data, usually by cloning. Read more

    🔬 This is a nightly-only experimental API. (toowned_clone_into)

    Uses borrowed data to replace owned data, usually by cloning. Read more

    Attempts to convert self into T using TryInto<T>. Read more

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -

    The type returned in the event of a conversion error.

    +

    The type returned in the event of a conversion error.

    Performs the conversion.

    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/transcript/trait.EncodedChallenge.html b/rustdoc/latest/halo2_proofs/transcript/trait.EncodedChallenge.html index fe433ad7..766dc9b0 100644 --- a/rustdoc/latest/halo2_proofs/transcript/trait.EncodedChallenge.html +++ b/rustdoc/latest/halo2_proofs/transcript/trait.EncodedChallenge.html @@ -35,5 +35,5 @@ an input from the Blake2b hash would be a [u8; 64].

    Required Methods

    Get an encoded challenge from a given input challenge.

    Get a scalar field element from an encoded challenge.

    Provided Methods

    Cast an encoded challenge as a typed ChallengeScalar.

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/transcript/trait.Transcript.html b/rustdoc/latest/halo2_proofs/transcript/trait.Transcript.html index 03175b37..fbfb874e 100644 --- a/rustdoc/latest/halo2_proofs/transcript/trait.Transcript.html +++ b/rustdoc/latest/halo2_proofs/transcript/trait.Transcript.html @@ -32,5 +32,5 @@ treating it as a common input.

    Writing the scalar to the transcript without writing it to the proof, treating it as a common input.

    Provided Methods

    Squeeze a typed challenge (in the scalar field) from the transcript.

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/transcript/trait.TranscriptRead.html b/rustdoc/latest/halo2_proofs/transcript/trait.TranscriptRead.html index 0f7d34d3..e7f6a41c 100644 --- a/rustdoc/latest/halo2_proofs/transcript/trait.TranscriptRead.html +++ b/rustdoc/latest/halo2_proofs/transcript/trait.TranscriptRead.html @@ -26,5 +26,5 @@ input stream of data from the prover to the verifier.

    Required Methods

    Read a curve point from the prover.

    Read a curve scalar from the prover.

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/halo2_proofs/transcript/trait.TranscriptWrite.html b/rustdoc/latest/halo2_proofs/transcript/trait.TranscriptWrite.html index 9e88e1ce..4dd626d6 100644 --- a/rustdoc/latest/halo2_proofs/transcript/trait.TranscriptWrite.html +++ b/rustdoc/latest/halo2_proofs/transcript/trait.TranscriptWrite.html @@ -26,5 +26,5 @@ output stream of messages from the prover to the verifier.

    Required Methods

    Write a curve point to the proof and the transcript.

    Write a scalar to the proof and the transcript.

    -

    Implementors

    +

    Implementors

    \ No newline at end of file diff --git a/rustdoc/latest/index.html b/rustdoc/latest/index.html index 71c95af2..005d3509 100644 --- a/rustdoc/latest/index.html +++ b/rustdoc/latest/index.html @@ -18,5 +18,5 @@

    List of all crates

    +

    List of all crates

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for elliptic curve operations.","Gadget and chips for the Poseidon algebraic hash function.","Primitives used in the Orchard protocol.","Gadget and chips for the SHA-256 hash function.","Gadgets for the Sinsemilla hash function.","Utility gadgets.","Fixed points that can be multiplied by base field elements.","The set of circuit instructions required to use the ECC …","A constant elliptic curve point over the given curve, for …","A constant elliptic curve point over the given curve, used …","A constant elliptic curve point over the given curve, used …","Defines the fixed points for a given instantiation of the …","Enumeration of the set of fixed bases to be used in scalar …","Fixed points that can be used with full-width scalar …","A non-identity elliptic curve point over the given curve.","Variable representing a non-identity elliptic curve point.","An elliptic curve point over the given curve.","Variable representing an elliptic curve point.","A full-width element of the given elliptic curve’s …","Variable representing a full-width element of the elliptic …","A signed short element of the given elliptic curve’s …","Variable representing a signed short element of the …","An element of the given elliptic curve’s base field, …","Variable representing an element of the elliptic curve’s …","Fixed points that can be used with short scalar …","The affine short Weierstrass x-coordinate of an elliptic …","Variable representing the affine short Weierstrass …","Performs complete point addition, returning a + b.","Returns self + other using complete addition.","Returns self + other using complete addition.","Performs incomplete point addition, returning a + b.","Returns self + other using incomplete addition. The …","","","","","","","","","","","","","","","","","","","Chip implementations for the ECC gadgets.","","","","","","","","","","","Constrains point a to be equal in value to point b.","Constrains this point to be equal in value to another …","Constrains this point to be equal in value to another …","","","","","","","","","","","","","","","","","","","","","","","","","","","","Extracts the x-coordinate of a point.","Extracts the x-coordinate of a point.","Extracts the x-coordinate of a point.","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Wraps the given point (obtained directly from an …","Wraps the given point (obtained directly from an …","Wraps the given x-coordinate (obtained directly from an …","Wraps the given fixed base (obtained directly from an …","Wraps the given fixed base (obtained directly from an …","Wraps the given fixed base (obtained directly from an …","","","","","","","","","","Returns the inner point.","Returns the inner point.","Returns the inner x-coordinate.","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Performs variable-base scalar multiplication, returning …","Returns [by] self.","Returns [by] self.","Returns [by] self.","Returns [by] self.","Performs fixed-base scalar multiplication using a …","Performs fixed-base scalar multiplication using a base …","Performs fixed-base scalar multiplication using a short …","Constructs a new point with the given value.","Constructs a new point with the given value.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Witnesses the given point as a private input to the …","Witnesses the given point as a private input to the …","Type marker representing a base field element being used …","A chip implementing EccInstructions","Configuration for the ECC chip","A curve point represented in affine (x, y) coordinates, or …","A full-width scalar used for fixed-base scalar …","A signed short scalar used for fixed-base scalar …","Returns information about a fixed point that is required …","Type marker representing a full-width scalar for use in …","The number of windows that this scalar kind requires.","A non-identity point represented in affine (x, y) …","A trait representing the kind of scalar used with a …","The kind of scalar that this fixed point can be multiplied …","Type marker representing a signed 64-bit scalar for use in …","","","Advice columns needed by instructions in the ECC chip.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Side effects","Constants required for the ECC chip.","","Reconstructs this chip from the given config.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Returns the argument unchanged.","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the generator for this fixed point.","","","","","","","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Returns the Lagrange coefficients for this fixed point.","","Lookup range check using 10-bit lookup table","","","","","","","Returns the value of this curve point, if known.","Returns the value of this curve point, if known.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Returns the $u$ values for this fixed point.","","","","","","","","","","","","The cell containing the affine short-Weierstrass …","The cell containing the affine short-Weierstrass …","The cell containing the affine short-Weierstrass …","The cell containing the affine short-Weierstrass …","Returns the $z$ value for this fixed point.","Window size for fixed-base scalar multiplication","$2^{FIXED_BASE_WINDOW_SIZE}$","Number of windows for a full-width scalar","Number of windows for a short signed scalar","For each window, we interpolate the $x$-coordinate. Here, …","For each window, $z$ is a field element such that for each …","Test that Lagrange interpolation coefficients reproduce …","Test that the z-values and u-values satisfy the conditions:","A Poseidon hash function, built around a sponge.","A message word provided by the prover.","A word from the padded input to a Poseidon sponge.","A padding word, that will be fixed in the circuit …","The set of circuit instructions required to use the …","The set of circuit instructions required to use the Sponge …","A Poseidon chip using an $x^5$ S-Box.","Configuration for a Pow5Chip.","A Poseidon sponge.","A word in the Poseidon state.","A word over which the Poseidon permutation operates.","Variable representing the word over which the Poseidon …","Absorbs an element into the sponge.","Adds the given input to the state.","","","","","","","","","","","","","","","","","","","","","","","","Configures this chip for use in a circuit.","Construct a Pow5Chip.","","","","","","","","","","","","","","","","","","","","","","Transitions the sponge into its squeezing state.","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Construct a Word gadget from the inner word.","Extracts sponge output from the given state.","","Hashes the given input.","","","","","","","Initializes a new hasher.","","Returns the initial empty state for the given domain.","","The word contained in this gadget.","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","","Constructs a new duplex sponge for the given Poseidon …","Applies the Poseidon permutation to the given state.","","Squeezes an element from the sponge.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","The Poseidon algebraic hash function.","The Sinsemilla hash function and commitment scheme.","The absorbing state of the Sponge.","A Poseidon hash function used with constant input length.","A domain in which a Poseidon hash function is being used.","A Poseidon hash function, built around a sponge.","Poseidon-128 using the $x^5$ S-box, with a width of 3 …","Iterator that outputs padding field elements.","A specification for a Poseidon permutation.","The state of the Sponge.","The squeezing state of the Sponge.","","","","","","","","","","","","","Generates (round_constants, mds, mds^-1) corresponding to …","","","","","","","","","","","","","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","The number of full rounds for this specification.","","","Hashes the given input.","","Initializes a new hasher.","","","","","The initial capacity element, encoding this domain.","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","The name of this domain, for debug formatting purposes.","","Returns the padding to be appended to the input.","","The number of partial rounds for this specification.","","","The S-box for this specification.","","","Side-loaded index of the first correct and secure MDS that …","","","","","","","","","","","","","","","","","","","","","","","","The largest integer such that $2^c \\\\leq (r_P - 1) / 2$, …","A domain in which $\\\\mathsf{SinsemillaCommit}$ and …","A domain in which $\\\\mathsf{SinsemillaHashToPoint}$ and …","$\\\\frac{1}{2^K}$","Number of bits of each message piece in …","Returns the Sinsemilla $Q$ constant for this domain.","Returns the Sinsemilla $Q$ constant for this domain.","SWU hash-to-curve personalization for Sinsemilla $Q$ …","Returns the Sinsemilla $R$ constant for this domain.","The precomputed bases for the Sinsemilla hash function.","SWU hash-to-curve personalization for Sinsemilla $S$ …","","","","","","","$\\\\mathsf{SinsemillaCommit}$ from § 5.4.8.4.","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","$\\\\mathsf{SinsemillaHash}$ from § 5.4.1.9.","$\\\\mathsf{SinsemillaHashToPoint}$ from § 5.4.1.9.","","","Calls U::from(self).","Calls U::from(self).","Constructs a new HashDomain with a specific prefix string.","Constructs a new CommitDomain with a specific prefix …","$\\\\mathsf{SinsemillaShortCommit}$ from § 5.4.8.4.","","","","","","","","","","","The size of a SHA-256 block, in 32-bit words.","A word in a Table16 message block.","Variable representing a 32-bit word of the input block to …","A gadget that constrains a SHA-256 invocation. It supports …","The output of a SHA-256 circuit invocation.","The set of circuit instructions required to use the Sha256 …","Variable representing the SHA-256 internal state.","A chip that implements SHA-256 with a maximum lookup table …","Configuration for a Table16Chip.","","","","","","","","","","","","","","","","","Starting from the given initialized state, processes a …","","","Configures a circuit to include this chip.","Reconstructs this chip from the given config.","","","","","","","","","","","","Converts the given state into a message digest.","","Convenience function to compute hash of the data. It will …","","","","","","Retrieve result and consume hasher instance.","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","","","","","Creates an initial state from the output state of a …","","Places the SHA-256 IV in the circuit, returning the …","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Loads the lookup table required by this chip into the …","","Create a new hasher instance.","","","","","","","","","","","","","","","","","","","Digest data, updating the internal state.","","","","","","A variable in the circuit.","Gadget representing a domain in which …","Trait allowing circuit’s Sinsemilla CommitDomains to be …","CommitDomains used in this instruction.","A type enumerating the fixed points used in CommitDomains.","A domain in which $\\\\mathsf{SinsemillaHashToPoint}$ and …","Trait allowing circuit’s Sinsemilla HashDomains to be …","HashDomains used in this instruction.","A message to be hashed.","A message composed of Self::MessagePieces.","A message piece with a bitlength of some multiple of K.","A piece in a message containing a number of K-bit words. A …","A point output of Self::hash_to_point.","Returns the Q constant for this domain.","A cumulative sum z is used to decompose a Sinsemilla …","The set of circuit instructions required to use the …","The x-coordinate of a point output of Self::hash_to_point.","","","","","","","","","Chip implementations for the Sinsemilla gadgets.","","","","","$\\\\mathsf{SinsemillaCommit}$ from § 5.4.8.4.","","","","","","","","","","","","","Extracts the x-coordinate of the output of a Sinsemilla …","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Constructs a MessagePiece from a field element.","Constructs a message from a vector of MessagePieces.","$\\\\mathsf{SinsemillaHash}$ from § 5.4.1.9.","Returns the HashDomain contained in this CommitDomain","Hashes a message to an ECC curve point. This returns both …","$\\\\mathsf{SinsemillaHashToPoint}$ from § 5.4.1.9.","","","","","Returns the inner MessagePiece contained in this gadget.","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Gadgets for implementing a Merkle tree with Sinsemilla.","Constructs a new HashDomain for the given domain.","Constructs a new CommitDomain for the given domain.","Returns the fixed point corresponding to the R constant …","$\\\\mathsf{SinsemillaShortCommit}$ from § 5.4.8.4.","","","","","","","","","","","","","","","","","","","Witness a message piece given a field element. Returns a …","A chip that implements 10-bit Sinsemilla using a lookup …","Configuration for the Sinsemilla hash chip","","","","","","","","","","Side-effects","Reconstructs this chip from the given config.","","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","","Calls U::from(self).","Calls U::from(self).","Loads the lookup table required by this chip into the …","","Returns the lookup range check config used in this config.","","","","","","","","","","","","","","SWU hash-to-curve personalization for the Merkle CRH …","Instructions to check the validity of a Merkle path of a …","Gadget representing a Merkle path that proves a leaf …","","","Calculates the root of the tree containing the given leaf …","Chip implementing a Merkle hash using Sinsemilla as the …","","","Constructs a MerklePath.","","","","","Returns the argument unchanged.","Compute MerkleCRH for a given layer. The hash that …","","Calls U::from(self).","","","","","","Chip implementing MerkleInstructions.","Configuration for the MerkleChip implementation.","","","","","","","","","","Configures the MerkleChip.","Constructs a MerkleChip given a MerkleConfig.","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","","","Calls U::from(self).","Calls U::from(self).","","","","","","","","","","","","","","Trait for utilities used across circuits.","Trait for a variable in the circuit.","Variable in the circuit.","Takes a specified subsequence of the little-endian bit …","Checks that an expression is either 1 or 0.","The cell at which this variable was allocated.","Gadget and chip for a conditional swap utility.","Decomposes an $n$-bit field element $\\\\alpha$ into $W$ …","Decompose a word alpha into window_num_bits bits …","The sequence of bits representing a u64 in little-endian …","The u64 integer represented by an L-bit little-endian …","Load a variable.","Make use of a K-bit lookup table to decompose a field …","Check that an expression is in the small range [0..range), …","If a then b, else c. Returns (a * b) + (1 - a) * c.","The value allocated to this variable.","A chip implementing a conditional swap.","Configuration for the CondSwapChip.","Instructions for a conditional swap gadget.","","","","","","","","","","Configures this chip for use in a circuit.","Constructs a CondSwapChip given a CondSwapConfig.","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","Calls U::from(self).","Calls U::from(self).","","Given an input pair (a,b) and a swap boolean flag, returns …","","","","","","","","","","","","The running sum $[z_0, …, z_W]$. If created in strict …","Configuration that provides methods for running sum …","","","","","","","perm MUST include the advice column z.","Decompose an existing variable alpha that is copied into …","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","Calls U::from(self).","Calls U::from(self).","","","","","","","","","","","Decompose a field element alpha that is witnessed in this …","Configuration that provides methods for a lookup range …","The running sum $[z_0, …, z_W]$. If created in strict …","","","","","","","The running_sum advice column breaks the field element …","Range check on an existing cell that is copied into this …","Short range check on an existing cell that is copied into …","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","Calls U::from(self).","Calls U::from(self).","","","","","","","","","","","Range check on a value that is witnessed in this helper.","Short range check on value that is witnessed in this 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module provides common utilities, traits and …","Traits and structs for implementing circuit components.","Tools for developing circuits.","This module provides an implementation of a variant of …","Contains utilities for performing arithmetic over …","This module contains utilities and traits for dealing with …","The affine version of the curve","The base field over which this elliptic curve is …","The base field over which this elliptic curve is …","CURVE_ID used for hash-to-curve.","The affine coordinates of a point on an elliptic curve.","This trait is the affine counterpart to Curve and is used …","This trait is a common interface for dealing with elements …","The projective form of the curve","Generator of the $t-order$ multiplicative subgroup","This trait represents an element of a field.","This trait is a common interface for dealing with elements …","This represents an element of a group with basic …","Modulus of the field written as a string for display …","Inverse of PrimeField::root_of_unity()","The group is assumed to be of prime order $p$. Scalar is …","The scalar field of this elliptic curve.","The scalar field of this elliptic curve.","A trait that exposes additional operations related to …","Tables used for square root computation.","Inverse of $2$ in the field.","The value $(T-1)/2$ such that $2^S \\\\cdot T = p - 1$ with …","Element of multiplicative order $3$.","Returns the curve constant a.","Returns the curve constant $a$.","Returns the curve constant b.","Returns the curve constant $b$.","Performs a radix-$2$ Fast-Fourier Transformation (FFT) on …","Performs a multi-exponentiation operation.","","","","","","","This computes the inner product of two vectors a and b.","","Gets the coordinates of this point.","Cubes this element.","","","","","","Doubles this element.","","","Apply the curve endomorphism by multiplying the …","This evaluates a provided polynomial (in coefficient form) …","","","Returns the argument unchanged.","Returns the argument unchanged.","Obtains a field element that is congruent to the provided …","Obtains a field element congruent to the integer v.","Obtains a point given $(x, y)$, failing if it is not on the","Obtains a Coordinates value given $(x, y)$, failing if it …","Gets the lower 128 bits of this field element when …","Gets the lower 32 bits of this field element when expressed","Adds rhs to this group element.","Scales this group element by a scalar.","Subtracts rhs from this group element.","Returns the additive identity of the group.","Requests a hasher that accepts messages and returns …","","","Calls U::from(self).","Calls U::from(self).","Computes the multiplicative inverse of this element, …","Returns whether or not this element is on the curve; should","Returns whether or not this element is on the curve; should","Returns true iff this element is zero.","Returns true iff this element is zero.","Return the Jacobian coordinates of this point.","Divides polynomial a in X by X - b with no remainder.","Returns coefficients of an n - 1 degree polynomial given a …","Build tables given parameters for the perfect hash.","Obtains a point given Jacobian coordinates $X : Y : Z$, …","Returns the one element of the field, the multiplicative …","This simple utility function will parallelize an operation …","Exponentiates self by by, where by is a little-endian order","Raise this field element to the power Self::T_MINUS1_OVER2.","Exponentiates self by exp, where exp is a little-endian …","Returns an element chosen uniformly at random using a …","Performs a small multi-exponentiation operation. Uses the …","Returns the square root of the field element, if it is …","Equivalent to Self::sqrt_ratio(self, one()).","Same as sqrt_ratio(u, one()) but more efficient.","Computes:","Computes:","Squares this element.","","","","","","","","Returns the u-coordinate.","Returns the v-coordinate.","","","Returns the x-coordinate.","Returns the y-coordinate.","Returns the zero element of the field, the additive …","An assigned cell.","A pointer to a cell within a circuit.","A chip implements a set of instructions that can be used …","A type that holds the configuration for this chip, and any …","A layout strategy within a circuit. The layouter is …","A type that holds any general chip state that needs to be …","This is a “namespaced” layouter which borrows a …","A region of the circuit in which a Chip can assign cells.","Index of a region in a layouter","Starting row of a region in a layouter","Represents the type of the “root” of this layouter, so …","A simple FloorPlanner that performs minimal optimizations.","A lookup table in the circuit.","Assign an advice column value (witness).","Assigns a constant value to the column advice at offset …","Assign the value of the instance column’s cell at …","Assigns a fixed value to a table cell.","Assign a fixed value.","Assign a region of gates to an absolute row number.","","Assign a table region to an absolute row number.","","","","","","","","","","","","","","","","","","Returns the cell.","","","","","","","","","The chip holds its own configuration.","Constrains a cell to have a constant value.","Constrains two cells to have the same value.","Constrains a Cell to equal an instance column’s row …","","Copies the value to a given advice cell and constrains …","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Evaluates this assigned cell’s value directly, …","Implementations of common circuit floor planners.","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","Returns the argument unchanged.","","Returns the argument unchanged.","Returns the argument unchanged.","Gets the “root” of this assignment, bypassing the …","","","","","","","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Implementations of common circuit layouters.","Provides access to general chip state loaded at the …","Enters into a namespace.","","Exits out of the existing namespace.","","Creates a new (sub)namespace and enters into it.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Returns the value of the AssignedCell.","Returns the field element value of the AssignedCell.","","","","","","","","","The version 1 FloorPlanner provided by halo2.","A single pass of the V1 layouter.","","","","","","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","","Calls U::from(self).","Calls U::from(self).","","","","","","","","","","","","Concrete column","The virtual column involved in a region. This includes …","Helper trait for implementing a custom Layouter.","The shape of a region. For a region at a certain index, we …","Virtual column representing a (boolean) selector","Helper trait for implementing a custom Layouter.","Assign an advice column value (witness)","","Assigns a constant value to the column advice at offset …","","Assign the value of the instance column’s cell at …","","Assigns a fixed value to a table cell.","Assign a fixed value","","","","","","","","","","","Get a reference to the set of columns used in a RegionShape…","Constrains a cell to have a constant value.","","Constraint two cells to have the same value.","","","","","","","","Enables a selector at the given offset.","","","","","Returns the argument unchanged.","","","Returns the argument unchanged.","","","","Calls U::from(self).","Calls U::from(self).","","Create a new RegionShape for a region at region_index.","","Get the region_index of a RegionShape.","Get the row_count of a RegionShape.","","","","","","","","","","","","","A cell used in an active gate was not assigned to.","A struct for collecting and displaying the gates within a …","Graphical renderer for circuit layouts.","A constraint was not satisfied for a particular row.","A constraint was active on an unusable row, and is likely …","The location within the circuit at which a particular …","A location inside a region.","A lookup input did not exist in its corresponding table.","A test prover for debugging circuits.","A location outside of a region.","A permutation did not preserve the original value of a …","The reasons why a particular circuit is not satisfied.","Panics if the circuit being checked by this MockProver is …","","","","","","","","","","","","","Builds a dot graph string representing the given circuit.","Collects the gates from within the circuit.","","Developer tools for investigating the cost of a circuit.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Marks cells involved in equality constraints, in red.","Metadata about circuits.","","","","","Prints the queries in this circuit to a CSV grid.","","Renders the given circuit on the given drawing area.","Runs a synthetic keygen-and-prove operation on the given …","Draws red lines between equality-constrained cells.","Sets the visibility of region labels.","","","","","","","","","","","","","","","","","","","Returns Ok(()) if this MockProver is satisfied, or a list …","Sets the view height for this layout, as a number of rows.","Sets the view width for this layout, as a number of …","","","","","","The offset (relative to the start of the region) at which …","The region in which the failure occurred.","The circuit row on which the failure occurred.","The values of the virtual cells used by this constraint.","The column in which this cell should be assigned.","The column in which this permutation is not satisfied.","The polynomial constraint that is not satisfied.","The polynomial constraint that is not satisfied.","The index of the active gate.","The offset (relative to the start of the region) at which …","The location at which this constraint is not satisfied.","The location at which the lookup is not satisfied.","The location at which the permutation is not satisfied.","The index of the lookup that is not satisfied. These …","The offset (relative to the start of the region) at which …","The region in which this cell should be assigned.","Measures a circuit to determine its costs, and explain …","The marginal size of a Halo 2 proof, broken down into its …","The size of a Halo 2 proof, broken down into its …","","","","","","","","","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Returns the marginal proof size per instance of this …","Measures a circuit with parameter constant k.","Returns the proof size for the given number of instances …","","","","","","","","","","","","","Metadata about a column within a circuit.","Metadata about a configured constraint within a circuit.","Metadata about a configured gate within a circuit.","Metadata about an assigned region within a circuit.","A “virtual cell” is a PLONK cell that has been queried …","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Returns the argument unchanged.","","","","Returns the argument unchanged.","","Returns the argument unchanged.","","Returns the argument unchanged.","Returns the argument unchanged.","","","","","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","An advice column","An Advice variant","This is an advice (witness) column queried at a certain …","An enum over the Advice, Fixed, Instance structs","A value assigned to a cell within a circuit.","This trait allows a Circuit to direct some backend to …","A verifier that checks multiple proofs in a batch.","Out of bounds index passed to a backend","This is a trait that circuits provide implementations for …","A column with an index and type","The instance sets up a copy constraint involving a column …","A column type","This is a configuration object that stores things like …","This is a constant polynomial","An individual polynomial constraint.","This is a description of the circuit environment, such as …","The constraint system is not satisfied.","A set of polynomial constraints with a common selector.","This is an error that could occur during proving or …","Low-degree expression representing an identity that must …","A fixed column","A Fixed variant","This is a fixed column queried at a certain relative …","A floor planning strategy for a circuit.","The floor planner used for this circuit. This is an …","An instance column","An Instance variant","This is an instance (external) column queried at a certain …","Instance provided exceeds number of available rows","The provided instances do not match the circuit parameters.","This is a negated polynomial","Circuit synthesis requires global constants, but circuit …","k is too small for the given circuit.","Opening error","The output type of this verification strategy after …","Represents the minimal parameters that determine a …","Minimal representation of a verification key that can be …","This is the product of two polynomials","This is a proving key which allows for the creation of …","A value stored as a fraction to enable batch inversion.","This is a scaled polynomial","A selector, representing a fixed boolean value per row of …","This is a virtual selector","A verifier that checks a single proof at a time.","This is the sum of two polynomials","This is an error that can occur during synthesis of the …","A fixed column of a lookup table.","Transcript error","A value that does not require inversion to evaluate.","Trait representing a strategy for verifying Halo 2 proofs.","This is a verifying key which allows for the verification …","Exposes the “virtual cells” that can be queried while …","The field element zero.","","","","","","","Allocate a new advice column","Assign an advice column value (witness)","Assign a fixed value","Compute the number of blinding factors necessary to …","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Type of this column.","Allocate a new complex selector that can appear anywhere …","The circuit is given an opportunity to describe the exact …","Assign two cells to have the same value","Creates a new gate.","This creates a proof for the provided circuit when given …","Cubes this element.","","Compute the degree of this polynomial","Compute the degree of the constraint system (the maximum …","Returns the denominator, if non-trivial.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Doubles this element.","","","","","","","","","","","","","","","","","","","","","Enable this selector at the given offset within the given …","Enables this fixed column to be used for global constant …","Enable the ability to enforce equality over cells in this …","Enables a selector at the given row.","Creates a new region and enters into it.","","","","","","","","","Evaluates this assigned value directly, performing an …","Evaluate the polynomial using the provided closures to …","Exits the current region.","Fills a fixed column starting from the given row with …","Finalizes the batch and checks its validity.","Allocate a new fixed column","","","","","","","","","","","","","","","","","","","","","","","","","","Returns the argument unchanged.","","Returns the argument unchanged.","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","","Returns the argument unchanged.","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Get the underlying EvaluationDomain.","Get the underlying VerifyingKey.","","","","","","","","Hashes a verification key into a transcript.","","","","","","","","","","","","","","","","","","","","","Allocate a new instance column","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","","Inverts this assigned value (taking the inverse of zero to …","Is this selector “simple”? Simple selectors can only …","Returns true iff this element is zero.","Generate a ProvingKey from a VerifyingKey and an instance …","Generate a VerifyingKey from an instance of Circuit.","Add a lookup argument for some input expressions and table …","Allocates a new fixed column that can be used in a lookup …","Returns the minimum necessary rows that need to exist in …","","","","","","","","","","","","","Constructs a new single proof verifier.","Constructs a new batch verifier.","Returns the numerator.","","","Obtain a pinned version of this constraint system; a …","Obtains a pinned representation of this verification key …","Exits out of the existing namespace.","Obtains an MSM from the verifier strategy and yields back …","","","Creates a new (sub)namespace and enters into it.","Query an advice column at a relative position","Query an Any column at a relative position","Query a fixed column at a relative position","Queries the cell of an instance column at a particular …","Query an instance column at a relative position","Query a selector at the current position.","Allocate a new (simple) selector. Simple selectors cannot …","Sets the minimum degree required by the circuit, which can …","","Squares this element.","Square this expression.","","","","","","","Given the provided cs, synthesize the given circuit.","Given the provided cs, synthesize the circuit. The …","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Returns a boolean indicating whether or not the proof is …","","","","","","","","","","","","","","","","","","","","","Constructs a set of constraints that are controlled by the …","Returns a copy of this circuit with no witness values …","","","","","","The current value of k being used.","","","","","","","","","","Column index","Column index","Column index","Query index","Query index","Query index","Rotation of this query","Rotation of this query","Rotation of this query","","The basis over which a polynomial is described.","The polynomial is defined as coefficients","This is an error that could occur during proving or …","This structure contains precomputed constants and other …","The polynomial is defined as coefficients of Lagrange …","The polynomial is defined as coefficients of Lagrange …","OpeningProof is not well-formed","Represents the minimal parameters that determine an …","Represents a univariate polynomial defined over a field …","Describes the relative rotation of a vector. Negative …","Caller needs to re-sample a point","","","","","","","","","","","","","","","","","","","","","","","","","","","","Obtains a polynomial in coefficient form when given a …","This takes us from an n-length coefficient vector into a …","This module contains an implementation of the polynomial …","Returns a constant polynomial in the extended Lagrange …","Returns a constant polynomial in the Lagrange coefficient …","The current location in the evaluation domain","","","","","","","","","","","","","","","","","","","This divides the polynomial (in the extended domain) by …","","","","","","","","","Returns an empty (zero) polynomial in the coefficient basis","Returns an empty (zero) polynomial in the extended …","Returns an empty (zero) polynomial in the Lagrange …","","Get the size of the extended domain","This takes us from the extended evaluation domain and gets …","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Get the generator of the extended domain’s …","Get $\\\\omega$, the generator of the $2^k$ order …","Get $\\\\omega^{-1}$, the inverse of the generator of the …","Gets the quotient polynomial’s degree (as a multiple of …","","","","","","","","","","","","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Iterate over the values, which are either in coefficient …","Iterate over the values mutably, which are either in …","Computes evaluations (at the point x, where xn = x^n) of …","Obtains a polynomial in Lagrange form when given a vector …","This takes us from an n-length vector into the coefficient …","","This module contains an optimisation of the polynomial …","","This constructs a new evaluation domain object based on …","The next location in the evaluation domain","Gets the size of this polynomial in terms of the number of …","Obtain a pinned version of this evaluation domain; a …","The previous location in the evaluation domain","Rotates the values in a Lagrange basis polynomial by …","Rotate the extended domain polynomial over the original …","Multiplies a value by some power of $\\\\omega$, essentially …","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","An accumulator instance consisting of an evaluation claim …","Wrapper type around a blinding factor.","A guard returned by the verifier","A multiscalar multiplication in the polynomial commitment …","These are the public parameters for the polynomial …","","","","Add a value to the first entry of g_scalars.","Add another multiexp into this one","Add a vector of scalars to g_scalars. This function will …","Add to u_scalar","Add to w_scalar","Add arbitrary term (the scalar and the point)","","","","","","","","","","","","","","","","","","","This computes a commitment to a polynomial described by …","This commits to a polynomial using its evaluations over …","Computes G = ⟨s, params.g⟩","Create a polynomial commitment opening proof for the …","","","","","","","","","","","","","","","","","Generates an empty multiscalar multiplication struct using …","","Perform multiexp and check that it results in zero","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","The claimed output of the linear-time polycommit opening …","Getter for g generators","","","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","","","","","Create a new, empty MSM using the provided parameters.","Initializes parameters for the curve, given a random …","Reads params from a buffer.","Scale all scalars in the MSM by some scaling factor","","","","","","","","","","","","","","","","","","","","A vector of challenges u_0, …, u_{k - 1} sampled by the …","Lets caller supply the challenges and obtain an MSM with …","Lets caller supply the purported G point and simply appends","Checks to see if the proof represented within transcript …","","","","","","Writes params to a buffer.","A polynomial query at a point","A polynomial query at a point","blinding factor of polynomial","","","","","","","","","Create a multi-opening proof","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","Calls U::from(self).","Calls U::from(self).","Create a new verifier query based on a commitment","Create a new verifier query based on a linear combination …","point at which polynomial is queried","coefficients of polynomial","","","","","","","","","Verify a multi-opening proof","","","We will replace BLAKE2b with an algebraic hash function in …","We will replace BLAKE2b with an algebraic hash function in …","A 255-bit challenge.","The scalar representation of a verifier challenge.","EncodedChallenge<C> defines a challenge encoding with a …","The Input type used to derive the challenge encoding. For …","Generic transcript view (from either the prover or verifier…","Transcript view from the perspective of a verifier that …","Transcript view from the perspective of a prover that has …","Cast an encoded challenge as a typed ChallengeScalar.","","","","","","","","","","","","","","","","","Writing the point to the transcript without writing it to …","","","Writing the scalar to the transcript without writing it to …","","","","","","","","","","","","","","","","","Conclude the interaction and return the output buffer …","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Get a scalar field element from an encoded challenge.","","Initialize a transcript given an input buffer.","","Initialize a transcript given an output buffer.","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Get an encoded challenge from a given input challenge.","","Read a curve point from the prover.","","Read a curve scalar from the prover.","","Squeeze an encoded verifier challenge from the transcript.","","","Squeeze a typed challenge (in the scalar field) from the …","","","","","","","","","","","","","","","","","","","","","","Write a curve point to the proof and the transcript.","","Write a scalar to the proof and the 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for elliptic curve operations.","Gadget and chips for the Poseidon algebraic hash function.","Primitives used in the Orchard protocol.","Gadget and chips for the SHA-256 hash function.","Gadgets for the Sinsemilla hash function.","Utility gadgets.","Fixed points that can be multiplied by base field elements.","The set of circuit instructions required to use the ECC …","A constant elliptic curve point over the given curve, for …","A constant elliptic curve point over the given curve, used …","A constant elliptic curve point over the given curve, used …","Defines the fixed points for a given instantiation of the …","Enumeration of the set of fixed bases to be used in scalar …","Fixed points that can be used with full-width scalar …","A non-identity elliptic curve point over the given curve.","Variable representing a non-identity elliptic curve point.","An elliptic curve point over the given curve.","Variable representing an elliptic curve point.","A full-width element of the given elliptic curve’s …","Variable representing a full-width element of the elliptic …","A signed short element of the given elliptic curve’s …","Variable representing a signed short element of the …","An element of the given elliptic curve’s base field, …","Variable representing an element of the elliptic curve’s …","Fixed points that can be used with short scalar …","The affine short Weierstrass x-coordinate of an elliptic …","Variable representing the affine short Weierstrass …","Performs complete point addition, returning a + b.","Returns self + other using complete addition.","Returns self + other using complete addition.","Performs incomplete point addition, returning a + b.","Returns self + other using incomplete addition. The …","","","","","","","","","","","","","","","","","","","Chip implementations for the ECC gadgets.","","","","","","","","","","","Constrains point a to be equal in value to point b.","Constrains this point to be equal in value to another …","Constrains this point to be equal in value to another …","","","","","","","","","","","","","","","","","","","","","","","","","","","","Extracts the x-coordinate of a point.","Extracts the x-coordinate of a point.","Extracts the x-coordinate of a point.","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Wraps the given point (obtained directly from an …","Wraps the given point (obtained directly from an …","Wraps the given x-coordinate (obtained directly from an …","Wraps the given fixed base (obtained directly from an …","Wraps the given fixed base (obtained directly from an …","Wraps the given fixed base (obtained directly from an …","","","","","","","","","","Returns the inner point.","Returns the inner point.","Returns the inner x-coordinate.","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Performs variable-base scalar multiplication, returning …","Returns [by] self.","Returns [by] self.","Returns [by] self.","Returns [by] self.","Performs fixed-base scalar multiplication using a …","Performs fixed-base scalar multiplication using a base …","Performs fixed-base scalar multiplication using a short …","Constructs a new point with the given value.","Constructs a new point with the given value.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Witnesses the given point as a private input to the …","Witnesses the given point as a private input to the …","Type marker representing a base field element being used …","A chip implementing EccInstructions","Configuration for the ECC chip","A curve point represented in affine (x, y) coordinates, or …","A full-width scalar used for fixed-base scalar …","A signed short scalar used for fixed-base scalar …","Returns information about a fixed point that is required …","Type marker representing a full-width scalar for use in …","The number of windows that this scalar kind requires.","A non-identity point represented in affine (x, y) …","A trait representing the kind of scalar used with a …","The kind of scalar that this fixed point can be multiplied …","Type marker representing a signed 64-bit scalar for use in …","","","Advice columns needed by instructions in the ECC chip.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Side effects","Constants required for the ECC chip.","","Reconstructs this chip from the given config.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the generator for this fixed point.","","","","","","","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Returns the Lagrange coefficients for this fixed point.","","Lookup range check using 10-bit lookup table","","","","","","","Returns the value of this curve point, if known.","Returns the value of this curve point, if known.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Returns the $u$ values for this fixed point.","","","","","","","","","","","","The cell containing the affine short-Weierstrass …","The cell containing the affine short-Weierstrass …","The cell containing the affine short-Weierstrass …","The cell containing the affine short-Weierstrass …","Returns the $z$ value for this fixed point.","Window size for fixed-base scalar multiplication","$2^{FIXED_BASE_WINDOW_SIZE}$","Number of windows for a full-width scalar","Number of windows for a short signed scalar","For each window, we interpolate the $x$-coordinate. Here, …","For each window, $z$ is a field element such that for each …","Test that Lagrange interpolation coefficients reproduce …","Test that the z-values and u-values satisfy the conditions:","A Poseidon hash function, built around a sponge.","A message word provided by the prover.","A word from the padded input to a Poseidon sponge.","A padding word, that will be fixed in the circuit …","The set of circuit instructions required to use the …","The set of circuit instructions required to use the Sponge …","A Poseidon chip using an $x^5$ S-Box.","Configuration for a Pow5Chip.","A Poseidon sponge.","A word in the Poseidon state.","A word over which the Poseidon permutation operates.","Variable representing the word over which the Poseidon …","Absorbs an element into the sponge.","Adds the given input to the state.","","","","","","","","","","","","","","","","","","","","","","","","Configures this chip for use in a circuit.","Construct a Pow5Chip.","","","","","","","","","","","","","","","","","","","","","","Transitions the sponge into its squeezing state.","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Construct a Word gadget from the inner word.","Extracts sponge output from the given state.","","Hashes the given input.","","","","","","","Initializes a new hasher.","","Returns the initial empty state for the given domain.","","The word contained in this gadget.","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","","Constructs a new duplex sponge for the given Poseidon …","Applies the Poseidon permutation to the given state.","","Squeezes an element from the sponge.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","The Poseidon algebraic hash function.","The Sinsemilla hash function and commitment scheme.","The absorbing state of the Sponge.","A Poseidon hash function used with constant input length.","A domain in which a Poseidon hash function is being used.","A Poseidon hash function, built around a sponge.","Poseidon-128 using the $x^5$ S-box, with a width of 3 …","Iterator that outputs padding field elements.","A specification for a Poseidon permutation.","The state of the Sponge.","The squeezing state of the Sponge.","","","","","","","","","","","","","Generates (round_constants, mds, mds^-1) corresponding to …","","","","","","","","","","","","","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","The number of full rounds for this specification.","","","Hashes the given input.","Initializes a new hasher.","","","","","","The initial capacity element, encoding this domain.","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","The name of this domain, for debug formatting purposes.","","Returns the padding to be appended to the input.","","The number of partial rounds for this specification.","","","The S-box for this specification.","","","Side-loaded index of the first correct and secure MDS that …","","","","","","","","","","","","","","","","","","","","","","","","The largest integer such that $2^c \\\\leq (r_P - 1) / 2$, …","A domain in which $\\\\mathsf{SinsemillaCommit}$ and …","A domain in which $\\\\mathsf{SinsemillaHashToPoint}$ and …","$\\\\frac{1}{2^K}$","Number of bits of each message piece in …","Returns the Sinsemilla $Q$ constant for this domain.","Returns the Sinsemilla $Q$ constant for this domain.","SWU hash-to-curve personalization for Sinsemilla $Q$ …","Returns the Sinsemilla $R$ constant for this domain.","The precomputed bases for the Sinsemilla hash function.","SWU hash-to-curve personalization for Sinsemilla $S$ …","","","","","","","$\\\\mathsf{SinsemillaCommit}$ from § 5.4.8.4.","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","$\\\\mathsf{SinsemillaHash}$ from § 5.4.1.9.","$\\\\mathsf{SinsemillaHashToPoint}$ from § 5.4.1.9.","","","Calls U::from(self).","Calls U::from(self).","Constructs a new HashDomain with a specific prefix string.","Constructs a new CommitDomain with a specific prefix …","$\\\\mathsf{SinsemillaShortCommit}$ from § 5.4.8.4.","","","","","","","","","","","The size of a SHA-256 block, in 32-bit words.","A word in a Table16 message block.","Variable representing a 32-bit word of the input block to …","A gadget that constrains a SHA-256 invocation. It supports …","The output of a SHA-256 circuit invocation.","The set of circuit instructions required to use the Sha256 …","Variable representing the SHA-256 internal state.","A chip that implements SHA-256 with a maximum lookup table …","Configuration for a Table16Chip.","","","","","","","","","","","","","","","","","Starting from the given initialized state, processes a …","","","Configures a circuit to include this chip.","Reconstructs this chip from the given config.","","","","","","","","","","","","Converts the given state into a message digest.","","Convenience function to compute hash of the data. It will …","","","","","","Retrieve result and consume hasher instance.","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","","","","","Creates an initial state from the output state of a …","","Places the SHA-256 IV in the circuit, returning the …","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Loads the lookup table required by this chip into the …","","Create a new hasher instance.","","","","","","","","","","","","","","","","","","","Digest data, updating the internal state.","","","","","","A variable in the circuit.","Gadget representing a domain in which …","Trait allowing circuit’s Sinsemilla CommitDomains to be …","CommitDomains used in this instruction.","A type enumerating the fixed points used in CommitDomains.","A domain in which $\\\\mathsf{SinsemillaHashToPoint}$ and …","Trait allowing circuit’s Sinsemilla HashDomains to be …","HashDomains used in this instruction.","A message to be hashed.","A message composed of Self::MessagePieces.","A message piece with a bitlength of some multiple of K.","A piece in a message containing a number of K-bit words. A …","A point output of Self::hash_to_point.","Returns the Q constant for this domain.","A cumulative sum z is used to decompose a Sinsemilla …","The set of circuit instructions required to use the …","The x-coordinate of a point output of Self::hash_to_point.","","","","","","","","","Chip implementations for the Sinsemilla gadgets.","","","","","$\\\\mathsf{SinsemillaCommit}$ from § 5.4.8.4.","","","","","","","","","","","","","Extracts the x-coordinate of the output of a Sinsemilla …","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Constructs a MessagePiece from a field element.","Constructs a message from a vector of MessagePieces.","$\\\\mathsf{SinsemillaHash}$ from § 5.4.1.9.","Returns the HashDomain contained in this CommitDomain","Hashes a message to an ECC curve point. This returns both …","$\\\\mathsf{SinsemillaHashToPoint}$ from § 5.4.1.9.","","","","","Returns the inner MessagePiece contained in this gadget.","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Gadgets for implementing a Merkle tree with Sinsemilla.","Constructs a new HashDomain for the given domain.","Constructs a new CommitDomain for the given domain.","Returns the fixed point corresponding to the R constant …","$\\\\mathsf{SinsemillaShortCommit}$ from § 5.4.8.4.","","","","","","","","","","","","","","","","","","","Witness a message piece given a field element. Returns a …","A chip that implements 10-bit Sinsemilla using a lookup …","Configuration for the Sinsemilla hash chip","","","","","","","","","","Side-effects","Reconstructs this chip from the given config.","","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","","Calls U::from(self).","Calls U::from(self).","Loads the lookup table required by this chip into the …","","Returns the lookup range check config used in this config.","","","","","","","","","","","","","","SWU hash-to-curve personalization for the Merkle CRH …","Instructions to check the validity of a Merkle path of a …","Gadget representing a Merkle path that proves a leaf …","","","Calculates the root of the tree containing the given leaf …","Chip implementing a Merkle hash using Sinsemilla as the …","","","Constructs a MerklePath.","","","","","Returns the argument unchanged.","Compute MerkleCRH for a given layer. The hash that …","","Calls U::from(self).","","","","","","Chip implementing MerkleInstructions.","Configuration for the MerkleChip implementation.","","","","","","","","","","Configures the MerkleChip.","Constructs a MerkleChip given a MerkleConfig.","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","","","Calls U::from(self).","Calls U::from(self).","","","","","","","","","","","","","","Trait for utilities used across circuits.","Trait for a variable in the circuit.","Variable in the circuit.","Takes a specified subsequence of the little-endian bit …","Checks that an expression is either 1 or 0.","The cell at which this variable was allocated.","Gadget and chip for a conditional swap utility.","Decomposes an $n$-bit field element $\\\\alpha$ into $W$ …","Decompose a word alpha into window_num_bits bits …","The sequence of bits representing a u64 in little-endian …","The u64 integer represented by an L-bit little-endian …","Load a variable.","Make use of a K-bit lookup table to decompose a field …","Check that an expression is in the small range [0..range), …","If a then b, else c. Returns (a * b) + (1 - a) * c.","The value allocated to this variable.","A chip implementing a conditional swap.","Configuration for the CondSwapChip.","Instructions for a conditional swap gadget.","","","","","","","","","","Configures this chip for use in a circuit.","Constructs a CondSwapChip given a CondSwapConfig.","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","Calls U::from(self).","Calls U::from(self).","","Given an input pair (a,b) and a swap boolean flag, returns …","","","","","","","","","","","","The running sum $[z_0, …, z_W]$. If created in strict …","Configuration that provides methods for running sum …","","","","","","","perm MUST include the advice column z.","Decompose an existing variable alpha that is copied into …","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","Calls U::from(self).","Calls U::from(self).","","","","","","","","","","","Decompose a field element alpha that is witnessed in this …","Configuration that provides methods for a lookup range …","The running sum $[z_0, …, z_W]$. If created in strict …","","","","","","","The running_sum advice column breaks the field element …","Range check on an existing cell that is copied into this …","Short range check on an existing cell that is copied into …","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","Calls U::from(self).","Calls U::from(self).","","","","","","","","","","","Range check on a value that is witnessed in this helper.","Short range check on value that is witnessed in this 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module provides common utilities, traits and …","Traits and structs for implementing circuit components.","Tools for developing circuits.","This module provides an implementation of a variant of …","Contains utilities for performing arithmetic over …","This module contains utilities and traits for dealing with …","The affine version of the curve","The base field over which this elliptic curve is …","The base field over which this elliptic curve is …","CURVE_ID used for hash-to-curve.","The affine coordinates of a point on an elliptic curve.","This trait is the affine counterpart to Curve and is used …","This trait is a common interface for dealing with elements …","The projective form of the curve","Generator of the $t-order$ multiplicative subgroup","This trait represents an element of a field.","This trait is a common interface for dealing with elements …","This represents an element of a group with basic …","Modulus of the field written as a string for display …","Inverse of PrimeField::root_of_unity()","The group is assumed to be of prime order $p$. Scalar is …","The scalar field of this elliptic curve.","The scalar field of this elliptic curve.","A trait that exposes additional operations related to …","Tables used for square root computation.","Inverse of $2$ in the field.","The value $(T-1)/2$ such that $2^S \\\\cdot T = p - 1$ with …","Element of multiplicative order $3$.","Returns the curve constant a.","Returns the curve constant $a$.","Returns the curve constant b.","Returns the curve constant $b$.","Performs a radix-$2$ Fast-Fourier Transformation (FFT) on …","Performs a multi-exponentiation operation.","","","","","","","This computes the inner product of two vectors a and b.","","Gets the coordinates of this point.","Cubes this element.","","","","","","Doubles this element.","","","Apply the curve endomorphism by multiplying the …","This evaluates a provided polynomial (in coefficient form) …","","","Returns the argument unchanged.","Returns the argument unchanged.","Obtains a field element that is congruent to the provided …","Obtains a field element congruent to the integer v.","Obtains a point given $(x, y)$, failing if it is not on the","Obtains a Coordinates value given $(x, y)$, failing if it …","Gets the lower 128 bits of this field element when …","Gets the lower 32 bits of this field element when expressed","Adds rhs to this group element.","Scales this group element by a scalar.","Subtracts rhs from this group element.","Returns the additive identity of the group.","Requests a hasher that accepts messages and returns …","","","Calls U::from(self).","Calls U::from(self).","Computes the multiplicative inverse of this element, …","Returns whether or not this element is on the curve; should","Returns whether or not this element is on the curve; should","Returns true iff this element is zero.","Returns true iff this element is zero.","Return the Jacobian coordinates of this point.","Divides polynomial a in X by X - b with no remainder.","Returns coefficients of an n - 1 degree polynomial given a …","Build tables given parameters for the perfect hash.","Obtains a point given Jacobian coordinates $X : Y : Z$, …","Returns the one element of the field, the multiplicative …","This simple utility function will parallelize an operation …","Exponentiates self by by, where by is a little-endian order","Raise this field element to the power Self::T_MINUS1_OVER2.","Exponentiates self by exp, where exp is a little-endian …","Returns an element chosen uniformly at random using a …","Performs a small multi-exponentiation operation. Uses the …","Returns the square root of the field element, if it is …","Equivalent to Self::sqrt_ratio(self, one()).","Same as sqrt_ratio(u, one()) but more efficient.","Computes:","Computes:","Squares this element.","","","","","","","","Returns the u-coordinate.","Returns the v-coordinate.","","","Returns the x-coordinate.","Returns the y-coordinate.","Returns the zero element of the field, the additive …","An assigned cell.","A pointer to a cell within a circuit.","A chip implements a set of instructions that can be used …","A type that holds the configuration for this chip, and any …","A layout strategy within a circuit. The layouter is …","A type that holds any general chip state that needs to be …","This is a “namespaced” layouter which borrows a …","A region of the circuit in which a Chip can assign cells.","Index of a region in a layouter","Starting row of a region in a layouter","Represents the type of the “root” of this layouter, so …","A simple FloorPlanner that performs minimal optimizations.","A lookup table in the circuit.","Assign an advice column value (witness).","Assigns a constant value to the column advice at offset …","Assign the value of the instance column’s cell at …","Assigns a fixed value to a table cell.","Assign a fixed value.","Assign a region of gates to an absolute row number.","","Assign a table region to an absolute row number.","","","","","","","","","","","","","","","","","","Returns the cell.","","","","","","","","","The chip holds its own configuration.","Constrains a cell to have a constant value.","Constrains two cells to have the same value.","Constrains a Cell to equal an instance column’s row …","","Copies the value to a given advice cell and constrains …","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Evaluates this assigned cell’s value directly, …","Implementations of common circuit floor planners.","","","","","","","","","Returns the argument unchanged.","","Returns the argument unchanged.","Returns the argument unchanged.","","Returns the argument unchanged.","Returns the argument unchanged.","","Returns the argument unchanged.","","Returns the argument unchanged.","Returns the argument unchanged.","Gets the “root” of this assignment, bypassing the …","","","","","","","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Implementations of common circuit layouters.","Provides access to general chip state loaded at the …","Enters into a namespace.","","Exits out of the existing namespace.","","Creates a new (sub)namespace and enters into it.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Returns the value of the AssignedCell.","Returns the field element value of the AssignedCell.","","","","","","","","","The version 1 FloorPlanner provided by halo2.","A single pass of the V1 layouter.","","","","","","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","","Calls U::from(self).","Calls U::from(self).","","","","","","","","","","","","Concrete column","The virtual column involved in a region. This includes …","Helper trait for implementing a custom Layouter.","The shape of a region. For a region at a certain index, we …","Virtual column representing a (boolean) selector","Helper trait for implementing a custom Layouter.","Assign an advice column value (witness)","","Assigns a constant value to the column advice at offset …","","Assign the value of the instance column’s cell at …","","Assigns a fixed value to a table cell.","Assign a fixed value","","","","","","","","","","","Get a reference to the set of columns used in a RegionShape…","Constrains a cell to have a constant value.","","Constraint two cells to have the same value.","","","","","","","","Enables a selector at the given offset.","","","","","Returns the argument unchanged.","","","Returns the argument unchanged.","","","","Calls U::from(self).","Calls U::from(self).","","Create a new RegionShape for a region at region_index.","","Get the region_index of a RegionShape.","Get the row_count of a RegionShape.","","","","","","","","","","","","","A cell used in an active gate was not assigned to.","A struct for collecting and displaying the gates within a …","Graphical renderer for circuit layouts.","A constraint was not satisfied for a particular row.","A constraint was active on an unusable row, and is likely …","The location within the circuit at which a particular …","A location inside a region.","A lookup input did not exist in its corresponding table.","A test prover for debugging circuits.","A location outside of a region.","A permutation did not preserve the original value of a …","The reasons why a particular circuit is not satisfied.","Panics if the circuit being checked by this MockProver is …","","","","","","","","","","","","","Builds a dot graph string representing the given circuit.","Collects the gates from within the circuit.","","Developer tools for investigating the cost of a circuit.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Marks cells involved in equality constraints, in red.","Metadata about circuits.","","","","","Prints the queries in this circuit to a CSV grid.","","Renders the given circuit on the given drawing area.","Runs a synthetic keygen-and-prove operation on the given …","Draws red lines between equality-constrained cells.","Sets the visibility of region labels.","","","","","","","","","","","","","","","","","","","Returns Ok(()) if this MockProver is satisfied, or a list …","Sets the view height for this layout, as a number of rows.","Sets the view width for this layout, as a number of …","","","","","","The offset (relative to the start of the region) at which …","The region in which the failure occurred.","The circuit row on which the failure occurred.","The values of the virtual cells used by this constraint.","The column in which this cell should be assigned.","The column in which this permutation is not satisfied.","The polynomial constraint that is not satisfied.","The polynomial constraint that is not satisfied.","The index of the active gate.","The offset (relative to the start of the region) at which …","The location at which this constraint is not satisfied.","The location at which the lookup is not satisfied.","The location at which the permutation is not satisfied.","The index of the lookup that is not satisfied. These …","The offset (relative to the start of the region) at which …","The region in which this cell should be assigned.","Measures a circuit to determine its costs, and explain …","The marginal size of a Halo 2 proof, broken down into its …","The size of a Halo 2 proof, broken down into its …","","","","","","","","","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Returns the marginal proof size per instance of this …","Measures a circuit with parameter constant k.","Returns the proof size for the given number of instances …","","","","","","","","","","","","","Metadata about a column within a circuit.","Metadata about a configured constraint within a circuit.","Metadata about a configured gate within a circuit.","Metadata about an assigned region within a circuit.","A “virtual cell” is a PLONK cell that has been queried …","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","","Returns the argument unchanged.","","Returns the argument unchanged.","Returns the argument unchanged.","","","","","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","An advice column","An Advice variant","This is an advice (witness) column queried at a certain …","An enum over the Advice, Fixed, Instance structs","A value assigned to a cell within a circuit.","This trait allows a Circuit to direct some backend to …","A verifier that checks multiple proofs in a batch.","Out of bounds index passed to a backend","This is a trait that circuits provide implementations for …","A column with an index and type","The instance sets up a copy constraint involving a column …","A column type","This is a configuration object that stores things like …","This is a constant polynomial","An individual polynomial constraint.","This is a description of the circuit environment, such as …","The constraint system is not satisfied.","A set of polynomial constraints with a common selector.","This is an error that could occur during proving or …","Low-degree expression representing an identity that must …","A fixed column","A Fixed variant","This is a fixed column queried at a certain relative …","A floor planning strategy for a circuit.","The floor planner used for this circuit. This is an …","An instance column","An Instance variant","This is an instance (external) column queried at a certain …","Instance provided exceeds number of available rows","The provided instances do not match the circuit parameters.","This is a negated polynomial","Circuit synthesis requires global constants, but circuit …","k is too small for the given circuit.","Opening error","The output type of this verification strategy after …","Represents the minimal parameters that determine a …","Minimal representation of a verification key that can be …","This is the product of two polynomials","This is a proving key which allows for the creation of …","A value stored as a fraction to enable batch inversion.","This is a scaled polynomial","A selector, representing a fixed boolean value per row of …","This is a virtual selector","A verifier that checks a single proof at a time.","This is the sum of two polynomials","This is an error that can occur during synthesis of the …","A fixed column of a lookup table.","Transcript error","A value that does not require inversion to evaluate.","Trait representing a strategy for verifying Halo 2 proofs.","This is a verifying key which allows for the verification …","Exposes the “virtual cells” that can be queried while …","The field element zero.","","","","","","","Allocate a new advice column","Assign an advice column value (witness)","Assign a fixed value","Compute the number of blinding factors necessary to …","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Type of this column.","Allocate a new complex selector that can appear anywhere …","The circuit is given an opportunity to describe the exact …","Assign two cells to have the same value","Creates a new gate.","This creates a proof for the provided circuit when given …","Cubes this element.","","Compute the degree of this polynomial","Compute the degree of the constraint system (the maximum …","Returns the denominator, if non-trivial.","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Doubles this element.","","","","","","","","","","","","","","","","","","","","","Enable this selector at the given offset within the given …","Enables this fixed column to be used for global constant …","Enable the ability to enforce equality over cells in this …","Enables a selector at the given row.","Creates a new region and enters into it.","","","","","","","","","Evaluates this assigned value directly, performing an …","Evaluate the polynomial using the provided closures to …","Exits the current region.","Fills a fixed column starting from the given row with …","Finalizes the batch and checks its validity.","Allocate a new fixed column","","","","","","","","","","","","","","","","","","","","","","","","","Returns the argument unchanged.","","","Returns the argument unchanged.","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","Returns the argument unchanged.","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Get the underlying EvaluationDomain.","Get the underlying VerifyingKey.","","","","","","","","Hashes a verification key into a transcript.","","","","","","","","","","","","","","","","","","","","","Allocate a new instance column","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","","Inverts this assigned value (taking the inverse of zero to …","Is this selector “simple”? Simple selectors can only …","Returns true iff this element is zero.","Generate a ProvingKey from a VerifyingKey and an instance …","Generate a VerifyingKey from an instance of Circuit.","Add a lookup argument for some input expressions and table …","Allocates a new fixed column that can be used in a lookup …","Returns the minimum necessary rows that need to exist in …","","","","","","","","","","","","","Constructs a new single proof verifier.","Constructs a new batch verifier.","Returns the numerator.","","","Obtain a pinned version of this constraint system; a …","Obtains a pinned representation of this verification key …","Exits out of the existing namespace.","Obtains an MSM from the verifier strategy and yields back …","","","Creates a new (sub)namespace and enters into it.","Query an advice column at a relative position","Query an Any column at a relative position","Query a fixed column at a relative position","Queries the cell of an instance column at a particular …","Query an instance column at a relative position","Query a selector at the current position.","Allocate a new (simple) selector. Simple selectors cannot …","Sets the minimum degree required by the circuit, which can …","","Squares this element.","Square this expression.","","","","","","","Given the provided cs, synthesize the given circuit.","Given the provided cs, synthesize the circuit. The …","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","Returns a boolean indicating whether or not the proof is …","","","","","","","","","","","","","","","","","","","","","Constructs a set of constraints that are controlled by the …","Returns a copy of this circuit with no witness values …","","","","","","The current value of k being used.","","","","","","","","","","Column index","Column index","Column index","Query index","Query index","Query index","Rotation of this query","Rotation of this query","Rotation of this query","","The basis over which a polynomial is described.","The polynomial is defined as coefficients","This is an error that could occur during proving or …","This structure contains precomputed constants and other …","The polynomial is defined as coefficients of Lagrange …","The polynomial is defined as coefficients of Lagrange …","OpeningProof is not well-formed","Represents the minimal parameters that determine an …","Represents a univariate polynomial defined over a field …","Describes the relative rotation of a vector. Negative …","Caller needs to re-sample a point","","","","","","","","","","","","","","","","","","","","","","","","","","","","Obtains a polynomial in coefficient form when given a …","This takes us from an n-length coefficient vector into a …","This module contains an implementation of the polynomial …","Returns a constant polynomial in the extended Lagrange …","Returns a constant polynomial in the Lagrange coefficient …","The current location in the evaluation domain","","","","","","","","","","","","","","","","","","","This divides the polynomial (in the extended domain) by …","","","","","","","","","Returns an empty (zero) polynomial in the coefficient basis","Returns an empty (zero) polynomial in the extended …","Returns an empty (zero) polynomial in the Lagrange …","","Get the size of the extended domain","This takes us from the extended evaluation domain and gets …","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Get the generator of the extended domain’s …","Get $\\\\omega$, the generator of the $2^k$ order …","Get $\\\\omega^{-1}$, the inverse of the generator of the …","Gets the quotient polynomial’s degree (as a multiple of …","","","","","","","","","","","","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Iterate over the values, which are either in coefficient …","Iterate over the values mutably, which are either in …","Computes evaluations (at the point x, where xn = x^n) of …","Obtains a polynomial in Lagrange form when given a vector …","This takes us from an n-length vector into the coefficient …","","This module contains an optimisation of the polynomial …","","This constructs a new evaluation domain object based on …","The next location in the evaluation domain","Gets the size of this polynomial in terms of the number of …","Obtain a pinned version of this evaluation domain; a …","The previous location in the evaluation domain","Rotates the values in a Lagrange basis polynomial by …","Rotate the extended domain polynomial over the original …","Multiplies a value by some power of $\\\\omega$, essentially …","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","An accumulator instance consisting of an evaluation claim …","Wrapper type around a blinding factor.","A guard returned by the verifier","A multiscalar multiplication in the polynomial commitment …","These are the public parameters for the polynomial …","","","","Add a value to the first entry of g_scalars.","Add another multiexp into this one","Add a vector of scalars to g_scalars. This function will …","Add to u_scalar","Add to w_scalar","Add arbitrary term (the scalar and the point)","","","","","","","","","","","","","","","","","","","This computes a commitment to a polynomial described by …","This commits to a polynomial using its evaluations over …","Computes G = ⟨s, params.g⟩","Create a polynomial commitment opening proof for the …","","","","","","","","","","","","","","","","","Generates an empty multiscalar multiplication struct using …","","Perform multiexp and check that it results in zero","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","The claimed output of the linear-time polycommit opening …","Getter for g generators","","","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","","","","","Create a new, empty MSM using the provided parameters.","Initializes parameters for the curve, given a random …","Reads params from a buffer.","Scale all scalars in the MSM by some scaling factor","","","","","","","","","","","","","","","","","","","","A vector of challenges u_0, …, u_{k - 1} sampled by the …","Lets caller supply the challenges and obtain an MSM with …","Lets caller supply the purported G point and simply appends","Checks to see if the proof represented within transcript …","","","","","","Writes params to a buffer.","A polynomial query at a point","A polynomial query at a point","blinding factor of polynomial","","","","","","","","","Create a multi-opening proof","","","","","","","","","Returns the argument unchanged.","Returns the argument unchanged.","","","Calls U::from(self).","Calls U::from(self).","Create a new verifier query based on a commitment","Create a new verifier query based on a linear combination …","point at which polynomial is queried","coefficients of polynomial","","","","","","","","","Verify a multi-opening proof","","","We will replace BLAKE2b with an algebraic hash function in …","We will replace BLAKE2b with an algebraic hash function in …","A 255-bit challenge.","The scalar representation of a verifier challenge.","EncodedChallenge<C> defines a challenge encoding with a …","The Input type used to derive the challenge encoding. For …","Generic transcript view (from either the prover or verifier…","Transcript view from the perspective of a verifier that …","Transcript view from the perspective of a prover that has …","Cast an encoded challenge as a typed ChallengeScalar.","","","","","","","","","","","","","","","","","Writing the point to the transcript without writing it to …","","","Writing the scalar to the transcript without writing it to …","","","","","","","","","","","","","","","","","Conclude the interaction and return the output buffer …","","","","","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Returns the argument unchanged.","Get a scalar field element from an encoded challenge.","","Initialize a transcript given an input buffer.","","Initialize a transcript given an output buffer.","","","","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Calls U::from(self).","Get an encoded challenge from a given input challenge.","","Read a curve point from the prover.","","Read a curve scalar from the prover.","","Squeeze an encoded verifier challenge from the transcript.","","","Squeeze a typed challenge (in the scalar field) from the …","","","","","","","","","","","","","","","","","","","","","","Write a curve point to the proof and the transcript.","","Write a scalar to the proof and the 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res=buildHrefAndPath(obj);obj.displayPath=pathSplitter(res[0]);obj.fullPath=obj.displayPath+obj.name;obj.fullPath+="|"+obj.ty;if(duplicates[obj.fullPath]){continue}duplicates[obj.fullPath]=true;obj.href=res[1];out.push(obj);if(out.length>=MAX_RESULTS){break}}}return out}function sortResults(results,isType){var userQuery=parsedQuery.userQuery;var ar=[];for(var entry in results){if(hasOwnPropertyRustdoc(results,entry)){var result=results[entry];result.word=searchWords[result.id];result.item=searchIndex[result.id]||{};ar.push(result)}}results=ar;if(results.length===0){return[]}results.sort(function(aaa,bbb){var a,b;a=(aaa.word!==userQuery);b=(bbb.word!==userQuery);if(a!==b){return a-b}a=(aaa.lev);b=(bbb.lev);if(a!==b){return a-b}a=(aaa.item.crate!==window.currentCrate);b=(bbb.item.crate!==window.currentCrate);if(a!==b){return a-b}a=aaa.word.length;b=bbb.word.length;if(a!==b){return a-b}a=aaa.word;b=bbb.word;if(a!==b){return(a>b?+1:-1)}a=(aaa.index<0);b=(bbb.index<0);if(a!==b){return a-b}a=aaa.index;b=bbb.index;if(a!==b){return a-b}if((aaa.item.ty===TY_PRIMITIVE&&bbb.item.ty!==TY_KEYWORD)||(aaa.item.ty===TY_KEYWORD&&bbb.item.ty!==TY_PRIMITIVE)){return-1}if((bbb.item.ty===TY_PRIMITIVE&&aaa.item.ty!==TY_PRIMITIVE)||(bbb.item.ty===TY_KEYWORD&&aaa.item.ty!==TY_KEYWORD)){return 1}a=(aaa.item.desc==="");b=(bbb.item.desc==="");if(a!==b){return a-b}a=aaa.item.ty;b=bbb.item.ty;if(a!==b){return a-b}a=aaa.item.path;b=bbb.item.path;if(a!==b){return(a>b?+1:-1)}return 0});var nameSplit=null;if(parsedQuery.elems.length===1){var hasPath=typeof parsedQuery.elems[0].path==="undefined";nameSplit=hasPath?null:parsedQuery.elems[0].path}for(var i=0,len=results.length;i0&&row[GENERICS_DATA][0][NAME]===""){if(row.length>GENERICS_DATA){return checkGenerics(row[GENERICS_DATA][0],elem,defaultLev)}return elem.generics.length===0?defaultLev:MAX_LEV_DISTANCE+1}var elem_name;if(elem.generics.length>0&&row[GENERICS_DATA].length>=elem.generics.length){var elems=Object.create(null);for(var x=0,length=row[GENERICS_DATA].length;xGENERICS_DATA){return checkIfInGenerics(row,elem)}return MAX_LEV_DISTANCE+1}var lev=levenshtein(row[NAME],elem.name);if(literalSearch){if(lev!==0){if(elem.generics.length===0){var checkGeneric=(row.length>GENERICS_DATA&&row[GENERICS_DATA].length>0);if(checkGeneric&&row[GENERICS_DATA].findIndex(function(tmp_elem){return tmp_elem[NAME]===elem.name})!==-1){return 0}}return MAX_LEV_DISTANCE+1}else if(elem.generics.length>0){return checkGenerics(row,elem,MAX_LEV_DISTANCE+1)}return 0}else if(row.length>GENERICS_DATA){if(elem.generics.length===0){if(lev===0){return 0}lev=checkIfInGenerics(row,elem);return lev+0.5}else if(lev>MAX_LEV_DISTANCE){return checkIfInGenerics(row,elem)}else{var tmp_lev=checkGenerics(row,elem,lev);if(tmp_lev>MAX_LEV_DISTANCE){return MAX_LEV_DISTANCE+1}return(tmp_lev+lev)/2}}else if(elem.generics.length>0){return MAX_LEV_DISTANCE+1}return lev}function findArg(row,elem,typeFilter){var lev=MAX_LEV_DISTANCE+1;if(row&&row.type&&row.type[INPUTS_DATA]&&row.type[INPUTS_DATA].length>0){var length=row.type[INPUTS_DATA].length;for(var i=0;iOUTPUT_DATA){var ret=row.type[OUTPUT_DATA];if(typeof ret[0]==="string"){ret=[ret]}for(var x=0,len=ret.length;xlength){return MAX_LEV_DISTANCE+1}for(var i=0;ilength){break}var lev_total=0;var aborted=false;for(var x=0;xMAX_LEV_DISTANCE){aborted=true;break}lev_total+=lev}if(!aborted){ret_lev=Math.min(ret_lev,Math.round(lev_total/clength))}}return ret_lev}function typePassesFilter(filter,type){if(filter<=NO_TYPE_FILTER||filter===type)return true;var name=itemTypes[type];switch(itemTypes[filter]){case"constant":return name==="associatedconstant";case"fn":return name==="method"||name==="tymethod";case"type":return name==="primitive"||name==="associatedtype";case"trait":return name==="traitalias"}return false}function createAliasFromItem(item){return{crate:item.crate,name:item.name,path:item.path,desc:item.desc,ty:item.ty,parent:item.parent,type:item.type,is_alias:true,}}function handleAliases(ret,query,filterCrates){var lowerQuery=query.toLowerCase();var aliases=[];var crateAliases=[];if(filterCrates!==null){if(ALIASES[filterCrates]&&ALIASES[filterCrates][lowerQuery]){var query_aliases=ALIASES[filterCrates][lowerQuery];var len=query_aliases.length;for(var i=0;iMAX_RESULTS){ret.others.pop()}};onEach(aliases,pushFunc);onEach(crateAliases,pushFunc)}function addIntoResults(results,fullId,id,index,lev){if(lev===0||(!parsedQuery.literalSearch&&lev<=MAX_LEV_DISTANCE)){if(results[fullId]!==undefined){var result=results[fullId];if(result.dontValidate||result.lev<=lev){return}}results[fullId]={id:id,index:index,dontValidate:parsedQuery.literalSearch,lev:lev,}}}function handleSingleArg(row,pos,elem,results_others,results_in_args,results_returned){if(!row||(filterCrates!==null&&row.crate!==filterCrates)){return}var lev,lev_add=0,index=-1;var fullId=row.id;var in_args=findArg(row,elem,parsedQuery.typeFilter);var returned=checkReturned(row,elem,parsedQuery.typeFilter);addIntoResults(results_in_args,fullId,pos,index,in_args);addIntoResults(results_returned,fullId,pos,index,returned);if(!typePassesFilter(parsedQuery.typeFilter,row.ty)){return}var searchWord=searchWords[pos];if(parsedQuery.literalSearch){if(searchWord===elem.name){addIntoResults(results_others,fullId,pos,-1,0)}return}if(elem.name.length===0){if(row.type!==null){lev=checkGenerics(row.type,elem,MAX_LEV_DISTANCE+1);addIntoResults(results_others,fullId,pos,index,lev)}return}if(elem.fullPath.length>1){lev=checkPath(elem.pathWithoutLast,elem.pathLast,row);if(lev>MAX_LEV_DISTANCE||(parsedQuery.literalSearch&&lev!==0)){return}else if(lev>0){lev_add=lev/10}}if(searchWord.indexOf(elem.pathLast)>-1||row.normalizedName.indexOf(elem.pathLast)>-1){if(!results_others[fullId]!==undefined){index=row.normalizedName.indexOf(elem.pathLast)}}lev=levenshtein(searchWord,elem.pathLast);lev+=lev_add;if(lev>0&&elem.pathLast.length>2&&searchWord.indexOf(elem.pathLast)>-1){if(elem.pathLast.length<6){lev=1}else{lev=0}}if(lev>MAX_LEV_DISTANCE){return}else if(index!==-1&&elem.fullPath.length<2){lev-=1}if(lev<0){lev=0}addIntoResults(results_others,fullId,pos,index,lev)}function handleArgs(row,pos,results){if(!row||(filterCrates!==null&&row.crate!==filterCrates)){return}var totalLev=0;var nbLev=0;var lev;function checkArgs(elems,callback){for(var i=0,len=elems.length;i0){var container=results_others;if(parsedQuery.returned.length!==0&&parsedQuery.elems.length===0){container=results_returned}for(i=0,nSearchWords=searchWords.length;i-1||path.indexOf(keys[i])>-1||(parent!==undefined&&parent.name!==undefined&&parent.name.toLowerCase().indexOf(keys[i])>-1)||levenshtein(name,keys[i])<=MAX_LEV_DISTANCE)){return false}}return true}function nextTab(direction){var next=(searchState.currentTab+direction+3)%searchState.focusedByTab.length;searchState.focusedByTab[searchState.currentTab]=document.activeElement;printTab(next);focusSearchResult()}function focusSearchResult(){var target=searchState.focusedByTab[searchState.currentTab]||document.querySelectorAll(".search-results.active a").item(0)||document.querySelectorAll("#titles > button").item(searchState.currentTab);if(target){target.focus()}}function buildHrefAndPath(item){var displayPath;var href;var type=itemTypes[item.ty];var name=item.name;var path=item.path;if(type==="mod"){displayPath=path+"::";href=window.rootPath+path.replace(/::/g,"/")+"/"+name+"/index.html"}else if(type==="primitive"||type==="keyword"){displayPath="";href=window.rootPath+path.replace(/::/g,"/")+"/"+type+"."+name+".html"}else if(type==="externcrate"){displayPath="";href=window.rootPath+name+"/index.html"}else if(item.parent!==undefined){var myparent=item.parent;var anchor="#"+type+"."+name;var parentType=itemTypes[myparent.ty];var pageType=parentType;var pageName=myparent.name;if(parentType==="primitive"){displayPath=myparent.name+"::"}else if(type==="structfield"&&parentType==="variant"){var enumNameIdx=item.path.lastIndexOf("::");var enumName=item.path.substr(enumNameIdx+2);path=item.path.substr(0,enumNameIdx);displayPath=path+"::"+enumName+"::"+myparent.name+"::";anchor="#variant."+myparent.name+".field."+name;pageType="enum";pageName=enumName}else{displayPath=path+"::"+myparent.name+"::"}href=window.rootPath+path.replace(/::/g,"/")+"/"+pageType+"."+pageName+".html"+anchor}else{displayPath=item.path+"::";href=window.rootPath+item.path.replace(/::/g,"/")+"/"+type+"."+name+".html"}return[displayPath,href]}function escape(content){var h1=document.createElement("h1");h1.textContent=content;return h1.innerHTML}function pathSplitter(path){var tmp=""+path.replace(/::/g,"::");if(tmp.endsWith("")){return tmp.slice(0,tmp.length-6)}return tmp}function addTab(array,query,display){var extraClass="";if(display===true){extraClass=" active"}var output=document.createElement("div");var length=0;if(array.length>0){output.className="search-results "+extraClass;array.forEach(function(item){var name=item.name;var type=itemTypes[item.ty];length+=1;var extra="";if(type==="primitive"){extra=" (primitive type)"}else if(type==="keyword"){extra=" (keyword)"}var link=document.createElement("a");link.className="result-"+type;link.href=item.href;var wrapper=document.createElement("div");var resultName=document.createElement("div");resultName.className="result-name";if(item.is_alias){var alias=document.createElement("span");alias.className="alias";var bold=document.createElement("b");bold.innerText=item.alias;alias.appendChild(bold);alias.insertAdjacentHTML("beforeend"," - see ");resultName.appendChild(alias)}resultName.insertAdjacentHTML("beforeend",item.displayPath+""+name+extra+"");wrapper.appendChild(resultName);var description=document.createElement("div");description.className="desc";var spanDesc=document.createElement("span");spanDesc.insertAdjacentHTML("beforeend",item.desc);description.appendChild(spanDesc);wrapper.appendChild(description);link.appendChild(wrapper);output.appendChild(link)})}else if(query.error===null){output.className="search-failed"+extraClass;output.innerHTML="No results :(
    "+"Try on DuckDuckGo?

    "+"Or try looking in one of these:"}return[output,length]}function makeTabHeader(tabNb,text,nbElems){if(searchState.currentTab===tabNb){return""}return""}function showResults(results,go_to_first,filterCrates){var search=searchState.outputElement();if(go_to_first||(results.others.length===1&&getSettingValue("go-to-only-result")==="true"&&(!search.firstChild||search.firstChild.innerText!==searchState.loadingText))){var elem=document.createElement("a");elem.href=results.others[0].href;removeClass(elem,"active");document.body.appendChild(elem);elem.click();return}if(results.query===undefined){results.query=parseQuery(searchState.input.value)}currentResults=results.query.userQuery;var ret_others=addTab(results.others,results.query,true);var ret_in_args=addTab(results.in_args,results.query,false);var ret_returned=addTab(results.returned,results.query,false);var currentTab=searchState.currentTab;if((currentTab===0&&ret_others[1]===0)||(currentTab===1&&ret_in_args[1]===0)||(currentTab===2&&ret_returned[1]===0)){if(ret_others[1]!==0){currentTab=0}else if(ret_in_args[1]!==0){currentTab=1}else if(ret_returned[1]!==0){currentTab=2}}let crates="";if(window.ALL_CRATES.length>1){crates=` in `}var typeFilter="";if(results.query.typeFilter!==NO_TYPE_FILTER){typeFilter=" (type: "+escape(itemTypes[results.query.typeFilter])+")"}var output=`
    `+`

    Results for ${escape(results.query.userQuery)}`+`${typeFilter}

    in ${crates}
    `;if(results.query.error!==null){output+=`

    Query parser error: "${results.query.error}".

    `}output+=`
    `+makeTabHeader(0,"In Names",ret_others[1])+makeTabHeader(1,"In Parameters",ret_in_args[1])+makeTabHeader(2,"In Return Types",ret_returned[1])+"
    ";var resultsElem=document.createElement("div");resultsElem.id="results";resultsElem.appendChild(ret_others[0]);resultsElem.appendChild(ret_in_args[0]);resultsElem.appendChild(ret_returned[0]);search.innerHTML=output;let crateSearch=document.getElementById("crate-search");if(crateSearch){crateSearch.addEventListener("input",updateCrate)}search.appendChild(resultsElem);searchState.focusedByTab=[null,null,null];searchState.showResults(search);var elems=document.getElementById("titles").childNodes;elems[0].onclick=function(){printTab(0)};elems[1].onclick=function(){printTab(1)};elems[2].onclick=function(){printTab(2)};printTab(currentTab)}function search(e,forced){var params=searchState.getQueryStringParams();var query=parseQuery(searchState.input.value.trim());if(e){e.preventDefault()}if(!forced&&query.userQuery===currentResults){if(query.userQuery.length>0){putBackSearch()}return}var filterCrates=getFilterCrates();if(filterCrates===null&¶ms["filter-crate"]!==undefined){filterCrates=params["filter-crate"]}searchState.title="Results for "+query.original+" - Rust";if(searchState.browserSupportsHistoryApi()){var newURL=buildUrl(query.original,filterCrates);if(!history.state&&!params.search){history.pushState(null,"",newURL)}else{history.replaceState(null,"",newURL)}}showResults(execQuery(query,searchWords,filterCrates),params.go_to_first,filterCrates)}function buildIndex(rawSearchIndex){searchIndex=[];var searchWords=[];var i,word;var currentIndex=0;var id=0;for(var crate in rawSearchIndex){if(!hasOwnPropertyRustdoc(rawSearchIndex,crate)){continue}var crateSize=0;var crateCorpus=rawSearchIndex[crate];searchWords.push(crate);var crateRow={crate:crate,ty:1,name:crate,path:"",desc:crateCorpus.doc,parent:undefined,type:null,id:id,normalizedName:crate.indexOf("_")===-1?crate:crate.replace(/_/g,""),};id+=1;searchIndex.push(crateRow);currentIndex+=1;var itemTypes=crateCorpus.t;var itemNames=crateCorpus.n;var itemPaths=crateCorpus.q;var itemDescs=crateCorpus.d;var itemParentIdxs=crateCorpus.i;var itemFunctionSearchTypes=crateCorpus.f;var paths=crateCorpus.p;var aliases=crateCorpus.a;var len=paths.length;for(i=0;i0?paths[itemParentIdxs[i]-1]:undefined,type:itemFunctionSearchTypes[i],id:id,normalizedName:word.indexOf("_")===-1?word:word.replace(/_/g,""),};id+=1;searchIndex.push(row);lastPath=row.path;crateSize+=1}if(aliases){ALIASES[crate]={};var j,local_aliases;for(var alias_name in aliases){if(!hasOwnPropertyRustdoc(aliases,alias_name)){continue}if(!hasOwnPropertyRustdoc(ALIASES[crate],alias_name)){ALIASES[crate][alias_name]=[]}local_aliases=aliases[alias_name];for(j=0,len=local_aliases.length;j0){searchState.input.value=params.search;search(e)}else{searchState.input.value="";searchState.hideResults()}})}window.onpageshow=function(){var qSearch=searchState.getQueryStringParams().search;if(searchState.input.value===""&&qSearch){searchState.input.value=qSearch}search()}}function updateCrate(ev){if(ev.target.value==="All crates"){var params=searchState.getQueryStringParams();var query=searchState.input.value.trim();if(!history.state&&!params.search){history.pushState(null,"",buildUrl(query,null))}else{history.replaceState(null,"",buildUrl(query,null))}}currentResults=null;search(undefined,true)}searchWords=buildIndex(rawSearchIndex);registerSearchEvents();function runSearchIfNeeded(){if(searchState.getQueryStringParams().search){search()}}runSearchIfNeeded()};if(window.searchIndex!==undefined){initSearch(window.searchIndex)}})() \ No newline at end of file +(function(){const itemTypes=["mod","externcrate","import","struct","enum","fn","type","static","trait","impl","tymethod","method","structfield","variant","macro","primitive","associatedtype","constant","associatedconstant","union","foreigntype","keyword","existential","attr","derive","traitalias",];const TY_PRIMITIVE=itemTypes.indexOf("primitive");const TY_KEYWORD=itemTypes.indexOf("keyword");function printTab(nb){if(nb===0||nb===1||nb===2){searchState.currentTab=nb}let nb_copy=nb;onEachLazy(document.getElementById("titles").childNodes,function(elem){if(nb_copy===0){addClass(elem,"selected")}else{removeClass(elem,"selected")}nb_copy-=1});onEachLazy(document.getElementById("results").childNodes,function(elem){if(nb===0){addClass(elem,"active")}else{removeClass(elem,"active")}nb-=1})}const levenshtein_row2=[];function levenshtein(s1,s2){if(s1===s2){return 0}const s1_len=s1.length,s2_len=s2.length;if(s1_len&&s2_len){let i1=0,i2=0,a,b,c,c2;const row=levenshtein_row2;while(i1-".indexOf(c)!==-1}function isStopCharacter(c){return isWhitespace(c)||isEndCharacter(c)}function isErrorCharacter(c){return"()".indexOf(c)!==-1}function itemTypeFromName(typename){for(let i=0,len=itemTypes.length;i0){throw new Error("Cannot use literal search when there is more than one element")}parserState.pos+=1;const start=parserState.pos;const end=getIdentEndPosition(parserState);if(parserState.pos>=parserState.length){throw new Error("Unclosed `\"`")}else if(parserState.userQuery[end]!=="\""){throw new Error(`Unexpected \`${parserState.userQuery[end]}\` in a string element`)}else if(start===end){throw new Error("Cannot have empty string element")}parserState.pos+=1;query.literalSearch=true}function isPathStart(parserState){return parserState.userQuery.slice(parserState.pos,parserState.pos+2)=='::'}function isReturnArrow(parserState){return parserState.userQuery.slice(parserState.pos,parserState.pos+2)=='->'}function isIdentCharacter(c){return(c==='_'||(c>='0'&&c<='9')||(c>='a'&&c<='z')||(c>='A'&&c<='Z'))}function isSeparatorCharacter(c){return c===","||isWhitespaceCharacter(c)}function isWhitespaceCharacter(c){return c===" "||c==="\t"}function createQueryElement(query,parserState,name,generics,isInGenerics){if(name==='*'||(name.length===0&&generics.length===0)){return}if(query.literalSearch&&parserState.totalElems-parserState.genericsElems>0){throw new Error("You cannot have more than one element if you use quotes")}const pathSegments=name.split("::");if(pathSegments.length>1){for(let i=0,len=pathSegments.length;i=end){throw new Error("Found generics without a path")}parserState.pos+=1;getItemsBefore(query,parserState,generics,">")}if(start>=end&&generics.length===0){return}elems.push(createQueryElement(query,parserState,parserState.userQuery.slice(start,end),generics,isInGenerics))}function getItemsBefore(query,parserState,elems,endChar){let foundStopChar=true;while(parserState.pos"){extra="`<`"}else if(endChar===""){extra="`->`"}throw new Error("Unexpected `"+c+"` after "+extra)}if(!foundStopChar){if(endChar!==""){throw new Error(`Expected \`,\`, \` \` or \`${endChar}\`, found \`${c}\``)}throw new Error(`Expected \`,\` or \` \`, found \`${c}\``)}const posBefore=parserState.pos;getNextElem(query,parserState,elems,endChar===">");if(posBefore===parserState.pos){parserState.pos+=1}foundStopChar=false}parserState.pos+=1}function checkExtraTypeFilterCharacters(parserState){const query=parserState.userQuery;for(let pos=0;pos"){if(isReturnArrow(parserState)){break}throw new Error(`Unexpected \`${c}\` (did you mean \`->\`?)`)}throw new Error(`Unexpected \`${c}\``)}else if(c===":"&&!isPathStart(parserState)){if(parserState.typeFilter!==null){throw new Error("Unexpected `:`")}if(query.elems.length===0){throw new Error("Expected type filter before `:`")}else if(query.elems.length!==1||parserState.totalElems!==1){throw new Error("Unexpected `:`")}else if(query.literalSearch){throw new Error("You cannot use quotes on type filter")}checkExtraTypeFilterCharacters(parserState);parserState.typeFilter=query.elems.pop().name;parserState.pos+=1;parserState.totalElems=0;query.literalSearch=false;foundStopChar=true;continue}if(!foundStopChar){if(parserState.typeFilter!==null){throw new Error(`Expected \`,\`, \` \` or \`->\`, found \`${c}\``)}throw new Error(`Expected \`,\`, \` \`, \`:\` or \`->\`, found \`${c}\``)}before=query.elems.length;getNextElem(query,parserState,query.elems,false);if(query.elems.length===before){parserState.pos+=1}foundStopChar=false}while(parserState.pos`")}break}else{parserState.pos+=1}}}function newParsedQuery(userQuery){return{original:userQuery,userQuery:userQuery.toLowerCase(),typeFilter:NO_TYPE_FILTER,elems:[],returned:[],foundElems:0,literalSearch:false,error:null,}}function buildUrl(search,filterCrates){let extra="?search="+encodeURIComponent(search);if(filterCrates!==null){extra+="&filter-crate="+encodeURIComponent(filterCrates)}return getNakedUrl()+extra+window.location.hash}function getFilterCrates(){const elem=document.getElementById("crate-search");if(elem&&elem.value!=="All crates"&&hasOwnPropertyRustdoc(rawSearchIndex,elem.value)){return elem.value}return null}function parseQuery(userQuery){userQuery=userQuery.trim();const parserState={length:userQuery.length,pos:0,totalElems:0,genericsElems:0,typeFilter:null,userQuery:userQuery.toLowerCase(),};let query=newParsedQuery(userQuery);try{parseInput(query,parserState);if(parserState.typeFilter!==null){let typeFilter=parserState.typeFilter;if(typeFilter==="const"){typeFilter="constant"}query.typeFilter=itemTypeFromName(typeFilter)}}catch(err){query=newParsedQuery(userQuery);query.error=err.message;query.typeFilter=-1;return query}if(!query.literalSearch){query.literalSearch=parserState.totalElems>1}query.foundElems=query.elems.length+query.returned.length;return query}function createQueryResults(results_in_args,results_returned,results_others,parsedQuery){return{"in_args":results_in_args,"returned":results_returned,"others":results_others,"query":parsedQuery,}}function execQuery(parsedQuery,searchWords,filterCrates){const results_others={},results_in_args={},results_returned={};function transformResults(results){const duplicates={};const out=[];for(const result of results){if(result.id>-1){const obj=searchIndex[result.id];obj.lev=result.lev;const res=buildHrefAndPath(obj);obj.displayPath=pathSplitter(res[0]);obj.fullPath=obj.displayPath+obj.name;obj.fullPath+="|"+obj.ty;if(duplicates[obj.fullPath]){continue}duplicates[obj.fullPath]=true;obj.href=res[1];out.push(obj);if(out.length>=MAX_RESULTS){break}}}return out}function sortResults(results,isType){const userQuery=parsedQuery.userQuery;const 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lev=MAX_LEV_DISTANCE+1;if(row&&row.type&&row.type[INPUTS_DATA]&&row.type[INPUTS_DATA].length>0){for(const input of row.type[INPUTS_DATA]){if(!typePassesFilter(typeFilter,input[1])){continue}lev=Math.min(lev,checkType(input,elem,parsedQuery.literalSearch));if(lev===0){return 0}}}return parsedQuery.literalSearch?MAX_LEV_DISTANCE+1:lev}function checkReturned(row,elem,typeFilter){let lev=MAX_LEV_DISTANCE+1;if(row&&row.type&&row.type.length>OUTPUT_DATA){let ret=row.type[OUTPUT_DATA];if(typeof ret[0]==="string"){ret=[ret]}for(const ret_ty of ret){if(!typePassesFilter(typeFilter,ret_ty[1])){continue}lev=Math.min(lev,checkType(ret_ty,elem,parsedQuery.literalSearch));if(lev===0){return 0}}}return parsedQuery.literalSearch?MAX_LEV_DISTANCE+1:lev}function checkPath(contains,lastElem,ty){if(contains.length===0){return 0}let ret_lev=MAX_LEV_DISTANCE+1;const path=ty.path.split("::");if(ty.parent&&ty.parent.name){path.push(ty.parent.name.toLowerCase())}const length=path.length;const 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handleSingleArg(row,pos,elem,results_others,results_in_args,results_returned){if(!row||(filterCrates!==null&&row.crate!==filterCrates)){return}let lev,lev_add=0,index=-1;const fullId=row.id;const in_args=findArg(row,elem,parsedQuery.typeFilter);const returned=checkReturned(row,elem,parsedQuery.typeFilter);addIntoResults(results_in_args,fullId,pos,index,in_args);addIntoResults(results_returned,fullId,pos,index,returned);if(!typePassesFilter(parsedQuery.typeFilter,row.ty)){return}const searchWord=searchWords[pos];if(parsedQuery.literalSearch){if(searchWord===elem.name){addIntoResults(results_others,fullId,pos,-1,0)}return}if(elem.name.length===0){if(row.type!==null){lev=checkGenerics(row.type,elem,MAX_LEV_DISTANCE+1);addIntoResults(results_others,fullId,pos,index,lev)}return}if(elem.fullPath.length>1){lev=checkPath(elem.pathWithoutLast,elem.pathLast,row);if(lev>MAX_LEV_DISTANCE||(parsedQuery.literalSearch&&lev!==0)){return}else 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    \ No newline at end of file diff --git a/rustdoc/latest/settings.js b/rustdoc/latest/settings.js index 80471891..0b015fc2 100644 --- a/rustdoc/latest/settings.js +++ b/rustdoc/latest/settings.js @@ -1 +1 @@ -(function(){function changeSetting(settingName,value){updateLocalStorage(settingName,value);switch(settingName){case"theme":case"preferred-dark-theme":case"preferred-light-theme":case"use-system-theme":updateSystemTheme();updateLightAndDark();break}}function handleKey(ev){if(ev.ctrlKey||ev.altKey||ev.metaKey){return}switch(getVirtualKey(ev)){case"Enter":case"Return":case"Space":ev.target.checked=!ev.target.checked;ev.preventDefault();break}}function showLightAndDark(){addClass(document.getElementById("theme").parentElement,"hidden");removeClass(document.getElementById("preferred-light-theme").parentElement,"hidden");removeClass(document.getElementById("preferred-dark-theme").parentElement,"hidden")}function 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handleKey(ev){if(ev.ctrlKey||ev.altKey||ev.metaKey){return}switch(getVirtualKey(ev)){case"Enter":case"Return":case"Space":ev.target.checked=!ev.target.checked;ev.preventDefault();break}}function showLightAndDark(){addClass(document.getElementById("theme").parentElement,"hidden");removeClass(document.getElementById("preferred-light-theme").parentElement,"hidden");removeClass(document.getElementById("preferred-dark-theme").parentElement,"hidden")}function hideLightAndDark(){addClass(document.getElementById("preferred-light-theme").parentElement,"hidden");addClass(document.getElementById("preferred-dark-theme").parentElement,"hidden");removeClass(document.getElementById("theme").parentElement,"hidden")}function updateLightAndDark(){if(getSettingValue("use-system-theme")!=="false"){showLightAndDark()}else{hideLightAndDark()}}function setEvents(){updateLightAndDark();onEachLazy(document.getElementsByClassName("slider"),function(elem){const toggle=elem.previousElementSibling;const settingId=toggle.id;const settingValue=getSettingValue(settingId);if(settingValue!==null){toggle.checked=settingValue==="true"}toggle.onchange=function(){changeSetting(this.id,this.checked)};toggle.onkeyup=handleKey;toggle.onkeyrelease=handleKey});onEachLazy(document.getElementsByClassName("select-wrapper"),function(elem){const select=elem.getElementsByTagName("select")[0];const settingId=select.id;const settingValue=getSettingValue(settingId);if(settingValue!==null){select.value=settingValue}select.onchange=function(){changeSetting(this.id,this.value)}});onEachLazy(document.querySelectorAll("input[type=\"radio\"]"),function(elem){const settingId=elem.name;const settingValue=getSettingValue(settingId);if(settingValue!==null&&settingValue!=="null"){elem.checked=settingValue===elem.value}elem.addEventListener("change",function(ev){changeSetting(ev.target.name,ev.target.value)})});document.getElementById("back").addEventListener("click",function(){history.back()})}window.addEventListener("DOMContentLoaded",setEvents)})() \ No newline at end of file diff --git a/rustdoc/latest/src/halo2/lib.rs.html b/rustdoc/latest/src/halo2/lib.rs.html index b4384f84..5beb985c 100644 --- a/rustdoc/latest/src/halo2/lib.rs.html +++ b/rustdoc/latest/src/halo2/lib.rs.html @@ -33,5 +33,5 @@ #![deny(missing_docs)] #![deny(unsafe_code)]     -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/ecc.rs.html b/rustdoc/latest/src/halo2_gadgets/ecc.rs.html index 23e4d5fd..1b046f09 100644 --- a/rustdoc/latest/src/halo2_gadgets/ecc.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/ecc.rs.html @@ -1681,5 +1681,5 @@ } }     -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/ecc/chip.rs.html b/rustdoc/latest/src/halo2_gadgets/ecc/chip.rs.html index f7125aca..221e6dbc 100644 --- a/rustdoc/latest/src/halo2_gadgets/ecc/chip.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/ecc/chip.rs.html @@ -1071,5 +1071,5 @@ } }     -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/ecc/chip/add.rs.html b/rustdoc/latest/src/halo2_gadgets/ecc/chip/add.rs.html index 7520fb2e..ff2df911 100644 --- a/rustdoc/latest/src/halo2_gadgets/ecc/chip/add.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/ecc/chip/add.rs.html @@ -520,10 +520,7 @@ 500 501 502 -503 -
    use std::array;
-
-use super::EccPoint;
+
    use super::EccPoint;
 use ff::{BatchInvert, Field};
 use halo2_proofs::{
     circuit::Region,
@@ -594,7 +591,7 @@
     }
 
     pub(crate) fn advice_columns(&self) -> HashSet<Column<Advice>> {
-        core::array::IntoIter::new([
+        [
             self.x_p,
             self.y_p,
             self.x_qr,
@@ -604,12 +601,13 @@
             self.beta,
             self.gamma,
             self.delta,
-        ])
+        ]
+        .into_iter()
         .collect()
     }
 
     pub(crate) fn output_columns(&self) -> HashSet<Column<Advice>> {
-        core::array::IntoIter::new([self.x_qr, self.y_qr]).collect()
+        [self.x_qr, self.y_qr].into_iter().collect()
     }
 
     fn create_gate(&self, meta: &mut ConstraintSystem<pallas::Base>) {
@@ -729,10 +727,10 @@
 
             Constraints::with_selector(
                 q_add,
-                array::IntoIter::new([
+                [
                     poly1, poly2, poly3, poly4, poly5, poly6, poly7, poly8, poly9, poly10, poly11,
                     poly12,
-                ]),
+                ],
             )
         });
     }
@@ -1025,5 +1023,5 @@
     }
 }
    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/ecc/chip/add_incomplete.rs.html b/rustdoc/latest/src/halo2_gadgets/ecc/chip/add_incomplete.rs.html index 1c03687c..23db9b2b 100644 --- a/rustdoc/latest/src/halo2_gadgets/ecc/chip/add_incomplete.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/ecc/chip/add_incomplete.rs.html @@ -223,8 +223,7 @@ 203 204 205 -206 -
    use std::{array, collections::HashSet};
+
    use std::collections::HashSet;
 
 use super::NonIdentityEccPoint;
 use ff::Field;
@@ -276,7 +275,9 @@
     }
 
     pub(crate) fn advice_columns(&self) -> HashSet<Column<Advice>> {
-        core::array::IntoIter::new([self.x_p, self.y_p, self.x_qr, self.y_qr]).collect()
+        [self.x_p, self.y_p, self.x_qr, self.y_qr]
+            .into_iter()
+            .collect()
     }
 
     fn create_gate(&self, meta: &mut ConstraintSystem<pallas::Base>) {
@@ -300,10 +301,7 @@
             // (y_r + y_q)(x_p − x_q) − (y_p − y_q)(x_q − x_r) = 0
             let poly2 = (y_r + y_q.clone()) * (x_p - x_q.clone()) - (y_p - y_q) * (x_q - x_r);
 
-            Constraints::with_selector(
-                q_add_incomplete,
-                array::IntoIter::new([("x_r", poly1), ("y_r", poly2)]),
-            )
+            Constraints::with_selector(q_add_incomplete, [("x_r", poly1), ("y_r", poly2)])
         });
     }
 
@@ -431,5 +429,5 @@
     }
 }
    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/ecc/chip/constants.rs.html b/rustdoc/latest/src/halo2_gadgets/ecc/chip/constants.rs.html index b02b0598..69a37098 100644 --- a/rustdoc/latest/src/halo2_gadgets/ecc/chip/constants.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/ecc/chip/constants.rs.html @@ -583,5 +583,5 @@ } }     -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul.rs.html b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul.rs.html index 5983285b..ef9b459f 100644 --- a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul.rs.html @@ -745,11 +745,11 @@ Constraints::with_selector( q_mul_lsb, - std::array::IntoIter::new([ + [ ("bool_check", bool_check), ("lsb_x", lsb_x), ("lsb_y", lsb_y), - ]), + ], ) }); } @@ -1163,5 +1163,5 @@ } }     -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul/complete.rs.html b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul/complete.rs.html index 2c65d91d..cd8d44ad 100644 --- a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul/complete.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul/complete.rs.html @@ -291,7 +291,7 @@ Constraints::with_selector( q_mul_decompose_var, - std::array::IntoIter::new([("bool_check", bool_check), ("y_switch", y_switch)]), + [("bool_check", bool_check), ("y_switch", y_switch)], ) }, ); @@ -413,5 +413,5 @@ } }     -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul/incomplete.rs.html b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul/incomplete.rs.html index 1db71baf..0a0a6904 100644 --- a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul/incomplete.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul/incomplete.rs.html @@ -713,5 +713,5 @@ } }     -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul/overflow.rs.html b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul/overflow.rs.html index 0ef2c3f2..88072ea8 100644 --- a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul/overflow.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul/overflow.rs.html @@ -441,5 +441,5 @@ } }     -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed.rs.html b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed.rs.html index e84ca327..5a06f58c 100644 --- a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed.rs.html @@ -1033,5 +1033,5 @@ } }     -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed/base_field_elem.rs.html b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed/base_field_elem.rs.html index f5b45c6c..be6b8e17 100644 --- a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed/base_field_elem.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed/base_field_elem.rs.html @@ -1057,5 +1057,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed/full_width.rs.html b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed/full_width.rs.html index 42d17cb9..e8e2ec50 100644 --- a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed/full_width.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed/full_width.rs.html @@ -597,5 +597,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed/short.rs.html b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed/short.rs.html index d1d66c4e..70958e3c 100644 --- a/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed/short.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/ecc/chip/mul_fixed/short.rs.html @@ -640,7 +640,7 @@ 620 621 622 -
    use std::{array, convert::TryInto};
+
    use std::convert::TryInto;
 
 use super::super::{EccPoint, EccScalarFixedShort, FixedPoints, L_SCALAR_SHORT, NUM_WINDOWS_SHORT};
 use crate::{ecc::chip::MagnitudeSign, utilities::bool_check};
@@ -701,12 +701,12 @@
 
             Constraints::with_selector(
                 q_mul_fixed_short,
-                array::IntoIter::new([
+                [
                     ("last_window_check", last_window_check),
                     ("sign_check", sign_check),
                     ("y_check", y_check),
                     ("negation_check", negation_check),
-                ]),
+                ],
             )
         });
     }
@@ -1263,5 +1263,5 @@
     }
 }
    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/ecc/chip/witness_point.rs.html b/rustdoc/latest/src/halo2_gadgets/ecc/chip/witness_point.rs.html index d3450b93..8997019f 100644 --- a/rustdoc/latest/src/halo2_gadgets/ecc/chip/witness_point.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/ecc/chip/witness_point.rs.html @@ -195,11 +195,7 @@ 175 176 177 -178 -179 -
    use std::array;
-
-use super::{EccPoint, NonIdentityEccPoint};
+
    use super::{EccPoint, NonIdentityEccPoint};
 
 use group::prime::PrimeCurveAffine;
 
@@ -267,10 +263,10 @@
             // of the form `q_point * (x * curve_eqn)`, but this was implemented without
             // parentheses, and thus evaluates as `(q_point * x) * curve_eqn`, which is
             // structurally different in the pinned verifying key.
-            array::IntoIter::new([
+            [
                 ("x == 0 v on_curve", q_point.clone() * x * curve_eqn(meta)),
                 ("y == 0 v on_curve", q_point * y * curve_eqn(meta)),
-            ])
+            ]
         });
 
         meta.create_gate("witness non-identity point", |meta| {
@@ -377,5 +373,5 @@
     }
 }
    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/lib.rs.html b/rustdoc/latest/src/halo2_gadgets/lib.rs.html index 30f54e78..2eb1deaa 100644 --- a/rustdoc/latest/src/halo2_gadgets/lib.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/lib.rs.html @@ -57,5 +57,5 @@ pub mod primitives;     -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/poseidon.rs.html b/rustdoc/latest/src/halo2_gadgets/poseidon.rs.html index 9d63a671..41a16616 100644 --- a/rustdoc/latest/src/halo2_gadgets/poseidon.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/poseidon.rs.html @@ -318,7 +318,6 @@ 298 
    //! Gadget and chips for the Poseidon algebraic hash function.
 
-use std::array;
 use std::convert::TryInto;
 use std::fmt;
 use std::marker::PhantomData;
@@ -601,7 +600,8 @@
         mut layouter: impl Layouter<F>,
         message: [AssignedCell<F, F>; L],
     ) -> Result<AssignedCell<F, F>, Error> {
-        for (i, value) in array::IntoIter::new(message)
+        for (i, value) in message
+            .into_iter()
             .map(PaddedWord::Message)
             .chain(<ConstantLength<L> as Domain<F, RATE>>::padding(L).map(PaddedWord::Padding))
             .enumerate()
@@ -615,5 +615,5 @@
     }
 }
    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/poseidon/pow5.rs.html b/rustdoc/latest/src/halo2_gadgets/poseidon/pow5.rs.html index 9f00bcc2..ad998519 100644 --- a/rustdoc/latest/src/halo2_gadgets/poseidon/pow5.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/poseidon/pow5.rs.html @@ -1803,5 +1803,5 @@ } }     -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/primitives.rs.html b/rustdoc/latest/src/halo2_gadgets/primitives.rs.html index 2d231187..9d99d3d2 100644 --- a/rustdoc/latest/src/halo2_gadgets/primitives.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/primitives.rs.html @@ -35,5 +35,5 @@ pub mod poseidon; pub mod sinsemilla; -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/primitives/poseidon.rs.html b/rustdoc/latest/src/halo2_gadgets/primitives/poseidon.rs.html index 0a375d0b..4f243d09 100644 --- a/rustdoc/latest/src/halo2_gadgets/primitives/poseidon.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/primitives/poseidon.rs.html @@ -417,7 +417,6 @@ 397 
    //! The Poseidon algebraic hash function.
 
-use std::array;
 use std::convert::TryInto;
 use std::fmt;
 use std::iter;
@@ -780,8 +779,9 @@
 {
     /// Hashes the given input.
     pub fn hash(mut self, message: [F; L]) -> F {
-        for value in
-            array::IntoIter::new(message).chain(<ConstantLength<L> as Domain<F, RATE>>::padding(L))
+        for value in message
+            .into_iter()
+            .chain(<ConstantLength<L> as Domain<F, RATE>>::padding(L))
         {
             self.sponge.absorb(value);
         }
@@ -813,5 +813,5 @@
     }
 }
    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/fp.rs.html b/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/fp.rs.html index 4b4103a3..1879cef0 100644 --- a/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/fp.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/fp.rs.html @@ -2871,5 +2871,5 @@ ], ]; -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/fq.rs.html b/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/fq.rs.html index 0c38f67c..daf2903b 100644 --- a/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/fq.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/fq.rs.html @@ -2871,5 +2871,5 @@ ], ]; -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/grain.rs.html b/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/grain.rs.html index 559efd61..0601a354 100644 --- a/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/grain.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/grain.rs.html @@ -405,5 +405,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/mds.rs.html b/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/mds.rs.html index 0eef41bb..690978e2 100644 --- a/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/mds.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/mds.rs.html @@ -265,5 +265,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/p128pow5t3.rs.html b/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/p128pow5t3.rs.html index 440ef253..9793cb06 100644 --- a/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/p128pow5t3.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/primitives/poseidon/p128pow5t3.rs.html @@ -645,5 +645,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/primitives/sinsemilla.rs.html b/rustdoc/latest/src/halo2_gadgets/primitives/sinsemilla.rs.html index 7f6858b4..9d4cc6d9 100644 --- a/rustdoc/latest/src/halo2_gadgets/primitives/sinsemilla.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/primitives/sinsemilla.rs.html @@ -629,5 +629,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/primitives/sinsemilla/addition.rs.html b/rustdoc/latest/src/halo2_gadgets/primitives/sinsemilla/addition.rs.html index feb6bca4..bcb8133a 100644 --- a/rustdoc/latest/src/halo2_gadgets/primitives/sinsemilla/addition.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/primitives/sinsemilla/addition.rs.html @@ -165,5 +165,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/primitives/sinsemilla/sinsemilla_s.rs.html b/rustdoc/latest/src/halo2_gadgets/primitives/sinsemilla/sinsemilla_s.rs.html index 6b14ae07..de8e3c36 100644 --- a/rustdoc/latest/src/halo2_gadgets/primitives/sinsemilla/sinsemilla_s.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/primitives/sinsemilla/sinsemilla_s.rs.html @@ -28707,5 +28707,5 @@ ), ]; -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256.rs.html index eaa6c3b1..b2ffe0e2 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256.rs.html @@ -351,5 +351,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16.rs.html index 7e3fcdd3..37725cd6 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16.rs.html @@ -1049,5 +1049,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression.rs.html index 0a01a4fe..32bbd307 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression.rs.html @@ -2029,5 +2029,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/compression_gates.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/compression_gates.rs.html index cdbde1d6..d015d4e5 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/compression_gates.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/compression_gates.rs.html @@ -462,16 +462,12 @@ 442 443 444 -445 -446 -447 -448 
    use super::super::{util::*, Gate};
 use halo2_proofs::{
     arithmetic::FieldExt,
-    plonk::{Constraints, Expression},
+    plonk::{Constraint, Constraints, Expression},
 };
-use std::{array, marker::PhantomData};
+use std::marker::PhantomData;
 
 pub struct CompressionGate<F: FieldExt>(PhantomData<F>);
 
@@ -894,26 +890,22 @@
         lo_3: Expression<F>,
         hi_3: Expression<F>,
         word_3: Expression<F>,
-    ) -> Constraints<
-        F,
-        (&'static str, Expression<F>),
-        impl Iterator<Item = (&'static str, Expression<F>)>,
-    > {
+    ) -> impl IntoIterator<Item = Constraint<F>> {
         let check_lo_hi = |lo: Expression<F>, hi: Expression<F>, word: Expression<F>| {
             lo + hi * F::from(1 << 16) - word
         };
 
         Constraints::with_selector(
             s_digest,
-            array::IntoIter::new([
+            [
                 ("check_lo_hi_0", check_lo_hi(lo_0, hi_0, word_0)),
                 ("check_lo_hi_1", check_lo_hi(lo_1, hi_1, word_1)),
                 ("check_lo_hi_2", check_lo_hi(lo_2, hi_2, word_2)),
                 ("check_lo_hi_3", check_lo_hi(lo_3, hi_3, word_3)),
-            ]),
+            ],
         )
     }
 }
    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/compression_util.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/compression_util.rs.html index 0ea08dfb..0339f45e 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/compression_util.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/compression_util.rs.html @@ -2005,5 +2005,5 @@ (a, b, c, d, e, f, g, h) } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/subregion_digest.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/subregion_digest.rs.html index ecd1449c..e6b17d5a 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/subregion_digest.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/subregion_digest.rs.html @@ -241,5 +241,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/subregion_initial.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/subregion_initial.rs.html index c429f373..59d0528c 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/subregion_initial.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/subregion_initial.rs.html @@ -319,5 +319,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/subregion_main.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/subregion_main.rs.html index 6ff923a7..3fd8562e 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/subregion_main.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16/compression/subregion_main.rs.html @@ -275,5 +275,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16/gates.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16/gates.rs.html index a491f6fe..4e45b870 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16/gates.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16/gates.rs.html @@ -267,5 +267,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule.rs.html index d0a7d6d0..53c3d32d 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule.rs.html @@ -908,7 +908,7 @@ // Run message_scheduler to get W_[0..64] let (w, _) = config.message_schedule.process(&mut layouter, inputs)?; for (word, test_word) in w.iter().zip(MSG_SCHEDULE_TEST_OUTPUT.iter()) { - let word: u32 = lebs2ip(&word.value().unwrap()) as u32; + let word: u32 = lebs2ip(word.value().unwrap()) as u32; assert_eq!(word, *test_word); } Ok(()) @@ -925,5 +925,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/schedule_gates.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/schedule_gates.rs.html index 6fe324fb..b2f0106c 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/schedule_gates.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/schedule_gates.rs.html @@ -410,9 +410,13 @@ 390 391 392 +393 +394 +395 +396 
    use super::super::Gate;
 use halo2_proofs::{arithmetic::FieldExt, plonk::Expression};
-use std::{array, marker::PhantomData};
+use std::marker::PhantomData;
 
 pub struct ScheduleGate<F: FieldExt>(PhantomData<F>);
 
@@ -441,7 +445,8 @@
             + (word * (-F::one()));
         let carry_check = Gate::range_check(carry, 0, 3);
 
-        array::IntoIter::new([("word_check", word_check), ("carry_check", carry_check)])
+        [("word_check", word_check), ("carry_check", carry_check)]
+            .into_iter()
             .map(move |(name, poly)| (name, s_word.clone() * poly))
     }
 
@@ -477,11 +482,12 @@
         let range_check_tag_c = Gate::range_check(tag_c, 0, 2);
         let range_check_tag_d = Gate::range_check(tag_d, 0, 4);
 
-        array::IntoIter::new([
+        [
             ("decompose_check", decompose_check),
             ("range_check_tag_c", range_check_tag_c),
             ("range_check_tag_d", range_check_tag_d),
-        ])
+        ]
+        .into_iter()
         .map(move |(name, poly)| (name, s_decompose_1.clone() * poly))
     }
 
@@ -513,11 +519,12 @@
         let range_check_tag_d = Gate::range_check(tag_d, 0, 0);
         let range_check_tag_g = Gate::range_check(tag_g, 0, 3);
 
-        array::IntoIter::new([
+        [
             ("decompose_check", decompose_check),
             ("range_check_tag_g", range_check_tag_g),
             ("range_check_tag_d", range_check_tag_d),
-        ])
+        ]
+        .into_iter()
         .map(move |(name, poly)| (name, s_decompose_2.clone() * poly))
     }
 
@@ -542,11 +549,12 @@
         let range_check_tag_a = Gate::range_check(tag_a, 0, 1);
         let range_check_tag_d = Gate::range_check(tag_d, 0, 3);
 
-        array::IntoIter::new([
+        [
             ("decompose_check", decompose_check),
             ("range_check_tag_a", range_check_tag_a),
             ("range_check_tag_d", range_check_tag_d),
-        ])
+        ]
+        .into_iter()
         .map(move |(name, poly)| (name, s_decompose_3.clone() * poly))
     }
 
@@ -803,5 +811,5 @@
     }
 }
    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/schedule_util.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/schedule_util.rs.html index 1bd6121a..b0710f63 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/schedule_util.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/schedule_util.rs.html @@ -379,5 +379,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/subregion1.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/subregion1.rs.html index 39a56b22..c5142dcc 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/subregion1.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/subregion1.rs.html @@ -457,5 +457,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/subregion2.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/subregion2.rs.html index bd6d8671..40c06d33 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/subregion2.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/subregion2.rs.html @@ -999,5 +999,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/subregion3.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/subregion3.rs.html index fe5f4745..a376ba51 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/subregion3.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16/message_schedule/subregion3.rs.html @@ -665,5 +665,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16/spread_table.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16/spread_table.rs.html index 5237b306..c1297778 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16/spread_table.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16/spread_table.rs.html @@ -901,5 +901,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sha256/table16/util.rs.html b/rustdoc/latest/src/halo2_gadgets/sha256/table16/util.rs.html index f5450ab8..a0eb3d70 100644 --- a/rustdoc/latest/src/halo2_gadgets/sha256/table16/util.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sha256/table16/util.rs.html @@ -162,10 +162,10 @@ fn gen_const_array_with_default<Output: Copy, const LEN: usize>( default_value: Output, - mut closure: impl FnMut(usize) -> Output, + closure: impl FnMut(usize) -> Output, ) -> [Output; LEN] { let mut ret: [Output; LEN] = [default_value; LEN]; - for (bit, val) in ret.iter_mut().zip((0..LEN).map(|idx| closure(idx))) { + for (bit, val) in ret.iter_mut().zip((0..LEN).map(closure)) { *bit = val; } ret @@ -269,5 +269,5 @@ (sum, carry) } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sinsemilla.rs.html b/rustdoc/latest/src/halo2_gadgets/sinsemilla.rs.html index 1abe71bf..ed34ce19 100644 --- a/rustdoc/latest/src/halo2_gadgets/sinsemilla.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sinsemilla.rs.html @@ -1441,5 +1441,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sinsemilla/chip.rs.html b/rustdoc/latest/src/halo2_gadgets/sinsemilla/chip.rs.html index ccb25385..05a3bc41 100644 --- a/rustdoc/latest/src/halo2_gadgets/sinsemilla/chip.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sinsemilla/chip.rs.html @@ -348,9 +348,6 @@ 328 329 330 -331 -332 -333 
    //! Chip implementations for the Sinsemilla gadgets.
 
 use super::{
@@ -612,10 +609,7 @@
                 lhs - rhs
             };
 
-            Constraints::with_selector(
-                q_s1,
-                std::array::IntoIter::new([("Secant line", secant_line), ("y check", y_check)]),
-            )
+            Constraints::with_selector(q_s1, [("Secant line", secant_line), ("y check", y_check)])
         });
 
         config
@@ -685,5 +679,5 @@
     }
 }
    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sinsemilla/chip/generator_table.rs.html b/rustdoc/latest/src/halo2_gadgets/sinsemilla/chip/generator_table.rs.html index c4ca372a..46e4c520 100644 --- a/rustdoc/latest/src/halo2_gadgets/sinsemilla/chip/generator_table.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sinsemilla/chip/generator_table.rs.html @@ -227,5 +227,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sinsemilla/chip/hash_to_point.rs.html b/rustdoc/latest/src/halo2_gadgets/sinsemilla/chip/hash_to_point.rs.html index 15819d92..784ba3ee 100644 --- a/rustdoc/latest/src/halo2_gadgets/sinsemilla/chip/hash_to_point.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sinsemilla/chip/hash_to_point.rs.html @@ -466,8 +466,6 @@ 446 447 448 -449 -450 
    use super::super::{CommitDomains, HashDomains, SinsemillaInstructions};
 use super::{NonIdentityEccPoint, SinsemillaChip};
 
@@ -602,16 +600,14 @@
                 // Get message as a bitstring.
                 let bitstring: Vec<bool> = message
                     .iter()
-                    .map(|piece: &MessagePiece<pallas::Base, K>| {
+                    .flat_map(|piece: &MessagePiece<pallas::Base, K>| {
                         piece
                             .field_elem()
                             .unwrap()
                             .to_le_bits()
                             .into_iter()
                             .take(K * piece.num_words())
-                            .collect::<Vec<_>>()
                     })
-                    .flatten()
                     .collect();
 
                 let hasher_S = pallas::Point::hash_to_curve(S_PERSONALIZATION);
@@ -721,7 +717,7 @@
         let words: Option<Vec<u32>> = bitstring.map(|bitstring| {
             bitstring
                 .chunks_exact(sinsemilla::K)
-                .map(|word| lebs2ip_k(word))
+                .map(lebs2ip_k)
                 .collect()
         });
 
@@ -919,5 +915,5 @@
     }
 }
    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sinsemilla/merkle.rs.html b/rustdoc/latest/src/halo2_gadgets/sinsemilla/merkle.rs.html index 7ce9d3f0..e3939966 100644 --- a/rustdoc/latest/src/halo2_gadgets/sinsemilla/merkle.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sinsemilla/merkle.rs.html @@ -787,5 +787,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sinsemilla/merkle/chip.rs.html b/rustdoc/latest/src/halo2_gadgets/sinsemilla/merkle/chip.rs.html index a39b40d5..01d8120e 100644 --- a/rustdoc/latest/src/halo2_gadgets/sinsemilla/merkle/chip.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sinsemilla/merkle/chip.rs.html @@ -517,7 +517,6 @@ 497 498 499 -500 
    //! Chip implementing a Merkle hash using Sinsemilla as the hash function.
 
 use halo2_proofs::{
@@ -545,7 +544,6 @@
     },
 };
 use group::ff::PrimeField;
-use std::array;
 
 /// Configuration for the `MerkleChip` implementation.
 #[derive(Clone, Debug)]
@@ -685,12 +683,12 @@
 
             Constraints::with_selector(
                 q_decompose,
-                array::IntoIter::new([
+                [
                     ("l_check", l_check),
                     ("left_check", left_check),
                     ("right_check", right_check),
                     ("b1_b2_check", b1_b2_check),
-                ]),
+                ],
             )
         });
 
@@ -1019,5 +1017,5 @@
     }
 }
    -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/sinsemilla/message.rs.html b/rustdoc/latest/src/halo2_gadgets/sinsemilla/message.rs.html index 1cbd6431..5fc12529 100644 --- a/rustdoc/latest/src/halo2_gadgets/sinsemilla/message.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/sinsemilla/message.rs.html @@ -155,5 +155,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/utilities.rs.html b/rustdoc/latest/src/halo2_gadgets/utilities.rs.html index 4d4321cb..a2037da9 100644 --- a/rustdoc/latest/src/halo2_gadgets/utilities.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/utilities.rs.html @@ -462,7 +462,7 @@ plonk::{Advice, Column, Error, Expression}, }; use pasta_curves::arithmetic::FieldExt; -use std::{array, ops::Range}; +use std::ops::Range; pub mod cond_swap; pub mod decompose_running_sum; @@ -520,7 +520,7 @@ ) -> [Option<T>; LEN] { let mut ret = [None; LEN]; if let Some(arr) = option_array { - for (entry, value) in ret.iter_mut().zip(array::IntoIter::new(arr)) { + for (entry, value) in ret.iter_mut().zip(arr) { *entry = Some(value); } } @@ -623,10 +623,10 @@ /// Takes in an FnMut closure and returns a constant-length array with elements of /// type `Output`. fn gen_const_array<Output: Copy + Default, const LEN: usize>( - mut closure: impl FnMut(usize) -> Output, + closure: impl FnMut(usize) -> Output, ) -> [Output; LEN] { let mut ret: [Output; LEN] = [Default::default(); LEN]; - for (bit, val) in ret.iter_mut().zip((0..LEN).map(|idx| closure(idx))) { + for (bit, val) in ret.iter_mut().zip((0..LEN).map(closure)) { *bit = val; } ret @@ -891,5 +891,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/utilities/cond_swap.rs.html b/rustdoc/latest/src/halo2_gadgets/utilities/cond_swap.rs.html index 6254a3a1..0e356183 100644 --- a/rustdoc/latest/src/halo2_gadgets/utilities/cond_swap.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/utilities/cond_swap.rs.html @@ -339,7 +339,7 @@ poly::Rotation, }; use pasta_curves::arithmetic::FieldExt; -use std::{array, marker::PhantomData}; +use std::marker::PhantomData; /// Instructions for a conditional swap gadget. pub trait CondSwapInstructions<F: FieldExt>: UtilitiesInstructions<F> { @@ -523,11 +523,11 @@ Constraints::with_selector( q_swap, - array::IntoIter::new([ + [ ("a check", a_check), ("b check", b_check), ("swap is bool", bool_check), - ]), + ], ) }); @@ -643,5 +643,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/utilities/decompose_running_sum.rs.html b/rustdoc/latest/src/halo2_gadgets/utilities/decompose_running_sum.rs.html index 52b3440a..f24f5b70 100644 --- a/rustdoc/latest/src/halo2_gadgets/utilities/decompose_running_sum.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/utilities/decompose_running_sum.rs.html @@ -827,5 +827,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_gadgets/utilities/lookup_range_check.rs.html b/rustdoc/latest/src/halo2_gadgets/utilities/lookup_range_check.rs.html index 8566987a..4dd4ab91 100644 --- a/rustdoc/latest/src/halo2_gadgets/utilities/lookup_range_check.rs.html +++ b/rustdoc/latest/src/halo2_gadgets/utilities/lookup_range_check.rs.html @@ -1285,5 +1285,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/arithmetic.rs.html b/rustdoc/latest/src/halo2_proofs/arithmetic.rs.html index a43184c6..9c9d3cda 100644 --- a/rustdoc/latest/src/halo2_proofs/arithmetic.rs.html +++ b/rustdoc/latest/src/halo2_proofs/arithmetic.rs.html @@ -877,5 +877,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/circuit.rs.html b/rustdoc/latest/src/halo2_proofs/circuit.rs.html index ff2d0d4b..17ffdb0d 100644 --- a/rustdoc/latest/src/halo2_proofs/circuit.rs.html +++ b/rustdoc/latest/src/halo2_proofs/circuit.rs.html @@ -1129,5 +1129,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/circuit/floor_planner.rs.html b/rustdoc/latest/src/halo2_proofs/circuit/floor_planner.rs.html index d06895b7..09e900cc 100644 --- a/rustdoc/latest/src/halo2_proofs/circuit/floor_planner.rs.html +++ b/rustdoc/latest/src/halo2_proofs/circuit/floor_planner.rs.html @@ -31,5 +31,5 @@ mod v1; pub use v1::{V1Pass, V1}; -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/circuit/floor_planner/single_pass.rs.html b/rustdoc/latest/src/halo2_proofs/circuit/floor_planner/single_pass.rs.html index 07752c42..296ea99c 100644 --- a/rustdoc/latest/src/halo2_proofs/circuit/floor_planner/single_pass.rs.html +++ b/rustdoc/latest/src/halo2_proofs/circuit/floor_planner/single_pass.rs.html @@ -1033,5 +1033,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/circuit/floor_planner/v1.rs.html b/rustdoc/latest/src/halo2_proofs/circuit/floor_planner/v1.rs.html index d8705f50..3cf2aa3b 100644 --- a/rustdoc/latest/src/halo2_proofs/circuit/floor_planner/v1.rs.html +++ b/rustdoc/latest/src/halo2_proofs/circuit/floor_planner/v1.rs.html @@ -1103,5 +1103,5 @@ } } -
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    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/circuit/floor_planner/v1/strategy.rs.html b/rustdoc/latest/src/halo2_proofs/circuit/floor_planner/v1/strategy.rs.html index 1238c6be..8a7c39f4 100644 --- a/rustdoc/latest/src/halo2_proofs/circuit/floor_planner/v1/strategy.rs.html +++ b/rustdoc/latest/src/halo2_proofs/circuit/floor_planner/v1/strategy.rs.html @@ -549,5 +549,5 @@ ); } -
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    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/circuit/layouter.rs.html b/rustdoc/latest/src/halo2_proofs/circuit/layouter.rs.html index fc02e42e..0a5e7b5e 100644 --- a/rustdoc/latest/src/halo2_proofs/circuit/layouter.rs.html +++ b/rustdoc/latest/src/halo2_proofs/circuit/layouter.rs.html @@ -593,5 +593,5 @@ } } -
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    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/dev/failure/emitter.rs.html b/rustdoc/latest/src/halo2_proofs/dev/failure/emitter.rs.html index 665e91a2..b96b360b 100644 --- a/rustdoc/latest/src/halo2_proofs/dev/failure/emitter.rs.html +++ b/rustdoc/latest/src/halo2_proofs/dev/failure/emitter.rs.html @@ -347,5 +347,5 @@ ) } -
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    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/dev/gates.rs.html b/rustdoc/latest/src/halo2_proofs/dev/gates.rs.html index 3edc134d..fe962dd1 100644 --- a/rustdoc/latest/src/halo2_proofs/dev/gates.rs.html +++ b/rustdoc/latest/src/halo2_proofs/dev/gates.rs.html @@ -575,5 +575,5 @@ } } -
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    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/dev/graph.rs.html b/rustdoc/latest/src/halo2_proofs/dev/graph.rs.html index 2be3528a..1cfbdcb7 100644 --- a/rustdoc/latest/src/halo2_proofs/dev/graph.rs.html +++ b/rustdoc/latest/src/halo2_proofs/dev/graph.rs.html @@ -391,5 +391,5 @@ } } -
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    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/dev/util.rs.html b/rustdoc/latest/src/halo2_proofs/dev/util.rs.html index 81c10bea..3f1a1812 100644 --- a/rustdoc/latest/src/halo2_proofs/dev/util.rs.html +++ b/rustdoc/latest/src/halo2_proofs/dev/util.rs.html @@ -237,5 +237,5 @@ cell_values.into_iter().collect() } -
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    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/helpers.rs.html b/rustdoc/latest/src/halo2_proofs/helpers.rs.html index fb7f0482..9ee509ec 100644 --- a/rustdoc/latest/src/halo2_proofs/helpers.rs.html +++ b/rustdoc/latest/src/halo2_proofs/helpers.rs.html @@ -51,5 +51,5 @@ impl<C: CurveAffine> CurveRead for C {} -
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    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/lib.rs.html b/rustdoc/latest/src/halo2_proofs/lib.rs.html index 5ee33444..9b2c8bdf 100644 --- a/rustdoc/latest/src/halo2_proofs/lib.rs.html +++ b/rustdoc/latest/src/halo2_proofs/lib.rs.html @@ -91,5 +91,5 @@ pub mod dev; mod helpers; -
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    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/multicore.rs.html b/rustdoc/latest/src/halo2_proofs/multicore.rs.html index bba26a8c..8bc2fcc8 100644 --- a/rustdoc/latest/src/halo2_proofs/multicore.rs.html +++ b/rustdoc/latest/src/halo2_proofs/multicore.rs.html @@ -29,5 +29,5 @@ pub use rayon::{current_num_threads, scope, Scope}; -
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    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/plonk.rs.html b/rustdoc/latest/src/halo2_proofs/plonk.rs.html index fe02329e..3f21a9ab 100644 --- a/rustdoc/latest/src/halo2_proofs/plonk.rs.html +++ b/rustdoc/latest/src/halo2_proofs/plonk.rs.html @@ -303,5 +303,5 @@ struct X; type ChallengeX<F> = ChallengeScalar<F, X>; -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/plonk/assigned.rs.html b/rustdoc/latest/src/halo2_proofs/plonk/assigned.rs.html index 2cd7a46c..6ccb2a47 100644 --- a/rustdoc/latest/src/halo2_proofs/plonk/assigned.rs.html +++ b/rustdoc/latest/src/halo2_proofs/plonk/assigned.rs.html @@ -1239,5 +1239,5 @@ } } -
    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/plonk/circuit.rs.html b/rustdoc/latest/src/halo2_proofs/plonk/circuit.rs.html index 99888edc..0aef6c36 100644 --- a/rustdoc/latest/src/halo2_proofs/plonk/circuit.rs.html +++ b/rustdoc/latest/src/halo2_proofs/plonk/circuit.rs.html @@ -3001,5 +3001,5 @@ } } -
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    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/plonk/circuit/compress_selectors.rs.html b/rustdoc/latest/src/halo2_proofs/plonk/circuit/compress_selectors.rs.html index 637172aa..f548ed70 100644 --- a/rustdoc/latest/src/halo2_proofs/plonk/circuit/compress_selectors.rs.html +++ b/rustdoc/latest/src/halo2_proofs/plonk/circuit/compress_selectors.rs.html @@ -727,5 +727,5 @@ } } -
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    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/plonk/lookup/verifier.rs.html b/rustdoc/latest/src/halo2_proofs/plonk/lookup/verifier.rs.html index a9d5f559..d60e1706 100644 --- a/rustdoc/latest/src/halo2_proofs/plonk/lookup/verifier.rs.html +++ b/rustdoc/latest/src/halo2_proofs/plonk/lookup/verifier.rs.html @@ -435,5 +435,5 @@ } } -
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    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/plonk/permutation/keygen.rs.html b/rustdoc/latest/src/halo2_proofs/plonk/permutation/keygen.rs.html index 9c862f8d..e7d4450b 100644 --- a/rustdoc/latest/src/halo2_proofs/plonk/permutation/keygen.rs.html +++ b/rustdoc/latest/src/halo2_proofs/plonk/permutation/keygen.rs.html @@ -439,5 +439,5 @@ } } -
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    +
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    +
    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/plonk/vanishing.rs.html b/rustdoc/latest/src/halo2_proofs/plonk/vanishing.rs.html index cea915f6..db09cca1 100644 --- a/rustdoc/latest/src/halo2_proofs/plonk/vanishing.rs.html +++ b/rustdoc/latest/src/halo2_proofs/plonk/vanishing.rs.html @@ -41,5 +41,5 @@ _marker: PhantomData<C>, } -
    +
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    \ No newline at end of file diff --git a/rustdoc/latest/src/halo2_proofs/plonk/verifier.rs.html b/rustdoc/latest/src/halo2_proofs/plonk/verifier.rs.html index 055cca41..571778f5 100644 --- a/rustdoc/latest/src/halo2_proofs/plonk/verifier.rs.html +++ b/rustdoc/latest/src/halo2_proofs/plonk/verifier.rs.html @@ -797,5 +797,5 @@ }) } -
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Typically, though, we will use them via gadgets . This indirection is useful because, for reasons of efficiency and limitations imposed by PLONKish circuits, the chip interfaces will often be dependent on low-level implementation details. The gadget interface can provide a more convenient and stable API that abstracts away from extraneous detail. For example, consider a hash function such as SHA-256. The interface of a chip supporting SHA-256 might be dependent on internals of the hash function design such as the separation between message schedule and compression function. The corresponding gadget interface can provide a more convenient and familiar update/finalize API, and can also handle parts of the hash function that do not need chip support, such as padding. This is similar to how accelerated instructions for cryptographic primitives on CPUs are typically accessed via software libraries, rather than directly. 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When adding points in a prime-order group, we can rely on Theorem 3 from Appendix C of the Halo paper , which says that if we have two such points with nonzero indices wrt a given odd-prime order base, where the indices taken in the range −(q−1)/2..(q−1)/2 are distinct disregarding sign, then they have different x-coordinates. This is helpful, because it is easier to reason about the indices of points occurring in the scalar multiplication algorithm than it is to reason about their x-coordinates directly. So, the required check is equivalent to saying that the following \"indexed version\" of the above algorithm never asserts: acc := 2\nfor i from n-1 down to 0 { p = k_{i+1} ? 1 : −1 assert acc ≠ ± p assert (acc + p) ≠ acc // X acc := (acc + p) + acc assert 0 < acc ≤ (q-1)/2\n}\nif k_0 = 0 { assert acc ≠ 1 acc := acc - 1\n} The maximum value of acc is: <--- n 1s ---> 1011111...111111\n= 1100000...000000 - 1 = 2n+1+2n−1 The assertion labelled X obviously cannot fail because p=0. It is possible to see that acc is monotonically increasing except in the last conditional. It reaches its largest value when k is maximal, i.e. 2n+1+2n−1. So to entirely avoid exceptional cases, we would need 2n+1+2n−1<(q−1)/2. But we can use n larger by c if the last c iterations use complete addition . The first i for which the algorithm using only incomplete addition fails is going to be 252, since 2252+1+2252−1>(q−1)/2. We need n=254 to make the wraparound technique above work. sage: q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001\nsage: 2^253 + 2^252 - 1 < (q-1)//2\nFalse\nsage: 2^252 + 2^251 - 1 < (q-1)//2\nTrue So the last three iterations of the loop (i=2..0) need to use complete addition , as does the conditional subtraction at the end. Writing this out using ⸭ for incomplete addition (as we do in the spec), we have: Acc := [2] T\nfor i from 253 down to 3 { P := k_{i+1} ? T : −T Acc := (Acc ⸭ P) ⸭ Acc\n}\nfor i from 2 down to 0 { P := k_{i+1} ? T : −T Acc := (Acc + P) + Acc // complete addition\n}\nreturn (k_0 = 0) ? (Acc + (-T)) : Acc // complete addition","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Variable-base scalar multiplication","id":"100","title":"Variable-base scalar multiplication"},"101":{"body":"Define a running sum zj​=∑i=jn​(ki​⋅2i−j), where n=254 and: ​zn+1​=0,zn​=kn​,(most significant bit)z0​=k.​ Initialize A254​=[2]T.for i from 254 down to 4:bool_check(ki​)=0zi​=2zi+1​+ki​xP,i​=xT​yP,i​=(2ki​−1)⋅yT​(conditionally negate)λ1,i​⋅(xA,i​−xP,i​)=yA,i​−yP,i​λ1,i2​=xR,i​+xA,i​+xP,i​(λ1,i​+λ2,i​)⋅(xA,i​−xR,i​)=2yA,i​λ2,i2​=xA,i−1​+xR,i​+xA,i​λ2,i​⋅(xA,i​−xA,i−1​)=yA,i​+yA,i−1​,​ where xR,i​=(λ1,i2​−xA,i​−xT​). The helper bool_check(x)=x⋅(1−x). After substitution of xP,i​,yP,i​,xR,i​,yA,i​, and yA,i−1​, this becomes: Initialize A254​=[2]T.for i from 254 down to 4:// let ki​=zi​−2zi+1​// let yA,i​=2(λ1,i​+λ2,i​)⋅(xA,i​−(λ1,i2​−xA,i​−xT​))​bool_check(ki​)=0λ1,i​⋅(xA,i​−xT​)=yA,i​−(2ki​−1)⋅yT​λ2,i2​=xA,i−1​+λ1,i2​−xT​{λ2,i​⋅(xA,i​−xA,i−1​)=yA,i​+yA,i−1​,λ2,4​⋅(xA,4​−xA,3​)=yA,4​+yA,3witnessed​,​if i>4if i=4.​​ Here, yA,3witnessed​ is assigned to a cell. This is unlike previous yA,i​'s, which were implicitly derived from λ1,i​,λ2,i​,xA,i​,xT​, but never actually assigned. The bits k3…1​ are used in three further steps, using complete addition : for i from 3 down to 1:// let ki​=zi​−2zi+1​bool_check(ki​)=0(xA,i−1​,yA,i−1​)=((xA,i​,yA,i​)+(xT​,yT​))+(xA,i​,yA,i​)​ If the least significant bit k0​=1, we set B=O, otherwise we set B=−T. Then we return A+B using complete addition. Let B={(0,0),(xT​,−yT​),​ if k0​=1, otherwise.​ Output (xA,0​,yA,0​)+B. (Note that (0,0) represents O.)","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Constraint program for optimized double-and-add (incomplete addition)","id":"101","title":"Constraint program for optimized double-and-add (incomplete addition)"},"102":{"body":"We need six advice columns to witness (xT​,yT​,λ1​,λ2​,xA,i​,zi​). However, since (xT​,yT​) are the same, we can perform two incomplete additions in a single row, reusing the same (xT​,yT​). We split the scalar bits used in incomplete addition into hi and lo halves and process them in parallel. This means that we effectively have two for loops: the first, covering the hi half for i from 254 down to 130, with a special case at i=130; and the second, covering the lo half for the remaining i from 129 down to 4, with a special case at i=4. xT​xT​xT​⋮xT​​yT​yT​yT​⋮yT​​zhiz255​=0z254​z253​⋮z130​​xAhi​xA,254​=2[T]x​xA,253​⋮xA,130​xA,129​​λ1hi​yA,254​=2[T]y​λ1,254​λ1,253​⋮λ1,130​yA,129​​λ2hi​λ2,254​λ2,253​⋮λ2,130​​q1hi​100⋮0​q2hi​011⋮0​q3hi​000⋮1​zloz130​z129​z128​⋮z5​z4​​xAlo​xA,129​xA,128​⋮xA,5​xA,4​xA,3​​λ1lo​yA,129​λ1,129​λ1,128​⋮λ1,5​λ1,4​yA,3​​λ2lo​λ2,129​λ2,128​⋮λ2,5​λ2,4​​q1lo​100⋮00​q2lo​011⋮10​q3lo​000⋮01​​ For each hi and lo half, we have three sets of gates. Note that i is going from 255..=3; i is NOT indexing the rows. q1​=1 This gate is only used on the first row (before the for loop). We check that λ1​,λ2​ are initialized to values consistent with the initial yA​. Degree3​Constraintq1​⋅(yA,nwitnessed​−yA,n​)=0​​ where yA,n​yA,nwitnessed​​=2(λ1,n​+λ2,n​)⋅(xA,n​−(λ1,n2​−xA,n​−xT​))​, is witnessed.​ q2​=1 This gate is used on all rows corresponding to the for loop except the last. Degree223433​Constraintq2​⋅(xT,cur​−xT,next​)=0q2​⋅(yT,cur​−yT,next​)=0q2​⋅bool_check(ki​)=0, where ki​=zi​−2zi+1​q2​⋅(λ1,i​⋅(xA,i​−xT,i​)−yA,i​+(2ki​−1)⋅yT,i​)=0q2​⋅(λ2,i2​−xA,i−1​−λ1,i2​+xT,i​)=0q2​⋅(λ2,i​⋅(xA,i​−xA,i−1​)−yA,i​−yA,i−1​)=0​​ where yA,i​yA,i−1​​=2(λ1,i​+λ2,i​)⋅(xA,i​−(λ1,i2​−xA,i​−xT​))​,=2(λ1,i−1​+λ2,i−1​)⋅(xA,i−1​−(λ1,i−12​−xA,i−1​−xT​))​,​ q3​=1 This gate is used on the final iteration of the for loop, handling the special case where we check that the output yA​ has been witnessed correctly. Degree3433​Constraintq3​⋅bool_check(ki​)=0, where ki​=zi​−2zi+1​q3​⋅(λ1,i​⋅(xA,i​−xT,i​)−yA,i​+(2ki​−1)⋅yT,i​)=0q3​⋅(λ2,i2​−xA,i−1​−λ1,i2​+xT,i​)=0q3​⋅(λ2,i​⋅(xA,i​−xA,i−1​)−yA,i​−yA,i−1witnessed​)=0​​ where yA,i​yA,i−1witnessed​​=2(λ1,i​+λ2,i​)⋅(xA,i​−(λ1,i2​−xA,i​−xT​))​, is witnessed.​","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Circuit design","id":"102","title":"Circuit design"},"103":{"body":"zi​ cannot overflow for any i≥1, because it is a weighted sum of bits only up to 2n−1=2253, which is smaller than p (and also q). However, z0​=α+tq​ can overflow [0,p). Since overflow can only occur in the final step that constrains z0​=2⋅z1​+k0​, we have z0​=k(modp). It is then sufficient to also check that z0​=α+tq​(modp) (so that k=α+tq​(modp)) and that k∈[tq​,p+tq​). These conditions together imply that k=α+tq​ as an integer, and so 2254+k=α(modq) as required. Note: the bits k254..0​ do not represent a value reduced modulo q, but rather a representation of the unreduced α+tq​.","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Overflow check","id":"103","title":"Overflow check"},"104":{"body":"Since tp​+tq​<2130, we have [tq​,p+tq​)=[tq​,tq​+2130)∪[2130,2254)∪([2254,2254+2130)∩[p+tq​−2130,p+tq​)). We may assume that k=α+tq​(modp). Therefore, k∈[tq​,p+tq​)​⇔⇔⇔​(k∈[tq​,tq​+2130)∨k∈[2130,2254))∨(k∈[2254,2254+2130)∧k∈[p+tq​−2130,p+tq​))(k254​=0⟹(k∈[tq​,tq​+2130)∨k∈[2130,2254)))∧(k254​=1⟹(k∈[2254,2254+2130)∧k∈[p+tq​−2130,p+tq​))(k254​=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254​=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))Ⓐ​ Given k∈[2254,2254+2130), we prove equivalence of k∈[p+tq​−2130,p+tq​) and (α+2130)modp∈[0,2130) as follows: shift the range by 2130−p−tq​ to give k+2130−p−tq​∈[0,2130); observe that k+2130−p−tq​ is guaranteed to be in [2130−tp​−tq​,2131−tp​−tq​) and therefore cannot overflow or underflow modulo p; using the fact that k=α+tq​(modp), observe that (k+2130−p−tq​)modp=(α+tq​+2130−p−tq​)modp=(α+2130)modp. (We can see in a different way that this is correct by observing that it checks whether αmodp∈[p−2130,p), so the upper bound is aligned as we would expect.) Now, we can continue optimizing from Ⓐ: k∈[tq​,p+tq​)​⇔⇔​(k254​=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254​=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))(k254​=0⟹(α∈[0,2130)∨k253..130​ are not all 0))∧(k254​=1⟹(k253..130​ are all 0∧(α+2130)modp∈[0,2130)))​ Constraining k253..130​ to be all-0 or not-all-0 can be implemented almost \"for free\", as follows. Recall that zi​=∑h=in​(kh​⋅2h−i), so we have: z130​z130​z130​−k254​⋅2124​===​∑h=130254​(kh​⋅2h−130)k254​⋅2254−130+∑h=130253​(kh​⋅2h−130)∑h=130253​(kh​⋅2h−130)​ So k253..130​ are all 0 exactly when z130​=k254​⋅2124. Finally, we can merge the 130-bit decompositions for the k254​=0 and k254​=1 cases by checking that (α+k254​⋅2130)modp∈[0,2130).","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Optimized check for k∈[tq​,p+tq​)","id":"104","title":"Optimized check for k∈[tq​,p+tq​)"},"105":{"body":"Let s=α+k254​⋅2130. The constraints for the overflow check are: z0​k254​=1⟹(z130​k254​=0⟹(z130​​=α+tq​(modp)=2124∧smodp∈[0,2130))=0∨smodp∈[0,2130))​ Define inv0(x)={0,1/x,​if x=0otherwise.​ Witness η=inv0(z130​), and decompose smodp as s129..0​. Then the needed gates are: Degree22335​Constraintq_muloverflow⋅(s−(α+k254​⋅2130))=0q_muloverflow⋅(z0​−α−tq​)=0q_muloverflow⋅(k254​⋅(z130​−2124))=0q_muloverflow⋅(k254​⋅(s−i=0∑129​2i⋅si​)/2130)=0q_muloverflow⋅((1−k254​)⋅(1−z130​⋅η)⋅(s−i=0∑129​2i⋅si​)/2130)=0​​ where (s−i=0∑129​2i⋅si​)/2130 can be computed by another running sum. Note that the factor of 1/2130 has no effect on the constraint, since the RHS is zero. Running sum range check We make use of a 10-bit lookup range check in the circuit to subtract the low 130 bits of s. The range check subtracts the first 13⋅10 bits of s, and right-shifts the result to give (s−i=0∑129​2i⋅si​)/2130.","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Overflow check constraints","id":"105","title":"Overflow check constraints"},"106":{"body":"","breadcrumbs":"Design » Gadgets » Sinsemilla » Sinsemilla","id":"106","title":"Sinsemilla"},"107":{"body":"Sinsemilla is a collision-resistant hash function and commitment scheme designed to be efficient in algebraic circuit models that support lookups , such as PLONK or Halo 2. The security properties of Sinsemilla are similar to Pedersen hashes; it is not designed to be used where a random oracle, PRF, or preimage-resistant hash is required. The only claimed security property of the hash function is collision-resistance for fixed-length inputs. Sinsemilla is roughly 4 times less efficient than the algebraic hashes Rescue and Poseidon inside a circuit, but around 19 times more efficient than Rescue outside a circuit. Unlike either of these hashes, the collision resistance property of Sinsemilla can be proven based on cryptographic assumptions that have been well-established for at least 20 years. Sinsemilla can also be used as a computationally binding and perfectly hiding commitment scheme. The general approach is to split the message into k-bit pieces, and for each piece, select from a table of 2k bases in our cryptographic group. We combine the selected bases using a double-and-add algorithm. This ends up being provably as secure as a vector Pedersen hash, and makes advantageous use of the lookup facility supported by Halo 2.","breadcrumbs":"Design » Gadgets » Sinsemilla » Overview","id":"107","title":"Overview"},"108":{"body":"This section is an outline of how Sinsemilla works: for the normative specification, refer to §5.4.1.9 Sinsemilla Hash Function in the protocol spec. The incomplete point addition operator, ⸭, that we use below is also defined there. Let G be a cryptographic group of prime order q. We write G additively, with identity O, and using [m]P for scalar multiplication of P by m. Let k≥1 be an integer chosen based on efficiency considerations (the table size will be 2k). Let n be an integer, fixed for each instantiation, such that messages are kn bits, where 2n≤2q−1​. We use zero-padding to the next multiple of k bits if necessary. Setup: Choose Q and P[0..2k−1] as 2k+1 independent, verifiably random generators of G, using a suitable hash into G, such that none of Q or P[0..2k−1] are O. In Orchard, we define Q to be dependent on a domain separator D. The protocol specification uses Q(D) in place of Q and S(m) in place of P[m]. Hash(M): Split M into n groups of k bits. Interpret each group as a k-bit little-endian integer mi​. let Acc0​:=Q for i from 0 up to n−1: let Acci+1​:=(Acci​⸭P[mi+1​])⸭Acci​ return Accn​ Let ShortHash(M) be the x-coordinate of Hash(M). (This assumes that G is a prime-order elliptic curve in short Weierstrass form, as is the case for Pallas and Vesta.) It is slightly more efficient to express a double-and-add [2]A+R as (A+R)+A. We also use incomplete additions: it is shown in the Sinsemilla security argument that in the case where G is a prime-order short Weierstrass elliptic curve, an exceptional case for addition would lead to finding a discrete logarithm, which can be assumed to occur with negligible probability even for adversarial input.","breadcrumbs":"Design » Gadgets » Sinsemilla » Description","id":"108","title":"Description"},"109":{"body":"Choose another generator H independently of Q and P[0..2k−1]. The randomness r for a commitment is chosen uniformly on [0,q). Let Commitr​(M)=Hash(M)⸭[r]H. Let ShortCommitr​(M) be the x-coordinate of Commitr​(M). (This again assumes that G is a prime-order elliptic curve in short Weierstrass form.) Note that unlike a simple Pedersen commitment, this commitment scheme (Commit or ShortCommit) is not additively homomorphic.","breadcrumbs":"Design » Gadgets » Sinsemilla » Use as a commitment scheme","id":"109","title":"Use as a commitment scheme"},"11":{"body":"You're probably here because you want to write circuits? Excellent! This section will guide you through the process of creating circuits with halo2.","breadcrumbs":"User Documentation » User Documentation","id":"11","title":"User Documentation"},"110":{"body":"The aim of the design is to optimize the number of bits that can be processed for each step of the algorithm (which requires a doubling and addition in G) for a given table size. Using a single table of size 2k group elements, we can process k bits at a time.","breadcrumbs":"Design » Gadgets » Sinsemilla » Efficient implementation","id":"110","title":"Efficient implementation"},"111":{"body":"Let P={(j,xP[j]​,yP[j]​) for j∈{0..2k−1}}. Input: m1..=n​. (The message words are 1-indexed here, as in the protocol spec , but we start the loop from i=0 so that (xA,i​,yA,i​) corresponds to Acci​ in the protocol spec.) Output: (xA,n​,yA,n​). (xA,0​,yA,0​)=Q for i from 0 up to n−1: yP,i​=yA,i​−λ1,i​⋅(xA,i​−xP,i​) xR,i​=λ1,i2​−xA,i​−xP,i​ 2⋅yA,i​=(λ1,i​+λ2,i​)⋅(xA,i​−xR,i​) (mi+1​,xP,i​,yP,i​)∈P λ2,i2​=xA,i+1​+xR,i​+xA,i​ λ2,i​⋅(xA,i​−xA,i+1​)=yA,i​+yA,i+1​","breadcrumbs":"Design » Gadgets » Sinsemilla » Constraint program","id":"111","title":"Constraint program"},"112":{"body":"","breadcrumbs":"Design » Gadgets » Sinsemilla » PLONK / Halo 2 constraints","id":"112","title":"PLONK / Halo 2 constraints"},"113":{"body":"We have an n-bit message m=m1​+2km2​+...+2k⋅(n−1)mn​. (Note that the message words are 1-indexed as in the protocol spec .) Initialise the running sum z0​=α and define zi+1​:=2Kzi​−mi+1​​. We will end up with zn​=0. Rearranging gives us an expression for each word of the original message mi+1​=zi​−2k⋅zi+1​, which we can look up in the table. In other words, zn−i​=h=0∑i−1​2kh⋅mh+1​. For a little-endian decomposition as used here, the running sum is initialized to the scalar and ends at 0. For a big-endian decomposition as used in variable-base scalar multiplication , the running sum would start at 0 and end with recovering the original scalar. The running sum only applies to message words within a single field element, i.e. if n≥PrimeField::NUM_BITS then we will have several disjoint running sums. A longer message can be constructed by splitting the message words across several field elements, and then running several instances of the constraints below. An additional qS2​ selector is set to 0 for the last step of each element, except for the last element where it is set to 2. In order to support chaining multiple field elements without a gap, we will use a slightly more complicated expression for mi+1​ that effectively forces zn​ to zero for the last step of each element, as indicated by qS2​. This allows the cell that would have been zn​ to be used to reinitialize the running sum for the next element.","breadcrumbs":"Design » Gadgets » Sinsemilla » Message decomposition","id":"113","title":"Message decomposition"},"114":{"body":"The Sinsemilla circuit makes use of 210 pre-computed random generators. These are loaded into a lookup table: tableidx​012⋮210−1​tablex​xP[0]​xP[1]​xP[2]​⋮xP[210−1]​​tabley​yP[0]​yP[1]​yP[2]​⋮yP[210−1]​​​","breadcrumbs":"Design » Gadgets » Sinsemilla » Generator lookup table","id":"114","title":"Generator lookup table"},"115":{"body":"Note: qS3​ is synthesized from qS1​ and qS2​; it is shown here only for clarity. Step012⋮n−10′1′2′⋮n−1′n′​xA​xQ​xA,1​xA,2​⋮xA,n−1​xA,0′​xA,1′​xA,2′​⋮xA,n−1′​xA,n′​​xP​xP[m1​]​xP[m2​]​xP[m3​]​⋮xP[mn​]​xP[m1′​]​xP[m2′​]​xP[m3′​]​⋮xP[mn′​]​​bitsz0​z1​z2​⋮zn−1​z0′​z1′​z2′​⋮zn−1′​​λ1​λ1,0​λ1,1​λ1,2​⋮λ1,n−1​λ1,0′​λ1,1′​λ1,2′​⋮λ1,n−1′​yA,n​​λ2​λ2,0​λ2,1​λ2,2​⋮λ2,n−1​λ2,0′​λ2,1′​λ2,2′​⋮λ2,n−1′​​qS1​11111111110​qS2​11110111120​qS3​00000000020​qS4​10000000000​fixed_y_QyQ​0000000000​​ xQ​, z0​, z0′​, etc. would be copied in using equality constraints.","breadcrumbs":"Design » Gadgets » Sinsemilla » Layout","id":"115","title":"Layout"},"116":{"body":"For i∈[0,n), let​xR,i​YA,i​yP,i​mi+1​qS3​​=====​λ1,i2​−xA,i​−xP,i​(λ1,i​+λ2,i​)⋅(xA,i​−xR,i​)YA,i​/2−λ1,i​⋅(xA,i​−xP,i​)zi​−2k⋅(qS2,i​−qS3,i​)⋅zi+1​qS2​⋅(qS2​−1)​ The Halo 2 circuit API can automatically substitute yP,i​, xR,i​, yA,i​, and yA,i+1​, so we don't need to do that manually. xA,0​=xQ​ 2⋅yQ​=YA,0​ for i from 0 up to n−1: (mi+1​,xP,i​,yP,i​)∈P λ2,i2​=xA,i+1​+xR,i​+xA,i​ 4⋅λ2,i​⋅(xA,i​−xA,i+1​)=2⋅YA,i​+(2−qS3​)⋅YA,i+1​+2qS3​⋅yA,n​ Note that each term of the last constraint is multiplied by 4 relative to the constraint program given earlier. This is a small optimization that avoids divisions by 2. By gating the lookup expression on qS1​, we avoid the need to fill in unused cells with dummy values to pass the lookup argument. The optimized lookup value (using a default index of 0) is: (​qS1​⋅mi+1​,qS1​⋅xP,i​+(1−qS1​)⋅xP,0​,qS1​⋅yP,i​+(1−qS1​)⋅yP,0​)​ This increases the degree of the lookup argument to 6. Degree4635​ConstraintqS4​⋅(2⋅yQ​−YA,0​)=0qS1,i​⇒(mi+1​,xP,i​,yP,i​)∈PqS1,i​⋅(λ2,i2​−(xA,i+1​+xR,i​+xA,i​))qS1,i​⋅(4⋅λ2,i​⋅(xA,i​−xA,i+1​)−(2⋅YA,i​+(2−qS3,i​)⋅YA,i+1​+2⋅qS3,i​⋅yA,n​))=0​​","breadcrumbs":"Design » Gadgets » Sinsemilla » Optimized Sinsemilla gate","id":"116","title":"Optimized Sinsemilla gate"},"117":{"body":"","breadcrumbs":"Design » Gadgets » Sinsemilla » MerkleCRH » MerkleCRH","id":"117","title":"MerkleCRH"},"118":{"body":"SinsemillaHash is used in the MerkleCRHOrchard hash function . The input to SinsemillaHash is: l⋆∣∣left⋆∣∣right⋆, where: l⋆=I2LEBSP10​(l)=I2LEBSP10​(MerkleDepthOrchard−1−layer), left⋆=I2LEBSPℓMerkleOrchard​​(left), right⋆=I2LEBSPℓMerkleOrchard​​(right), with ℓMerkleOrchard​=255. left and right are allowed to be non-canonical 255-bit encodings. We break these inputs into the following MessagePieces: a (250 bits)b (20 bits)c (250 bits)​=a0​∣∣a1​=l⋆∣∣(bits 0..=239 of left )=b0​∣∣b1​∣∣b2​=(bits 240..=249 of left)∣∣(bits 250..=254 of left)∣∣(bits 0..=4 of right)=bits 5..=254 of right​ a,b,c are constrained by the SinsemillaHash to be 250 bits, 20 bits, and 250 bits respectively. In a custom gate, we check this message decomposition by enforcing the following constraints: a0​=l z1,a​, the index-1 running sum output of SinsemillaHash(a), is copied into the gate. z1,a​ has been constrained by the SinsemillaHash to be 240 bits. We recover the subpieces a0​,a1​ using a,z1,a​: z1,a​⟹a0​​=210a−a0​​=a1​=a−z1,a​⋅210.​ l+1 is loaded into a fixed column at each layer of the hash. It is used both as a gate selector, and to fix the value of l. We check that a0​=(l+1)−1. Note: The reason for using l+1 instead of l is that l=0 when layer=31 (hashing two leaves). We cannot have a zero-valued selector, since a constraint gated by a zero-valued selector is never checked. b1​+25⋅b2​=z1,b​ z1,b​, the index-1 running sum output of SinsemillaHash(b), is copied into the gate. z1,b​ has been constrained by the SinsemillaHash to be 10 bits. We witness the subpieces b1​,b2​ outside this gate, and constrain them each to be 5 bits. Inside the gate, we check that b1​+25⋅b2​=z1,b​. We also recover the subpiece b0​ using (b,z1,b​): z1,b​⟹b0​​=210b−b0..=10​​=b−(z1,b​⋅210).​ We have now derived or witnessed every subpiece, and range-constrained every subpiece: a0​ (10 bits), derived as a0​=a−210⋅z1,a​; a1​ (240 bits), equal to z1,a​; b0​ (10 bits), derived as b0​=b−210⋅z1,b​; b1​ (5 bits) is witnessed and constrained outside the gate; b2​ (5 bits) is witnessed and constrained outside the gate; b1​+25⋅b2​ is constrained to equal z1,b​, and we use them to reconstruct the original field element inputs: left=a1​+2240⋅b0​+2254⋅b1​ right=b2​+25⋅c","breadcrumbs":"Design » Gadgets » Sinsemilla » MerkleCRH » Message decomposition","id":"118","title":"Message decomposition"},"119":{"body":"The Orchard circuit spans 10 advice columns while the Sinsemilla chip only uses 5 advice columns. We distribute the path hashing evenly across two Sinsemilla chips to make better use of the available circuit area. Since the output from the previous layer hash is copied into the next layer hash, we maintain continuity even when moving from one chip to the other.","breadcrumbs":"Design » Gadgets » Sinsemilla » MerkleCRH » Circuit components","id":"119","title":"Circuit components"},"12":{"body":"The halo2 crate includes several utilities to help you design and implement your circuits.","breadcrumbs":"User Documentation » Developer tools » Developer tools","id":"12","title":"Developer tools"},"120":{"body":"Given a field element α, these gadgets decompose it into W K-bit windows α=k0​+2K⋅k1​+22K⋅k2​+⋯+2(W−1)K⋅kW−1​ where each ki​ a K-bit value. This is done using a running sum zi​,i∈[0..W). We initialize the running sum z0​=α, and compute subsequent terms zi+1​=2Kzi​−ki​​. This gives us: z0​z1​z2​↓zW​​=α=k0​+2K⋅k1​+22K⋅k2​+23K⋅k3​+⋯,=(z0​−k0​)/2K=k1​+2K⋅k2​+22K⋅k3​+⋯,=(z1​−k1​)/2K=k2​+2K⋅k3​+⋯,⋮ (in strict mode)=(zW−1​−kW−1​)/2K=0 (because zW−1​=kW−1​)​","breadcrumbs":"Design » Gadgets » Decomposition » Decomposition","id":"120","title":"Decomposition"},"121":{"body":"Strict mode constrains the running sum output zW​ to be zero, thus range-constraining the field element to be within W⋅K bits. In strict mode, we are also assured that zW−1​=kW−1​ gives us the last window in the decomposition.","breadcrumbs":"Design » Gadgets » Decomposition » Strict mode","id":"121","title":"Strict mode"},"122":{"body":"This gadget makes use of a K-bit lookup table to decompose a field element α into K-bit words. Each K-bit word ki​=zi​−2K⋅zi+1​ is range-constrained by a lookup in the K-bit table. The region layout for the lookup decomposition uses a single advice column z, and two selectors qlookup​ and qrunning​. zz0​z1​⋮zn−1​zn​​qlookup​11⋮10​qrunning​11⋮10​​","breadcrumbs":"Design » Gadgets » Decomposition » Lookup decomposition","id":"122","title":"Lookup decomposition"},"123":{"body":"Using two K-bit lookups, we can range-constrain a field element α to be n bits, where n≤K. To do this: Constrain 0≤α<2K to be within K bits using a K-bit lookup. Constrain 0≤α⋅2K−n<2K to be within K bits using a K-bit lookup. The short variant of the lookup decomposition introduces a qbitshift​ selector. The same advice column z has here been renamed to word for clarity: wordαα′2K−n​qlookup​110​qrunning​000​qbitshift​010​​ where α′=α⋅2K−n. Note that 2K−n is assigned to a fixed column at keygen, and copied in at proving time. This is used in the gate enabled by the qbitshift​ selector to check that α was shifted correctly: Degree2​Constraintqbitshift​⋅(α′−(α⋅2K−n))​​","breadcrumbs":"Design » Gadgets » Decomposition » Short range check","id":"123","title":"Short range check"},"124":{"body":"Since the lookup decomposition and its short variant both make use of the same lookup table, we combine their lookup input expressions into a single one: qlookup​⋅(qrunning​⋅(zi​−2K⋅zi+1​)+(1−qrunning​)⋅word) where zi​ and word are the same cell (but distinguished here for clarity of usage).","breadcrumbs":"Design » Gadgets » Decomposition » Combined lookup expression","id":"124","title":"Combined lookup expression"},"125":{"body":"For a short range (for instance, [0,range) where range≤8), we can range-constrain each word using a degree-range polynomial constraint instead of a lookup: range_check(word,range)=word⋅(1−word)⋯(range−1−word).","breadcrumbs":"Design » Gadgets » Decomposition » Short range decomposition","id":"125","title":"Short range decomposition"},"126":{"body":"","breadcrumbs":"Design » Gadgets » SHA-256 » SHA-256","id":"126","title":"SHA-256"},"127":{"body":"SHA-256 is specified in NIST FIPS PUB 180-4 . Unlike the specification, we use ⊞ for addition modulo 232, and + for field addition. ⊕ is used for XOR.","breadcrumbs":"Design » Gadgets » SHA-256 » Specification","id":"127","title":"Specification"},"128":{"body":"SHA-256 maintains state in eight 32-bit variables. It processes input as 512-bit blocks, but internally splits these blocks into 32-bit chunks. We therefore designed the SHA-256 gadget to consume input in 32-bit chunks.","breadcrumbs":"Design » Gadgets » SHA-256 » Gadget interface","id":"128","title":"Gadget interface"},"129":{"body":"The SHA-256 gadget requires a chip with the following instructions: # extern crate halo2_proofs;\n# use halo2_proofs::plonk::Error;\n# use std::fmt;\n#\n# trait Chip: Sized {}\n# trait Layouter {}\nconst BLOCK_SIZE: usize = 16;\nconst DIGEST_SIZE: usize = 8; pub trait Sha256Instructions: Chip { /// Variable representing the SHA-256 internal state. type State: Clone + fmt::Debug; /// Variable representing a 32-bit word of the input block to the SHA-256 compression /// function. type BlockWord: Copy + fmt::Debug; /// Places the SHA-256 IV in the circuit, returning the initial state variable. fn initialization_vector(layouter: &mut impl Layouter) -> Result; /// Starting from the given initial state, processes a block of input and returns the /// final state. fn compress( layouter: &mut impl Layouter, initial_state: &Self::State, input: [Self::BlockWord; BLOCK_SIZE], ) -> Result; /// Converts the given state into a message digest. fn digest( layouter: &mut impl Layouter, state: &Self::State, ) -> Result<[Self::BlockWord; DIGEST_SIZE], Error>;\n} TODO: Add instruction for computing padding. This set of instructions was chosen to strike a balance between the reusability of the instructions, and the scope for chips to internally optimise them. In particular, we considered splitting the compression function into its constituent parts (Ch, Maj etc), and providing a compression function gadget that implemented the round logic. However, this would prevent chips from using relative references between the various parts of a compression round. Having an instruction that implements all compression rounds is also similar to the Intel SHA extensions, which provide an instruction that performs multiple compression rounds.","breadcrumbs":"Design » Gadgets » SHA-256 » Chip instructions","id":"129","title":"Chip instructions"},"13":{"body":"halo2_proofs::dev::MockProver is a tool for debugging circuits, as well as cheaply verifying their correctness in unit tests. The private and public inputs to the circuit are constructed as would normally be done to create a proof, but MockProver::run instead creates an object that will test every constraint in the circuit directly. It returns granular error messages that indicate which specific constraint (if any) is not satisfied.","breadcrumbs":"User Documentation » Developer tools » Mock prover","id":"13","title":"Mock prover"},"130":{"body":"This chip implementation is based around a single 16-bit lookup table. It requires a minimum of 216 circuit rows, and is therefore suitable for use in larger circuits. We target a maximum constraint degree of 9. That will allow us to handle constraining carries and \"small pieces\" to a range of up to {0..7} in one row.","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » 16-bit table chip for SHA-256","id":"130","title":"16-bit table chip for SHA-256"},"131":{"body":"There are 64 compression rounds. Each round takes 32-bit values A,B,C,D,E,F,G,H as input, and performs the following operations: Ch(E,F,G)Maj(A,B,C)Σ0​(A)Σ1​(E)H′Enew​Anew​​========​(E∧F)⊕(¬E∧G)(A∧B)⊕(A∧C)⊕(B∧C)count(A,B,C)≥2(A⋙2)⊕(A⋙13)⊕(A⋙22)(E⋙6)⊕(E⋙11)⊕(E⋙25)H+Ch(E,F,G)+Σ1​(E)+Kt​+Wt​reduce6​(H′+D)reduce7​(H′+Maj(A,B,C)+Σ0​(A))​ where reducei​ must handle a carry 0≤carry = MyCircuit { a: None, lookup_table, }; // Create the area you want to draw on. // Use SVGBackend if you want to render to .svg instead. use plotters::prelude::*; let root = BitMapBackend::new(\"layout.png\", (1024, 768)).into_drawing_area(); root.fill(&WHITE).unwrap(); let root = root .titled(\"Example Circuit Layout\", (\"sans-serif\", 60)) .unwrap(); halo2_proofs::dev::CircuitLayout::default() // You can optionally render only a section of the circuit. .view_width(0..2) .view_height(0..16) // You can hide labels, which can be useful with smaller areas. .show_labels(false) // Render the circuit onto your area! // The first argument is the size parameter for the circuit. .render(5, &circuit, &root) .unwrap();\n} Columns are laid out from left to right as instance, advice and fixed. The order of columns is otherwise without meaning. Instance columns have a white background. Advice columns have a red background. Fixed columns have a blue background. Regions are shown as labelled green boxes (overlaying the background colour). A region may appear as multiple boxes if some of its columns happen to not be adjacent. Cells that have been assigned to by the circuit will be shaded in grey. If any cells are assigned to more than once (which is usually a mistake), they will be shaded darker than the surrounding cells.","breadcrumbs":"User Documentation » Developer tools » Circuit layout","id":"15","title":"Circuit layout"},"150":{"body":"v1 v1 of the σ1​ gate takes in a word that's split into (10,7,2,13)-bit chunks (already constrained by message scheduling). We refer to these chunks respectively as (a(10),b(7),c(2),d(13)). b(7) is further split into (2,2,3)-bit chunks b(7)lo,b(7)mid,b(7)hi. We witness the spread versions of the small chunks. We already have spread(a(10)) and spread(d(13)) from the message scheduling. (X⋙17)⊕(X⋙19)⊕(X≫10) is equivalent to (X⋘15)⊕(X⋘13)⊕(X≫10). s_low_sigma_1 a0​ a1​ a2​ a3​ a4​ a5​ a6​ 0 {0,1,2,3,4,5} R0even​ spread(R0even​) b(7)lo spread(b(7)lo) b(7)mid spread(b(7)mid) 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(a(10)) spread(d(13)) b(7) 0 {0,1,2,3,4,5} R1even​ spread(R1even​) c(2) spread(c(2)) b(7)hi spread(b(7)hi) 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) Constraints: s_low_sigma_1 (σ1​ v1 constraint): LHS−RHS=0 LHS​=​spread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​)​ RHS=​429b(7)hi430c(2)​++​49d(13)427b(7)mid427b(7)hi​+++​47c(2)425b(7)lo425b(7)mid​+++​44b(7)hi415a(10)423b(7)lo​+++​42b(7)mid42d(13)413a(10)​+++​b(7)loc(2)d(13)​++​ check that b was properly split into subsections for 7-bit pieces. Wb(7)lo+22Wb(7)mid+24Wb(7)hi−W=0 2-bit range check and 2-bit spread check on b(7)lo,b(7)mid,c(2) 3-bit range check and 3-bit spread check on b(7)hi v2 v2 of the σ1​ gate takes in a word that's split into (3,4,3,7,1,1,13)-bit chunks (already constrained by message scheduling). We refer to these chunks respectively as (a(3),b(4),c(3),d(7),e(1),f(1),g(13)). We already have spread(d(7)),spread(g(13)) from the message scheduling. The 1-bit e(1),f(1) remain unchanged by the spread operation and can be used directly. We further split b(4) into two 2-bit chunks b(4)lo,b(4)hi. We witness the spread versions of the small chunks. (X⋙17)⊕(X⋙19)⊕(X≫10) is equivalent to (X⋘15)⊕(X⋘13)⊕(X≫10). s_low_sigma_1_v2 a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ 0 {0,1,2,3,4,5} R0even​ spread(R0even​) b(4)lo spread(b(4)lo) b(4)hi spread(b(4)hi) 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(7)) spread(g(13)) b(4) e(1) 0 {0,1,2,3,4,5} R1even​ spread(R1even​) a(3) spread(a(3)) c(3) spread(c(3)) f(1) 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) Constraints: s_low_sigma_1_v2 (σ1​ v2 constraint): LHS−RHS=0 LHS​=​spread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​)​ RHS=​425d(7)431f(1)​++​422c(3)430e(1)​++​420b(4)hi423d(7)​++​49g(13)418b(4)lo420c(3)​+++​48f(1)415a418b(4)hi​+++​47e(1)42g(13)416b(4)lo​+++​d(7)41f(1)413a​+++​e(1)g(13)​+​ check that b was properly split into subsections for 4-bit pieces. Wb(4)lo+22Wb(4)hi−W=0 2-bit range check and 2-bit spread check on b(4)lo,b(4)hi 3-bit range check and 3-bit spread check on a(3),c(3)","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » σ_1 gate","id":"150","title":"σ_1 gate"},"151":{"body":"Small range constraints Let constrainn​(x)=∏i=0n​(x−i). Constraining this expression to equal zero enforces that x is in [0..n]. 2-bit range check (a−3)(a−2)(a−1)(a)=0 sr2 a0​ 1 a 2-bit spread l1​(a)+4∗l2​(a)+5∗l3​(a)−a′=0 ss2 a0​ a1​ 1 a a' with interpolation polynomials: l0​(a)=(−3)(−2)(−1)(a−3)(a−2)(a−1)​ (spread(00)=0000) l1​(a)=(−2)(−1)(1)(a−3)(a−2)(a)​ (spread(01)=0001) l2​(a)=(−1)(1)(2)(a−3)(a−1)(a)​ (spread(10)=0100) l3​(a)=(1)(2)(3)(a−2)(a−1)(a)​ (spread(11)=0101) 3-bit range check (a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)(a)=0 sr3 a0​ 1 a 3-bit spread l1​(a)+4∗l2​(a)+5∗l3​(a)+16∗l4​(a)+17∗l5​(a)+20∗l6​(a)+21∗l7​(a)−a′=0 ss3 a0​ a1​ 1 a a' with interpolation polynomials: l0​(a)=(−7)(−6)(−5)(−4)(−3)(−2)(−1)(a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)​ (spread(000)=000000) l1​(a)=(−6)(−5)(−4)(−3)(−2)(−1)(1)(a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a)​ (spread(001)=000001) l2​(a)=(−5)(−4)(−3)(−2)(−1)(1)(2)(a−7)(a−6)(a−5)(a−4)(a−3)(a−1)(a)​ (spread(010)=000100) l3​(a)=(−4)(−3)(−2)(−1)(1)(2)(3)(a−7)(a−6)(a−5)(a−4)(a−2)(a−1)(a)​ (spread(011)=000101) l4​(a)=(−3)(−2)(−1)(1)(2)(3)(4)(a−7)(a−6)(a−5)(a−3)(a−2)(a−1)(a)​ (spread(100)=010000) l5​(a)=(−2)(−1)(1)(2)(3)(4)(5)(a−7)(a−6)(a−4)(a−3)(a−2)(a−1)(a)​ (spread(101)=010001) l6​(a)=(−1)(1)(2)(3)(4)(5)(6)(a−7)(a−5)(a−4)(a−3)(a−2)(a−1)(a)​ (spread(110)=010100) l7​(a)=(1)(2)(3)(4)(5)(6)(7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)(a)​ (spread(111)=010101) reduce_6 gate Addition (mod232) of 6 elements Input: E {eilo​,eihi​}i=05​ carry Check: E=e0​+e1​+e2​+e3​+e4​+e5​(mod32) Assume inputs are constrained to 16 bits. Addition gate (sa): a0​+a1​+a2​+a3​+a4​+a5​+a6​−a7​=0 Carry gate (sc): 216a6​ω−1+a6​+[(a6​−5)(a6​−4)(a6​−3)(a6​−2)(a6​−1)(a6​)]=0 sa sc a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ 1 0 e0lo​ e1lo​ e2lo​ e3lo​ e4lo​ e5lo​ −carry∗216 Elo 1 1 e0hi​ e1hi​ e2hi​ e3hi​ e4hi​ e5hi​ carry Ehi Assume inputs are constrained to 16 bits. Addition gate (sa): a0​ω−1+a1​ω−1+a2​ω−1+a0​+a1​+a2​+a3​ω−1−a3​=0 Carry gate (sc): 216a3​ω+a3​ω−1=0 sa sc a0​ a1​ a2​ a3​ 0 0 e0lo​ e1lo​ e2lo​ −carry∗216 1 1 e3lo​ e4lo​ e5lo​ Elo 0 0 e0hi​ e1hi​ e2hi​ carry 1 0 e3hi​ e4hi​ e5hi​ Ehi reduce_7 gate Addition (mod232) of 7 elements Input: E {eilo​,eihi​}i=06​ carry Check: E=e0​+e1​+e2​+e3​+e4​+e5​+e6​(mod32) Assume inputs are constrained to 16 bits. Addition gate (sa): a0​+a1​+a2​+a3​+a4​+a5​+a6​+a7​−a8​=0 Carry gate (sc): 216a7​ω−1+a7​+[(a7​−6)(a7​−5)(a7​−4)(a7​−3)(a7​−2)(a7​−1)(a7​)]=0 sa sc a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ 1 0 e0lo​ e1lo​ e2lo​ e3lo​ e4lo​ e5lo​ e6lo​ −carry∗216 Elo 1 1 e0hi​ e1hi​ e2hi​ e3hi​ e4hi​ e5hi​ e6hi​ carry Ehi","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » Helper gates","id":"151","title":"Helper gates"},"152":{"body":"For each block M∈{0,1}512 of the padded message, 64 words of 32 bits each are constructed as follows: the first 16 are obtained by splitting M into 32-bit blocks M=W0​∣∣W1​∣∣⋯∣∣W14​∣∣W15​; the remaining 48 words are constructed using the formula: Wi​=σ1​(Wi−2​)⊞Wi−7​⊞σ0​(Wi−15​)⊞Wi−16​, for 16≤i<64. sw sd0 sd1 sd2 sd3 ss0 ss0_v2 ss1 ss1_v2 a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W0lo​ spread(W0lo​) W0lo​ W0hi​ W0​ σ0​(W1​)lo σ1​(W14​)lo W9lo​ 1 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W0hi​ spread(W0hi​) W16​ σ0​(W1​)hi σ1​(W14​)hi W9hi​ carry16​ 0 1 1 0 0 0 0 0 0 {0,1,2,3,4} W1d(14)​ spread(W1d(14)​) W1lo​ W1hi​ W1​ σ0​(W2​)lo σ1​(W15​)lo W10lo​ 1 0 0 0 0 0 0 0 0 {0,1,2} W1c(11)​ spread(W1c(11)​) W1a(3)​ W1b(4)​ W17​ σ0​(W2​)hi σ1​(W15​)hi W10hi​ carry17​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W1b(4)lo​ spread(W1b(4)lo​) W1b(4)hi​ spread(W1b(4)hi​) 0 0 0 0 0 1 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R0odd​) spread(R1odd​) spread(W1c(11)​) spread(W1d(14)​) W1b(4)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R1even​) 0 0 W1a(3)​ spread(W1a(3)​) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1odd​) σ0​v1R0​ σ0​v1R1​ σ0​v1R0even​ σ0​v1R0odd​ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 0 0 0 0 0 0 0 0 {0,1,2,3} W14g(13)​ spread(W14g(13)​) W14a(3)​ W14c(3)​ 0 1 0 1 0 0 0 0 0 0 W14d(7)​ spread(W14d(7)​) W14lo​ W14hi​ W14​ σ0​(W15​)lo σ1​(W28​)lo W23lo​ 1 0 0 0 0 0 0 0 0 0 W14b(4)​ spread(W14b(4)​) W14e(1)​ W14f(1)​ W30​ σ0​(W15​)hi σ1​(W28​)hi W23hi​ carry30​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W14b(4)lo​ spread(W14b(4)lo​) W14b(4)hi​ spread(W14b(4)hi​) 0 0 0 0 0 0 1 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(W14d(7)​) spread(W14g(13)​) W1b(14)​ W14e(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) W14a(3)​ spread(W14a(3)​) W14c(3)​ spread(W14c(3)​) W14f(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) σ0​v2R0​ σ0​v2R1​ σ0​v2R0even​ σ0​v2R0odd​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W14b(4)lo​ spread(W14b(4)lo​) W14b(4)hi​ spread(W14b(4)hi​) 0 0 0 0 0 0 0 0 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d) spread(g) W14e(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) W14a(3)​ spread(W14a(3)​) W14c(3)​ spread(W14c(3)​) W14f(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) σ1​v2R0​ σ1​v2R1​ σ1​v2R0even​ σ1​v2R0odd​ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 1 0 0 1 0 0 0 0 {0,1,2,3} W49d(13)​ spread(W49d(13)​) W49lo​ W49hi​ W49​ 0 0 0 0 0 0 0 0 0 {0,1} W49a(10)​ spread(W49a(10)​) W49c(2)​ W49b(7)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W49b(7)lo​ spread(W49b(7)lo​) W49b(7)mid​ spread(W49b(7)mid​) 0 0 0 0 0 0 0 0 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(a) spread(d) W1b(49)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) W49c(2)​ spread(W49c(2)​) W49b(7)hi​ spread(W49b(7)hi​) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) σ1​v1R0​ σ1​v1R1​ σ1​v1R0even​ σ1​v1R0odd​ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W62lo​ spread(W62lo​) W62lo​ W62hi​ W62​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W62hi​ spread(W62hi​) 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W63lo​ spread(W63lo​) W63lo​ W63hi​ W63​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W63hi​ spread(W63hi​) Constraints: sw: construct word using reduce4​ sd0: decomposition gate for W0​,W62​,W63​ Wlo+216Whi−W=0 sd1: decomposition gate for W1..13​ (split into (3,4,11,14)-bit pieces) Wa(3)+23Wb(4)lo+25Wb(4)hi+27Wc(11)+218Wd(14)−W=0 sd2: decomposition gate for W14..48​ (split into (3,4,3,7,1,1,13)-bit pieces) Wa(3)+23Wb(4)lo+25Wb(4)hi+27Wc(11)+210Wd(14)+217We(1)+218Wf(1)+219Wg(13)−W=0 sd3: decomposition gate for W49..61​ (split into (10,7,2,13)-bit pieces) Wa(10)+210Wb(7)lo+212Wb(7)mid+215Wb(7)hi+217Wc(2)+219Wd(13)−W=0","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » Message scheduling region","id":"152","title":"Message scheduling region"},"153":{"body":"+----------------------------------------------------------+\n| |\n| decompose E, |\n| Σ_1(E) |\n| |\n| +---------------------------------------+\n| | |\n| | reduce_5() to get H' |\n| | |\n+----------------------------------------------------------+\n| decompose F, decompose G |\n| |\n| Ch(E,F,G) |\n| |\n+----------------------------------------------------------+\n| |\n| decompose A, |\n| Σ_0(A) |\n| |\n| |\n| +---------------------------------------+\n| | |\n| | reduce_7() to get A_new, |\n| | using H' |\n| | |\n+------------------+---------------------------------------+\n| decompose B, decompose C |\n| |\n| Maj(A,B,C) |\n| |\n| +---------------------------------------+\n| | reduce_6() to get E_new, |\n| | using H' |\n+------------------+---------------------------------------+ Initial round: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0​d(7) spread(E0​d(7)) E0​b(5)lo spread(E0​b(5)lo) E0​b(5)hi spread(E0​b(5)hi) E0lo​ spread(E0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} E0​c(14) spread(E0​c(14)) E0​a(6)lo spread(E0​a(6)lo) E0​a(6)hi spread(E0​a(6)hi) E0hi​ spread(E0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(E0​b(5)lo) spread(E0​b(5)hi) 0 0 0 0 1 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(E0​d(7)) spread(E0​c(14)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(E0​a(6)lo) spread(E0​a(6)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0​d(7) spread(F0​d(7)) F0​b(5)lo spread(F0​b(5)lo) F0​b(5)hi spread(F0​b(5)hi) F0lo​ spread(F0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} F0​c(14) spread(F0​c(14)) F0​a(6)lo spread(F0​a(6)lo) F0​a(6)hi spread(F0​a(6)hi) F0hi​ spread(F0hi​) 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} G0​d(7) spread(G0​d(7)) G0​b(5)lo spread(G0​b(5)lo) G0​b(5)hi spread(G0​b(5)hi) G0lo​ spread(G0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} G0​c(14) spread(G0​c(14)) G0​a(6)lo spread(G0​a(6)lo) G0​a(6)hi spread(G0​a(6)hi) G0hi​ spread(G0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P0even​ spread(P0even​) spread(Elo) spread(Ehi) Q0odd​ K0lo​ H0lo​ W0lo​ 0 0 0 0 0 0 0 1 0 0 1 {0,1,2,3,4,5} P0odd​ spread(P0odd​) spread(P1odd​) Σ1​(E0​)lo Σ1​(E0​)hi K0hi​ H0hi​ W0hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1even​ spread(P1even​) spread(Flo) spread(Fhi) Q1odd​ P1odd​ Hprime0lo​ Hprime0hi​ Hprime0​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1odd​ spread(P1odd​) D0lo​ E1lo​ 0 0 0 0 0 0 0 0 0 1 0 {0,1,2,3,4,5} Q0even​ spread(Q0even​) spread(Eneglo​) spread(Eneghi​) spread(Elo) D0hi​ E1hi​ E1​carry 0 0 0 0 0 0 1 0 0 0 0 {0,1,2,3,4,5} Q0odd​ spread(Q0odd​) spread(Q1odd​) spread(Ehi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1even​ spread(Q1even​) spread(Glo) spread(Ghi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1odd​ spread(Q1odd​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} A0​b(11) spread(A0​b(11)) A0​c(9)lo spread(A0​c(9)lo) A0​c(9)mid spread(A0​c(9)mid) A0lo​ spread(A0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} A0​d(10) spread(A0​d(10)) A0​a(2) spread(A0​a(2)) A0​c(9)hi spread(A0​c(9)hi) A0hi​ spread(A0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(c(9)lo) spread(c(9)mid) 0 0 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(10)) spread(b(11)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(a(2)) spread(c(9)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} B0​b(11) spread(B0​b(11)) B0​c(9)lo spread(B0​c(9)lo) B0​c(9)mid spread(B0​c(9)mid) B0lo​ spread(B0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} B0​d(10) spread(B0​d(10)) B0​a(2) spread(B0​a(2)) B0​c(9)hi spread(B0​c(9)hi) B0hi​ spread(B0hi​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} C0​b(11) spread(C0​b(11)) C0​c(9)lo spread(C0​c(9)lo) C0​c(9)mid spread(C0​c(9)mid) C0lo​ spread(C0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} C0​d(10) spread(C0​d(10)) C0​a(2) spread(C0​a(2)) C0​c(9)hi spread(C0​c(9)hi) C0hi​ spread(C0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M0even​ spread(M0even​) M1odd​ spread(A0lo​) spread(A0hi​) Hprime0lo​ Hprime0hi​ 0 0 0 0 0 1 0 0 1 0 0 {0,1,2,3,4,5} M0odd​ spread(M0odd​) spread(M1odd​) spread(B0lo​) spread(B0hi​) Σ0​(A0​)lo A1lo​ A1​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1even​ spread(M1even​) spread(C0lo​) spread(C0hi​) Σ0​(A0​)hi A1hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1odd​ spread(M1odd​) Steady-state: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0​d(7) spread(E0​d(7)) E0​b(5)lo spread(E0​b(5)lo) E0​b(5)hi spread(E0​b(5)hi) E0lo​ spread(E0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} E0​c(14) spread(E0​c(14)) E0​a(6)lo spread(E0​a(6)lo) E0​a(6)hi spread(E0​a(6)hi) E0hi​ spread(E0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(E0​b(5)lo) spread(E0​b(5)hi) 0 0 0 0 1 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(E0​d(7)) spread(E0​c(14)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(E0​a(6)lo) spread(E0​a(6)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P0even​ spread(P0even​) spread(Elo) spread(Ehi) Q0odd​ K0lo​ H0lo​ W0lo​ 0 0 0 0 0 0 0 1 0 0 1 {0,1,2,3,4,5} P0odd​ spread(P0odd​) spread(P1odd​) Σ1​(E0​)lo Σ1​(E0​)hi K0hi​ H0hi​ W0hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1even​ spread(P1even​) spread(Flo) spread(Fhi) Q1odd​ P1odd​ Hprime0lo​ Hprime0hi​ Hprime0​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1odd​ spread(P1odd​) D0lo​ E1lo​ 0 0 0 0 0 0 0 0 0 1 0 {0,1,2,3,4,5} Q0even​ spread(Q0even​) spread(Eneglo​) spread(Eneghi​) spread(Elo) D0hi​ E1hi​ E1​carry 0 0 0 0 0 0 1 0 0 0 0 {0,1,2,3,4,5} Q0odd​ spread(Q0odd​) spread(Q1odd​) spread(Ehi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1even​ spread(Q1even​) spread(Glo) spread(Ghi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1odd​ spread(Q1odd​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} A0​b(11) spread(A0​b(11)) A0​c(9)lo spread(A0​c(9)lo) A0​c(9)mid spread(A0​c(9)mid) A0lo​ spread(A0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} A0​d(10) spread(A0​d(10)) A0​a(2) spread(A0​a(2)) A0​c(9)hi spread(A0​c(9)hi) A0hi​ spread(A0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(c(9)lo) spread(c(9)mid) 0 0 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(10)) spread(b(11)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(a(2)) spread(c(9)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M0even​ spread(M0even​) M1odd​ spread(A0lo​) spread(A0hi​) Hprime0lo​ Hprime0hi​ 0 0 0 0 0 1 0 0 1 0 0 {0,1,2,3,4,5} M0odd​ spread(M0odd​) spread(M1odd​) spread(B0lo​) spread(B0hi​) Σ0​(A0​)lo A1lo​ A1​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1even​ spread(M1even​) spread(C0lo​) spread(C0hi​) Σ0​(A0​)hi A1hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1odd​ spread(M1odd​) Final digest: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 A63lo​ A63hi​ A63​ B63lo​ B63hi​ B63​ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C63lo​ C63hi​ C63​ C63lo​ C63hi​ C63​ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 E63lo​ E63hi​ E63​ G63lo​ G63hi​ G63​ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 F63lo​ F63hi​ F63​ H63lo​ H63hi​ H63​","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » Compression region","id":"153","title":"Compression region"},"154":{"body":"This section covers the background material required to understand the Halo 2 proving system. It is targeted at an ELI15 (Explain It Like I'm 15) level; if you think anything could do with additional explanation, let us know !","breadcrumbs":"Background Material » Background Material","id":"154","title":"Background Material"},"155":{"body":"A fundamental component of many cryptographic protocols is the algebraic structure known as a field . Fields are sets of objects (usually numbers) with two associated binary operators + and × such that various field axioms hold. The real numbers R are an example of a field with uncountably many elements. Halo makes use of finite fields which have a finite number of elements. Finite fields are fully classified as follows: if F is a finite field, it contains ∣F∣=pk elements for some integer k≥1 and some prime p; any two finite fields with the same number of elements are isomorphic. In particular, all of the arithmetic in a prime field Fp​ is isomorphic to addition and multiplication of integers modulo p, i.e. in Zp​. This is why we often refer to p as the modulus . We'll write a field as Fq​ where q=pk. The prime p is called its characteristic . In the cases where k>1 the field Fq​ is a k-degree extension of the field Fp​. (By analogy, the complex numbers C=R(i) are an extension of the real numbers.) However, in Halo we do not use extension fields. Whenever we write Fp​ we are referring to what we call a prime field which has a prime p number of elements, i.e. k=1. Important notes: There are two special elements in any field: 0, the additive identity, and 1, the multiplicative identity. The least significant bit of a field element, when represented as an integer in binary format, can be interpreted as its \"sign\" to help distinguish it from its additive inverse (negation). This is because for some nonzero element a which has a least significant bit 0 we have that −a=p−a has a least significant bit 1, and vice versa. We could also use whether or not an element is larger than (p−1)/2 to give it a \"sign.\" Finite fields will be useful later for constructing polynomials and elliptic curves . Elliptic curves are examples of groups, which we discuss next.","breadcrumbs":"Background Material » Fields » Fields","id":"155","title":"Fields"},"156":{"body":"Groups are simpler and more limited than fields; they have only one binary operator ⋅ and fewer axioms. They also have an identity, which we'll denote as 1. Any non-zero element a in a group has an inverse b=a−1, which is the unique element b such that a⋅b=1. For example, the set of nonzero elements of Fp​ forms a group, where the group operation is given by multiplication on the field. (aside) Additive vs multiplicative notation If ⋅ is written as × or omitted (i.e. a⋅b written as ab), the identity as 1, and inversion as a−1, as we did above, then we say that the group is \"written multiplicatively\". If ⋅ is written as +, the identity as 0 or O, and inversion as −a, then we say it is \"written additively\". It's conventional to use additive notation for elliptic curve groups, and multiplicative notation when the elements come from a finite field. When additive notation is used, we also write [k]A=k timesA+A+⋯+A​​ for nonnegative k and call this \"scalar multiplication\"; we also often use uppercase letters for variables denoting group elements. When multiplicative notation is used, we also write ak=k timesa×a×⋯×a​​ and call this \"exponentiation\". In either case we call the scalar k such that [k]g=a or gk=a the \"discrete logarithm\" of a to base g. We can extend scalars to negative integers by inversion, i.e. [−k]A+[k]A=O or a−k×ak=1. The order of an element a of a finite group is defined as the smallest positive integer k such that ak=1 (in multiplicative notation) or [k]a=O (in additive notation). The order of the group is the number of elements. Groups always have a generating set , which is a set of elements such that we can produce any element of the group as (in multiplicative terminology) a product of powers of those elements. So if the generating set is g1..k​, we can produce any element of the group as i=1∏k​giai​​. There can be many different generating sets for a given group. A group is called cyclic if it has a (not necessarily unique) generating set with only a single element — call it g. In that case we can say that g generates the group, and that the order of g is the order of the group. Any finite cyclic group G of order n is isomorphic to the integers modulo n (denoted Z/nZ), such that: the operation ⋅ in G corresponds to addition modulo n; the identity in G corresponds to 0; some generator g∈G corresponds to 1. Given a generator g, the isomorphism is always easy to compute in the Z/nZ→G direction; it is just a↦ga (or in additive notation, a↦[a]g). It may be difficult in general to compute in the G→Z/nZ direction; we'll discuss this further when we come to elliptic curves . If the order n of a finite group is prime, then the group is cyclic, and every non-identity element is a generator.","breadcrumbs":"Background Material » Fields » Groups","id":"156","title":"Groups"},"157":{"body":"We use the notation Fp×​ for the multiplicative group (i.e. the group operation is multiplication in Fp​) over the set Fp​−{0}. A quick way of obtaining the inverse in Fp×​ is a−1=ap−2. The reason for this stems from Fermat's little theorem , which states that ap=a(modp) for any integer a. If a is nonzero, we can divide by a twice to get ap−2=a−1. Let's assume that α is a generator of Fp×​, so it has order p−1 (equal to the number of elements in Fp×​). Therefore, for any element in a∈Fp×​ there is a unique integer i∈{0..p−2} such that a=αi. Notice that a×b where a,b∈Fp×​ can really be interpreted as αi×αj where a=αi and b=αj. Indeed, it holds that αi×αj=αi+j for all 0≤i,j = MyCircuit { a: None, lookup_table, }; // Generate the DOT graph string. let dot_string = halo2_proofs::dev::circuit_dot_graph(&circuit); // Now you can either handle it in Rust, or just // print it out to use with command-line tools. print!(\"{}\", dot_string);\n}","breadcrumbs":"User Documentation » Developer tools » Circuit structure","id":"16","title":"Circuit structure"},"160":{"body":"In a field Fp​ exactly half of all nonzero elements are squares; the remainder are non-squares or \"quadratic non-residues\". In order to see why, consider an α that generates the 2-order multiplicative subgroup of Fp×​ (this exists because p−1 is divisible by 2 since p is a prime greater than 2) and β that generates the t-order multiplicative subgroup of Fp×​ where p−1=2t. Then every element a∈Fp×​ can be written uniquely as αi⋅βj with i∈Z2​ and j∈Zt​. Half of all elements will have i=0 and the other half will have i=1. Let's consider the simple case where p≡3(mod4) and so t is odd (if t is even, then p−1 would be divisible by 4, which contradicts p being 3(mod4)). If a∈Fp×​ is a square, then there must exist b=αi⋅βj such that b2=a. But this means that a=(αi⋅βj)2=α2i⋅β2j=β2j. In other words, all squares in this particular field do not generate the 2-order multiplicative subgroup, and so since half of the elements generate the 2-order subgroup then at most half of the elements are square. In fact exactly half of the elements are square (since squaring each nonsquare element gives a unique square). This means we can assume all squares can be written as βm for some m, and therefore finding the square root is a matter of exponentiating by 2−1(modt). In the event that p≡1(mod4) then things get more complicated because 2−1(modt) does not exist. Let's write p−1 as 2k⋅t with t odd. The case k=0 is impossible, and the case k=1 is what we already described, so consider k≥2. α generates a 2k-order multiplicative subgroup and β generates the odd t-order multiplicative subgroup. Then every element a∈Fp×​ can be written as αi⋅βj for i∈Z2k​ and j∈Zt​. If the element is a square, then there exists some b=a​ which can be written b=αi′⋅βj′ for i′∈Z2k​ and j′∈Zt​. This means that a=b2=α2i′⋅β2j′, therefore we have i≡2i′(mod2k), and j≡2j′(modt). i would have to be even in this case because otherwise it would be impossible to have i≡2i′(mod2k) for any i′. In the case that a is not a square, then i is odd, and so half of all elements are squares. In order to compute the square root, we can first raise the element a=αi⋅βj to the power t to \"kill\" the t-order component, giving at=αit(mod2)k⋅βjt(modt)=αit(mod2)k and then raise this result to the power t−1(mod2k) to undo the effect of the original exponentiation on the 2k-order component: (αitmod2k)t−1(mod2k)=αi (since t is relatively prime to 2k). This leaves bare the αi value which we can trivially handle. We can similarly kill the 2k-order component to obtain βj⋅2−1(modt), and put the values together to obtain the square root. It turns out that in the cases k=2,3 there are simpler algorithms that merge several of these exponentiations together for efficiency. For other values of k, the only known way is to manually extract i by squaring until you obtain the identity for every single bit of i. This is the essence of the Tonelli-Shanks square root algorithm and describes the general strategy. (There is another square root algorithm that uses quadratic extension fields, but it doesn't pay off in efficiency until the prime becomes quite large.)","breadcrumbs":"Background Material » Fields » Square roots","id":"160","title":"Square roots"},"161":{"body":"In the previous sections we wrote p−1=2k⋅t with t odd, and stated that an element α∈Fp×​ generated the 2k-order subgroup. For convenience, let's denote n:=2k. The elements {1,α,…,αn−1} are known as the nth roots of unity . The primitive root of unity , ω, is an nth root of unity such that ωi=1 except when i≡0(modn). Important notes: If α is an nth root of unity, α satisfies αn−1=0. If α=1, then 1+α+α2+⋯+αn−1=0. Equivalently, the roots of unity are solutions to the equation Xn−1=(X−1)(X−α)(X−α2)⋯(X−αn−1). ω2n​+i=−ωi​ (\"Negation lemma\") . Proof: ωn=1​⟹ωn−1=0⟹(ωn/2+1)(ωn/2−1)=0.​ Since the order of ω is n, ωn/2=1. Therefore, ωn/2=−1. (ω2n​+i)2=(ωi)2​ (\"Halving lemma\") . Proof: (ω2n​+i)2=ωn+2i=ωn⋅ω2i=ω2i=(ωi)2. In other words, if we square each element in the nth roots of unity, we would get back only half the elements, {(ωni​)2}={ωn/2​} (i.e. the 2n​th roots of unity). There is a two-to-one mapping between the elements and their squares.","breadcrumbs":"Background Material » Fields » Roots of unity","id":"161","title":"Roots of unity"},"162":{"body":"Friedman, R. (n.d.) \"Cyclic Groups and Elementary Number Theory II\" (p. 5).","breadcrumbs":"Background Material » Fields » References","id":"162","title":"References"},"163":{"body":"Let A(X) be a polynomial over Fp​ with formal indeterminate X. As an example, A(X)=a0​+a1​X+a2​X2+a3​X3 defines a degree-3 polynomial. a0​ is referred to as the constant term. Polynomials of degree n−1 have n coefficients. We will often want to compute the result of replacing the formal indeterminate X with some concrete value x, which we denote by A(x). In mathematics this is commonly referred to as \"evaluating A(X) at a point x\". The word \"point\" here stems from the geometrical usage of polynomials in the form y=A(x), where (x,y) is the coordinate of a point in two-dimensional space. However, the polynomials we deal with are almost always constrained to equal zero, and x will be an element of some field . This should not be confused with points on an elliptic curve , which we also make use of, but never in the context of polynomial evaluation. Important notes: Multiplication of polynomials produces a product polynomial that is the sum of the degrees of its factors. Polynomial division subtracts from the degree. deg(A(X)B(X))=deg(A(X))+deg(B(X)), deg(A(X)/B(X))=deg(A(X))−deg(B(X)). Given a polynomial A(X) of degree n−1, if we obtain n evaluations of the polynomial at distinct points then these evaluations perfectly define the polynomial. In other words, given these evaluations we can obtain a unique polynomial A(X) of degree n−1 via polynomial interpolation. [a0​,a1​,⋯,an−1​] is the coefficient representation of the polynomial A(X). Equivalently, we could use its evaluation representation [(x0​,A(x0​)),(x1​,A(x1​)),⋯,(xn−1​,A(xn−1​))] at n distinct points. Either representation uniquely specifies the same polynomial. (aside) Horner's rule Horner's rule allows for efficient evaluation of a polynomial of degree n−1, using only n−1 multiplications and n−1 additions. It is the following identity: a0​​+a1​X+a2​X2+⋯+an−1​Xn−1=a0​+X(a1​+X(a2​+⋯+X(an−2​+Xan−1​))),​","breadcrumbs":"Background Material » Polynomials » Polynomials","id":"163","title":"Polynomials"},"164":{"body":"The FFT is an efficient way of converting between the coefficient and evaluation representations of a polynomial. It evaluates the polynomial at the nth roots of unity {ω0,ω1,⋯,ωn−1}, where ω is a primitive nth root of unity. By exploiting symmetries in the roots of unity, each round of the FFT reduces the evaluation into a problem only half the size. Most commonly we use polynomials of length some power of two, n=2k, and apply the halving reduction recursively.","breadcrumbs":"Background Material » Polynomials » Fast Fourier Transform (FFT)","id":"164","title":"Fast Fourier Transform (FFT)"},"165":{"body":"In the coefficient representation, it takes O(n2) operations to multiply two polynomials A(X)⋅B(X)=C(X): A(X)B(X)C(X)​=a0​+a1​X+a2​X2+⋯+an−1​Xn−1,=b0​+b1​X+b2​X2+⋯+bn−1​Xn−1,=a0​⋅(b0​+b1​X+b2​X2+⋯+bn−1​Xn−1)+a1​X⋅(b0​+b1​X+b2​X2+⋯+bn−1​Xn−1)+⋯+an−1​Xn−1⋅(b0​+b1​X+b2​X2+⋯+bn−1​Xn−1),​ where each of the n terms in the first polynomial has to be multiplied by the n terms of the second polynomial. In the evaluation representation, however, polynomial multiplication only requires O(n) operations: ABC​:{(x0​,A(x0​)),(x1​,A(x1​)),⋯,(xn−1​,A(xn−1​))},:{(x0​,B(x0​)),(x1​,B(x1​)),⋯,(xn−1​,B(xn−1​))},:{(x0​,A(x0​)B(x0​)),(x1​,A(x1​)B(x1​)),⋯,(xn−1​,A(xn−1​)B(xn−1​))},​ where each evaluation is multiplied pointwise. This suggests the following strategy for fast polynomial multiplication: Evaluate polynomials at all n points; Perform fast pointwise multiplication in the evaluation representation (O(n)); Convert back to the coefficient representation. The challenge now is how to evaluate and interpolate the polynomials efficiently. Naively, evaluating a polynomial at n points would require O(n2) operations (we use the O(n) Horner's rule at each point): ⎣⎡​A(1)A(ω)A(ω2)⋮A(ωn−1)​⎦⎤​=⎣⎡​111⋮1​1ωω2⋮ωn−1​1ω2ω2⋅2⋮ω2(n−1)​………⋯​1ωn−1ω2⋅(n−1)⋮ω(n−1)2​⎦⎤​⋅⎣⎡​a0​a1​a2​⋮an−1​​⎦⎤​. For convenience, we will denote the matrices above as: A^=Vω​⋅A. (A^ is known as the Discrete Fourier Transform of A; Vω​ is also called the Vandermonde matrix .)","breadcrumbs":"Background Material » Polynomials » Motivation: Fast polynomial multiplication","id":"165","title":"Motivation: Fast polynomial multiplication"},"166":{"body":"Our strategy is to divide a DFT of size n into two interleaved DFTs of size n/2. Given the polynomial A(X)=a0​+a1​X+a2​X2+⋯+an−1​Xn−1, we split it up into even and odd terms: Aeven​Aodd​​=a0​+a2​X+⋯+an−2​X2n​−1,=a1​+a3​X+⋯+an−1​X2n​−1.​ To recover the original polynomial, we do A(X)=Aeven​(X2)+XAodd​(X2). Trying this out on points ωni​ and ωn2n​+i​, i∈[0..2n​−1], we start to notice some symmetries: A(ωni​)A(ωn2n​+i​)​=Aeven​((ωni​)2)+ωni​Aodd​((ωni​)2),=Aeven​((ωn2n​+i​)2)+ωn2n​+i​Aodd​((ωn2n​+i​)2)=Aeven​((−ωni​)2)−ωni​Aodd​((−ωni​)2)←(negation lemma)=Aeven​((ωni​)2)−ωni​Aodd​((ωni​)2).​ Notice that we are only evaluating Aeven​(X) and Aodd​(X) over half the domain {(ωn0​)2,(ωn​)2,⋯,(ωn2n​−1​)2}={ωn/2i​},i=[0..2n​−1] (halving lemma). This gives us all the terms we need to reconstruct A(X) over the full domain {ω0,ω,⋯,ωn−1}: which means we have transformed a length-n DFT into two length-2n​ DFTs. We choose n=2k to be a power of two (by zero-padding if needed), and apply this divide-and-conquer strategy recursively. By the Master Theorem [1] , this gives us an evaluation algorithm with O(nlog2​n) operations, also known as the Fast Fourier Transform (FFT).","breadcrumbs":"Background Material » Polynomials » The (radix-2) Cooley-Tukey algorithm","id":"166","title":"The (radix-2) Cooley-Tukey algorithm"},"167":{"body":"So we've evaluated our polynomials and multiplied them pointwise. What remains is to convert the product from the evaluation representation back to coefficient representation. To do this, we simply call the FFT on the evaluation representation. However, this time we also: replace ωi by ω−i in the Vandermonde matrix, and multiply our final result by a factor of 1/n. In other words: A=n1​Vω−1​⋅A^. (To understand why the inverse FFT has a similar form to the FFT, refer to Slide 13-1 of [2] . The below image was also taken from [2] .)","breadcrumbs":"Background Material » Polynomials » Inverse FFT","id":"167","title":"Inverse FFT"},"168":{"body":"The Schwartz-Zippel lemma informally states that \"different polynomials are different at most points.\" Formally, it can be written as follows: Let p(x1​,x2​,⋯,xn​) be a nonzero polynomial of n variables with degree d. Let S be a finite set of numbers with at least d elements in it. If we choose random α1​,α1​,⋯,αn​ from S, Pr[p(α1​,α2​,⋯,αn​)=0]≤∣S∣d​. In the familiar univariate case p(X), this reduces to saying that a nonzero polynomial of degree d has at most d roots. The Schwartz-Zippel lemma is used in polynomial equality testing. Given two multi-variate polynomials p1​(x1​,⋯,xn​) and p2​(x1​,⋯,xn​) of degrees d1​,d2​ respectively, we can test if p1​(α1​,⋯,αn​)−p2​(α1​,⋯,αn​)=0 for random α1​,⋯,αn​←S, where the size of S is at least ∣S∣≥(d1​+d2​). If the two polynomials are identical, this will always be true, whereas if the two polynomials are different then the equality holds with probability at most ∣S∣max(d1​,d2​)​.","breadcrumbs":"Background Material » Polynomials » The Schwartz-Zippel lemma","id":"168","title":"The Schwartz-Zippel lemma"},"169":{"body":"Consider the order-n multiplicative subgroup H with primitive root of unity ω. For all ωi∈H,i∈[n−1], we have (ωi)n=(ωn)i=(ω0)i=1. In other words, every element of H fulfils the equation ZH​(X)​=Xn−1=(X−ω0)(X−ω1)(X−ω2)⋯(X−ωn−1),​ meaning every element is a root of ZH​(X). We call ZH​(X) the vanishing polynomial over H because it evaluates to zero on all elements of H. This comes in particularly handy when checking polynomial constraints. For instance, to check that A(X)+B(X)=C(X) over H, we simply have to check that A(X)+B(X)−C(X) is some multiple of ZH​(X). In other words, if dividing our constraint by the vanishing polynomial still yields some polynomial ZH​(X)A(X)+B(X)−C(X)​=H(X), we are satisfied that A(X)+B(X)−C(X)=0 over H.","breadcrumbs":"Background Material » Polynomials » Vanishing polynomial","id":"169","title":"Vanishing polynomial"},"17":{"body":"The cost-model binary takes high-level parameters for a circuit design, and estimates the verification cost, as well as resulting proof size. Usage: cargo run --example cost-model -- [OPTIONS] k Positional arguments: k 2^K bound on the number of rows. Optional arguments: -h, --help Print this message. -a, --advice R[,R..] An advice column with the given rotations. May be repeated. -i, --instance R[,R..] An instance column with the given rotations. May be repeated. -f, --fixed R[,R..] A fixed column with the given rotations. May be repeated. -g, --gate-degree D Maximum degree of the custom gates. -l, --lookup N,I,T A lookup over N columns with max input degree I and max table degree T. May be repeated. -p, --permutation N A permutation over N columns. May be repeated. For example, to estimate the cost of a circuit with three advice columns and one fixed column (with various rotations), and a maximum gate degree of 4: > cargo run --example cost-model -- -a 0,1 -a 0 -a-0,-1,1 -f 0 -g 4 11 Finished dev [unoptimized + debuginfo] target(s) in 0.03s Running `target/debug/examples/cost-model -a 0,1 -a 0 -a 0,-1,1 -f 0 -g 4 11`\nCircuit { k: 11, max_deg: 4, advice_columns: 3, lookups: 0, permutations: [], column_queries: 7, point_sets: 3, estimator: Estimator,\n}\nProof size: 1440 bytes\nVerification: at least 81.689ms","breadcrumbs":"User Documentation » Developer tools » Cost estimator","id":"17","title":"Cost estimator"},"170":{"body":"TODO: explain what a basis is in general (briefly). Polynomials are commonly written in the monomial basis (e.g. X,X2,...Xn). However, when working over a multiplicative subgroup of order n, we find a more natural expression in the Lagrange basis. Consider the order-n multiplicative subgroup H with primitive root of unity ω. The Lagrange basis corresponding to this subgroup is a set of functions {Li​}i=0n−1​, where Li​(ωj)={10​if i=j,otherwise.​ We can write this more compactly as Li​(ωj)=δij​, where δ is the Kronecker delta function. Now, we can write our polynomial as a linear combination of Lagrange basis functions, A(X)=i=0∑n−1​ai​Li​(X),X∈H, which is equivalent to saying that p(X) evaluates to a0​ at ω0, to a1​ at ω1, to a2​ at ω2,⋯, and so on. When working over a multiplicative subgroup, the Lagrange basis function has a convenient sparse representation of the form Li​(X)=X−ωici​⋅(Xn−1)​, where ci​ is the barycentric weight. (To understand how this form was derived, refer to [3] .) For i=0, we have c=1/n⟹L0​(X)=n1​X−1(Xn−1)​. Suppose we are given a set of evaluation points {x0​,x1​,⋯,xn−1​}. Since we cannot assume that the xi​'s form a multiplicative subgroup, we consider also the Lagrange polynomials Li​'s in the general case. Then we can construct: Li​(X)=j=i∏​xi​−xj​X−xj​​,i∈[0..n−1]. Here, every X=xj​=xi​ will produce a zero numerator term (xj​−xj​), causing the whole product to evaluate to zero. On the other hand, X=xi​ will evaluate to xi​−xj​xi​−xj​​ at every term, resulting in an overall product of one. This gives the desired Kronecker delta behaviour Li​(xj​)=δij​ on the set {x0​,x1​,⋯,xn−1​}.","breadcrumbs":"Background Material » Polynomials » Lagrange basis functions","id":"170","title":"Lagrange basis functions"},"171":{"body":"Given a polynomial in its evaluation representation A:{(x0​,A(x0​)),(x1​,A(x1​)),⋯,(xn−1​,A(xn−1​))}, we can reconstruct its coefficient form in the Lagrange basis: A(X)=i=0∑n−1​A(xi​)Li​(X), where X∈{x0​,x1​,⋯,x1−n​}.","breadcrumbs":"Background Material » Polynomials » Lagrange interpolation","id":"171","title":"Lagrange interpolation"},"172":{"body":"Dasgupta, S., Papadimitriou, C. H., & Vazirani, U. V. (2008). \"Algorithms\" (ch. 2). New York: McGraw-Hill Higher Education. Golin, M. (2016). \"The Fast Fourier Transform and Polynomial Multiplication\" [lecture notes], COMP 3711H Design and Analysis of Algorithms, Hong Kong University of Science and Technology. Berrut, J. and Trefethen, L. (2004). \"Barycentric Lagrange Interpolation.\"","breadcrumbs":"Background Material » Polynomials » References","id":"172","title":"References"},"173":{"body":"In the section Inverses and groups we introduced the concept of groups . A group has an identity and a group operation. In this section we will write groups additively, i.e. the identity is O and the group operation is +. Some groups can be used as cryptographic groups . At the risk of oversimplifying, this means that the problem of finding a discrete logarithm of a group element P to a given base G, i.e. finding x such that P=[x]G, is hard in general.","breadcrumbs":"Background Material » Cryptographic groups » Cryptographic groups","id":"173","title":"Cryptographic groups"},"174":{"body":"The Pedersen commitment [ P99 ] is a way to commit to a secret message in a verifiable way. It uses two random public generators G,H∈G, where G is a cryptographic group of order p. A random secret r is chosen in Zq​, and the message to commit to m is from any subset of Zq​. The commitment is c=Commit(m,r)=[m]G+[r]H. To open the commitment, the committer reveals m and r, thus allowing anyone to verify that c is indeed a commitment to m. Notice that the Pedersen commitment scheme is homomorphic: Commit(m,r)+Commit(m′,r′)​=[m]G+[r]H+[m′]G+[r′]H=[m+m′]G+[r+r′]H=Commit(m+m′,r+r′).​ Assuming the discrete log assumption holds, Pedersen commitments are also perfectly hiding and computationally binding: hiding : the adversary chooses messages m0​,m1​. The committer commits to one of these messages c=Commit(mb​;r),b∈{0,1}. Given c, the probability of the adversary guessing the correct b is no more than 21​. binding : the adversary cannot pick two different messages m0​=m1​, and randomness r0​,r1​, such that Commit(m0​,r0​)=Commit(m1​,r1​).","breadcrumbs":"Background Material » Cryptographic groups » Pedersen commitment","id":"174","title":"Pedersen commitment"},"175":{"body":"We can use a variant of the Pedersen commitment scheme to commit to multiple messages at once, m=(m1​,⋯,mn​). This time, we'll have to sample a corresponding number of random public generators G=(G0​,⋯,Gn−1​), along with a single random generator H as before (for use in hiding). Then, our commitment scheme is: Commit(m;r)​=Commit((m0​,⋯,mn−1​);r)=[r]H+[m0​]G0​+⋯+[mn−1​]Gn−1​=[r]H+i=0∑n−1​[mi​]Gi​.​ TODO: is this positionally binding?","breadcrumbs":"Background Material » Cryptographic groups » Vector Pedersen commitment","id":"175","title":"Vector Pedersen commitment"},"176":{"body":"An example of a protocol that uses cryptographic groups is Diffie--Hellman key agreement [ DH1976 ]. The Diffie--Hellman protocol is a method for two users, Alice and Bob, to generate a shared private key. It proceeds as follows: Alice and Bob publicly agree on two prime numbers, p and G, where p is large and G is a primitive root (modp). (Note that g is a generator of the group Fp×​.) Alice chooses a large random number a as her private key. She computes her public key A=[a]G(modp), and sends A to Bob. Similarly, Bob chooses a large random number b as his private key. He computes his public key B=[b]G(modp), and sends B to Alice. Now both Alice and Bob compute their shared key K=[ab]G(modp), which Alice computes as K=[a]B(modp)=[a]([b]G)(modp), and Bob computes as K=[b]A(modp)=[b]([a]G)(modp). A potential eavesdropper would need to derive K=[ab]g(modp) knowing only g,p,A=[a]G, and B=[b]G: in other words, they would need to either get the discrete logarithm a from A=[a]G or b from B=[b]G, which we assume to be computationally infeasible in Fp×​. More generally, protocols that use similar ideas to Diffie--Hellman are used throughout cryptography. One way of instantiating a cryptographic group is as an elliptic curve . Before we go into detail on elliptic curves, we'll describe some algorithms that can be used for any group.","breadcrumbs":"Background Material » Cryptographic groups » Diffie--Hellman","id":"176","title":"Diffie--Hellman"},"177":{"body":"","breadcrumbs":"Background Material » Cryptographic groups » Multiscalar multiplication","id":"177","title":"Multiscalar multiplication"},"178":{"body":"Reference: https://jbootle.github.io/Misc/pippenger.pdf","breadcrumbs":"Background Material » Cryptographic groups » TODO: Pippenger's algorithm","id":"178","title":"TODO: Pippenger's algorithm"},"179":{"body":"Elliptic curves constructed over finite fields are another important cryptographic tool. We use elliptic curves because they provide a cryptographic group , i.e. a group in which the discrete logarithm problem (discussed below) is hard. There are several ways to define the curve equation, but for our purposes, let Fp​ be a large (255-bit) field, and then let the set of solutions (x,y) to y2=x3+b for some constant b define the Fp​-rational points on an elliptic curve E(Fp​). These (x,y) coordinates are called \"affine coordinates\". Each of the Fp​-rational points, together with a \"point at infinity\" O that serves as the group identity, can be interpreted as an element of a group. By convention, elliptic curve groups are written additively. \"Three points on a line sum to zero, which is the point at infinity.\" The group addition law is simple: to add two points together, find the line that intersects both points and obtain the third point, and then negate its y-coordinate. The case that a point is being added to itself, called point doubling, requires special handling: we find the line tangent to the point, and then find the single other point that intersects this line and then negate. Otherwise, in the event that a point is being \"added\" to its negation, the result is the point at infinity. The ability to add and double points naturally gives us a way to scale them by integers, called scalars . The number of points on the curve is the group order. If this number is a prime q, then the scalars can be considered as elements of a scalar field , Fq​. Elliptic curves, when properly designed, have an important security property. Given two random elements G,H∈E(Fp​) finding a such that [a]G=H, otherwise known as the discrete log of H with respect to G, is considered computationally infeasible with classical computers. This is called the elliptic curve discrete log assumption. If an elliptic curve group G has prime order q (like the ones used in Halo 2), then it is a finite cyclic group. Recall from the section on groups that this implies it is isomorphic to Z/qZ, or equivalently, to the scalar field Fq​. Each possible generator G fixes the isomorphism; then an element on the scalar side is precisely the discrete log of the corresponding group element with respect to G. In the case of a cryptographically secure elliptic curve, the isomorphism is hard to compute in the G→Fq​ direction because the elliptic curve discrete log problem is hard. It is sometimes helpful to make use of this isomorphism by thinking of group-based cryptographic protocols and algorithms in terms of the scalars instead of in terms of the group elements. This can make proofs and notation simpler. For instance, it has become common in papers on proof systems to use the notation [x] to denote a group element with discrete log x, where the generator is implicit. We also used this idea in the \" distinct-x theorem \", in order to prove correctness of optimizations for elliptic curve scalar multiplication in Sapling, and an endomorphism-based optimization in Appendix C of the original Halo paper .","breadcrumbs":"Background Material » Elliptic curves » Elliptic curves","id":"179","title":"Elliptic curves"},"18":{"body":"Let's start with a simple circuit, to introduce you to the common APIs and how they are used. The circuit will take a public input c, and will prove knowledge of two private inputs a and b such that a2⋅b2=c.","breadcrumbs":"User Documentation » A simple example » A simple example","id":"18","title":"A simple example"},"180":{"body":"","breadcrumbs":"Background Material » Elliptic curves » Curve arithmetic","id":"180","title":"Curve arithmetic"},"181":{"body":"The simplest situation is doubling a point (x0​,y0​). Continuing with our example y2=x3+b, this is done first by computing the derivative λ=dxdy​=2y3x2​. To obtain expressions for (x1​,y1​)=(x0​,y0​)+(x0​,y0​), we consider x1​−x0​−y1​−y0​​=λ​⟹−y1​=λ(x1​−x0​)+y0​⟹y1​=λ(x0​−x1​)−y0​​.​ To get the expression for x1​, we substitute y=λ(x0​−x)−y0​ into the elliptic curve equation: y2=x3+b​⟹(λ(x0​−x)−y0​)2=x3+b⟹x3−λ2x2+⋯=0←(rearranging terms)=(x−x0​)(x−x0​)(x−x1​)←(known roots x0​,x0​,x1​)=x3−(x0​+x0​+x1​)x2+⋯.​ Comparing coefficients for the x2 term gives us λ2=x0​+x0​+x1​⟹x1​=λ2−2x0​​.","breadcrumbs":"Background Material » Elliptic curves » Point doubling","id":"181","title":"Point doubling"},"182":{"body":"This unfortunately requires an expensive inversion of 2y. We can avoid this by arranging our equations to \"defer\" the computation of the inverse, since we often do not need the actual affine (x′,y′) coordinate of the resulting point immediately after an individual curve operation. Let's introduce a third coordinate Z and scale our curve equation by Z3 like so: Z3y2=Z3x3+Z3b Our original curve is just this curve at the restriction Z=1. If we allow the affine point (x,y) to be represented by X=xZ, Y=yZ and Z=0 then we have the homogenous projective curve Y2Z=X3+Z3b. Obtaining (x,y) from (X,Y,Z) is as simple as computing (X/Z,Y/Z) when Z=0. (When Z=0, we are dealing with the point at infinity O:=(0:1:0).) In this form, we now have a convenient way to defer the inversion required by doubling a point. The general strategy is to express x′,y′ as rational functions using x=X/Z and y=Y/Z, rearrange to make their denominators the same, and then take the resulting point (X,Y,Z) to have Z be the shared denominator and X=x′Z,Y=y′Z. Projective coordinates are often, but not always, more efficient than affine coordinates. There may be exceptions to this when either we have a different way to apply Montgomery's trick, or when we're in the circuit setting where multiplications and inversions are about equally as expensive (at least in terms of circuit size). The following shows an example of doubling a point (X,Y,Z)=(xZ,yZ,Z) without an inversion. Substituting with X,Y,Z gives us λ=2y3x2​=2(Y/Z)3(X/Z)2​=2YZ3X2​ and gives us x′y′​=λ2−2x=λ2−Z2X​=4Y2Z29X4​−Z2X​=4Y2Z29X4−8XY2Z​=8Y3Z318X4YZ−16XY3Z2​=λ(x−x′)−y=λ(ZX​−4Y2Z29X4−8XY2Z​)−ZY​=2YZ3X2​(ZX​−4Y2Z29X4−8XY2Z​)−ZY​=2YZ23X3​−8Y3Z327X6−24X3Y2Z​−ZY​=8Y3Z312X3Y2Z−8Y4Z2−27X6+24X3Y2Z​​ Notice how the denominators of x′ and y′ are the same. Thus, instead of computing (x′,y′) we can compute (X,Y,Z) with Z=8Y3Z3 and X,Y set to the corresponding numerators such that X/Z=x′ and Y/Z=y′. This completely avoids the need to perform an inversion when doubling, and something analogous to this can be done when adding two distinct points.","breadcrumbs":"Background Material » Elliptic curves » Projective coordinates","id":"182","title":"Projective coordinates"},"183":{"body":"We now add two points with distinct x-coordinates, P=(x0​,y0​) and Q=(x1​,y1​), where x0​=x1​, to obtain R=P+Q=(x2​,y2​). The line PQ​ has slope λ=fracy1​−y0​x1​−x0​⟹y−y0​=λ⋅(x−x0​). Using the expression for PQ​, we compute y-coordinate −y2​ of −R as: −y2​−y0​=λ⋅(x2​−x0​)⟹y2​=(x0​−x2​)−y0​​. Plugging the expression for PQ​ into the curve equation y2=x3+b yields y2=x3+b​⟹(λ⋅(x−x0​)+y0​)2=x3+b⟹x3−λ2x2+⋯=0←(rearranging terms)=(x−x0​)(x−x1​)(x−x2​)←(known roots x0​,x1​,x2​)=x3−(x0​+x1​+x2​)x2+⋯.​ Comparing coefficients for the x2 term gives us λ2=x0​+x1​+x2​⟹x2​=λ2−x0​−x1​​. Important notes: There exist efficient formulae [1] for point addition that do not have edge cases (so-called \"complete\" formulae) and that unify the addition and doubling cases together. The result of adding a point to its negation using those formulae produces Z=0, which represents the point at infinity. In addition, there are other models like the Jacobian representation where (x,y)=(xZ2,yZ3,Z) where the curve is rescaled by Z6 instead of Z3, and this representation has even more efficient arithmetic but no unified/complete formulae. We can easily compare two curve points (X1​,Y1​,Z1​) and (X2​,Y2​,Z2​) for equality in the homogenous projective coordinate space by \"homogenizing\" their Z-coordinates; the checks become X1​Z2​=X2​Z1​ and Y1​Z2​=Y2​Z1​.","breadcrumbs":"Background Material » Elliptic curves » Point addition","id":"183","title":"Point addition"},"184":{"body":"Imagine that Fp​ has a primitive cube root of unity, or in other words that 3∣p−1 and so an element ζp​ generates a 3-order multiplicative subgroup. Notice that a point (x,y) on our example elliptic curve y2=x3+b has two cousin points: (ζp​x,ζp2​x), because the computation x3 effectively kills the ζ component of the x-coordinate. Applying the map (x,y)↦(ζp​x,y) is an application of an endomorphism over the curve. The exact mechanics involved are complicated, but when the curve has a prime q number of points (and thus a prime \"order\") the effect of the endomorphism is to multiply the point by a scalar in Fq​ which is also a primitive cube root ζq​ in the scalar field.","breadcrumbs":"Background Material » Elliptic curves » Curve endomorphisms","id":"184","title":"Curve endomorphisms"},"185":{"body":"Given a point on the curve P=(x,y), we know that its negation −P=(x,−y) is also on the curve. To uniquely specify a point, we need only encode its x-coordinate along with the sign of its y-coordinate.","breadcrumbs":"Background Material » Elliptic curves » Curve point compression","id":"185","title":"Curve point compression"},"186":{"body":"As mentioned in the Fields section, we can interpret the least significant bit of a field element as its \"sign\", since its additive inverse will always have the opposite LSB. So we record the LSB of the y-coordinate as sign. Pallas and Vesta are defined over the Fp​ and Fq​ fields, which elements can be expressed in 255 bits. This conveniently leaves one unused bit in a 32-byte representation. We pack the y-coordinate sign bit into the highest bit in the representation of the x-coordinate: <----------------------------------- x --------------------------------->\nEnc(P) = [_ _ _ _ _ _ _ _] [_ _ _ _ _ _ _ _] ... [_ _ _ _ _ _ _ _] [_ _ _ _ _ _ _ sign] ^ <-------------------------------------> ^ LSB 30 bytes MSB The \"point at infinity\" O that serves as the group identity, does not have an affine (x,y) representation. However, it turns out that there are no points on either the Pallas or Vesta curve with x=0 or y=0. We therefore use the \"fake\" affine coordinates (0,0) to encode O, which results in the all-zeroes 32-byte array.","breadcrumbs":"Background Material » Elliptic curves » Serialization","id":"186","title":"Serialization"},"187":{"body":"When deserializing a compressed curve point, we first read the most significant bit as ysign, the sign of the y-coordinate. Then, we set this bit to zero to recover the original x-coordinate. If x=0,y=0, we return the \"point at infinity\" O. Otherwise, we proceed to compute y=x3+b​. Here, we read the least significant bit of y as sign. If sign == ysign, we already have the correct sign and simply return the curve point (x,y). Otherwise, we negate y and return (x,−y).","breadcrumbs":"Background Material » Elliptic curves » Deserialization","id":"187","title":"Deserialization"},"188":{"body":"Let Ep​ be an elliptic curve over a finite field Fp​, where p is a prime. We denote this by Ep​/Fp​. and we denote the group of points of Ep​ over Fp​, with order q=#E(Fp​). For this curve, we call Fp​ the \"base field\" and Fq​ the \"scalar field\". We instantiate our proof system over the elliptic curve Ep​/Fp​. This allows us to prove statements about Fq​-arithmetic circuit satisfiability. (aside) If our curve Ep​ is over Fp​, why is the arithmetic circuit instead in Fq​? The proof system is basically working on encodings of the scalars in the circuit (or more precisely, commitments to polynomials whose coefficients are scalars). The scalars are in Fq​ when their encodings/commitments are elliptic curve points in Ep​/Fp​. However, most of the verifier's arithmetic computations are over the base field Fp​, and are thus efficiently expressed as an Fp​-arithmetic circuit. (aside) Why are the verifier's computations (mainly) over Fp​? The Halo 2 verifier actually has to perform group operations using information output by the circuit. Group operations like point doubling and addition use arithmetic in Fp​, because the coordinates of points are in Fp​. This motivates us to construct another curve with scalar field Fp​, which has an Fp​-arithmetic circuit that can efficiently verify proofs from the first curve. As a bonus, if this second curve had base field Eq​/Fq​, it would generate proofs that could be efficiently verified in the first curve's Fq​-arithmetic circuit. In other words, we instantiate a second proof system over Eq​/Fq​, forming a 2-cycle with the first:","breadcrumbs":"Background Material » Elliptic curves » Cycles of curves","id":"188","title":"Cycles of curves"},"189":{"body":"Reference: https://github.com/zcash/pasta","breadcrumbs":"Background Material » Elliptic curves » TODO: Pallas-Vesta curves","id":"189","title":"TODO: Pallas-Vesta curves"},"19":{"body":"Firstly, we need to define the instructions that our circuit will rely on. Instructions are the boundary between high-level gadgets and the low-level circuit operations. Instructions may be as coarse or as granular as desired, but in practice you want to strike a balance between an instruction being large enough to effectively optimize its implementation, and small enough that it is meaningfully reusable. For our circuit, we will use three instructions: Load a private number into the circuit. Multiply two numbers. Expose a number as a public input to the circuit. We also need a type for a variable representing a number. Instruction interfaces provide associated types for their inputs and outputs, to allow the implementations to represent these in a way that makes the most sense for their optimization goals. trait NumericInstructions: Chip { /// Variable representing a number. type Num; /// Loads a number into the circuit as a private input. fn load_private(&self, layouter: impl Layouter, a: Option) -> Result; /// Loads a number into the circuit as a fixed constant. fn load_constant(&self, layouter: impl Layouter, constant: F) -> Result; /// Returns `c = a * b`. fn mul( &self, layouter: impl Layouter, a: Self::Num, b: Self::Num, ) -> Result; /// Exposes a number as a public input to the circuit. fn expose_public( &self, layouter: impl Layouter, num: Self::Num, row: usize, ) -> Result<(), Error>;\n}","breadcrumbs":"User Documentation » A simple example » Define instructions","id":"19","title":"Define instructions"},"190":{"body":"Sometimes it is useful to be able to produce a random point on an elliptic curve Ep​/Fp​ corresponding to some input, in such a way that no-one will know its discrete logarithm (to any other base). This is described in detail in the Internet draft on Hashing to Elliptic Curves . Several algorithms can be used depending on efficiency and security requirements. The framework used in the Internet Draft makes use of several functions: hash_to_field: takes a byte sequence input and maps it to a element in the base field Fp​ map_to_curve: takes an Fp​ element and maps it to Ep​.","breadcrumbs":"Background Material » Elliptic curves » Hashing to curves","id":"190","title":"Hashing to curves"},"191":{"body":"Reference: https://eprint.iacr.org/2019/403.pdf","breadcrumbs":"Background Material » Elliptic curves » TODO: Simplified SWU","id":"191","title":"TODO: Simplified SWU"},"192":{"body":"Renes, J., Costello, C., & Batina, L. (2016, May). \"Complete addition formulas for prime order elliptic curves.\" In Annual International Conference on the Theory and Applications of Cryptographic Techniques (pp. 403-428). Springer, Berlin, Heidelberg.","breadcrumbs":"Background Material » Elliptic curves » References","id":"192","title":"References"},"193":{"body":"We want to commit to some polynomial p(X)∈Fp​[X], and be able to provably evaluate the committed polynomial at arbitrary points. The naive solution would be for the prover to simply send the polynomial's coefficients to the verifier: however, this requires O(n) communication. Our polynomial commitment scheme gets the job done using O(logn) communication.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Polynomial commitment using inner product argument","id":"193","title":"Polynomial commitment using inner product argument"},"194":{"body":"Given a parameter d=2k, we generate the common reference string σ=(G,G,H,Fp​) defining certain constants for this scheme: G is a group of prime order p; G∈Gd is a vector of d random group elements; H∈G is a random group element; and Fp​ is the finite field of order p.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Setup","id":"194","title":"Setup"},"195":{"body":"The Pedersen vector commitment Commit is defined as Commit(σ,p(X);r)=⟨a,G⟩+[r]H, for some polynomial p(X)∈Fp​[X] and some blinding factor r∈Fp​. Here, each element of the vector ai​∈Fp​ is the coefficient for the ith degree term of p(X), and p(X) is of maximal degree d−1.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Commit","id":"195","title":"Commit"},"196":{"body":"The modified inner product argument is an argument of knowledge for the relation {((P,x,v);(a,r)):P=⟨a,G⟩+[r]H,v=⟨a,b⟩}​, where b=(1,x,x2,⋯,xd−1) is composed of increasing powers of the evaluation point x. This allows a prover to demonstrate to a verifier that the polynomial contained “inside” the commitment P evaluates to v at x, and moreover, that the committed polynomial has maximum degree d−1. The inner product argument proceeds in k=log2​d rounds. For our purposes, it is sufficient to know about its final outputs, while merely providing intuition about the intermediate rounds. (Refer to Section 3 in the Halo paper for a full explanation.) Before beginning the argument, the verifier selects a random group element U and sends it to the prover. We initialize the argument at round k, with the vectors a(k):=a, G(k):=G and b(k):=b. In each round j=k,k−1,⋯,1: the prover computes two values Lj​ and Rj​ by taking some inner product of a(j) with G(j) and b(j). Note that are in some sense \"cross-terms\": the lower half of a is used with the higher half of G and b, and vice versa: Lj​Rj​​=⟨alo(j)​,Ghi(j)​⟩+[lj​]H+[⟨alo(j)​,bhi(j)​⟩]U=⟨ahi(j)​,Glo(j)​⟩+[lj​]H+[⟨ahi(j)​,blo(j)​⟩]U​ the verifier issues a random challenge uj​; the prover uses uj​ to compress the lower and higher halves of a(j), thus producing a new vector of half the original length a(j−1)=ahi(j)​⋅uj−1​+alo(j)​⋅uj​. The vectors G(j) and b(j) are similarly compressed to give G(j−1) and b(j−1). a(j−1), G(j−1) and b(j−1) are input to the next round j−1. Note that at the end of the last round j=1, we are left with a:=a(0), G:=G(0), b:=b(0), each of length 1. The intuition is that these final scalars, together with the challenges {uj​} and \"cross-terms\" {Lj​,Rj​} from each round, encode the compression in each round. Since the prover did not know the challenges U,{uj​} in advance, they would have been unable to manipulate the round compressions. Thus, checking a constraint on these final terms should enforce that the compression had been performed correctly, and that the original a satisfied the relation before undergoing compression. Note that G,b are simply rearrangements of the publicly known G,b, with the round challenges {uj​} mixed in: this means the verifier can compute G,b independently and verify that the prover had provided those same values.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Open (prover) and OpenVerify (verifier)","id":"196","title":"Open (prover) and OpenVerify (verifier)"},"197":{"body":"Alternative terms: Induction; Accumulation scheme; Proof-carrying data However, the computation of G requires a length-2k multiexponentiation ⟨G,s⟩, where s is composed of the round challenges u1​,⋯,uk​ arranged in a binary counting structure. This is the linear-time computation that we want to amortise across a batch of proof instances. Instead of computing G, notice that we can express G as a commitment to a polynomial G=Commit(σ,g(X,u1​,⋯,uk​)), where g(X,u1​,⋯,uk​):=∏i=1k​(ui​+ui−1​X2i−1) is a polynomial with degree 2k−1. Since G is a commitment, it can be checked in an inner product argument. The verifier circuit witnesses G and brings G,u1​,⋯,uk​ out as public inputs to the proof π. The next verifier instance checks π using the inner product argument; this includes checking that G=Commit(g(X,u1​,⋯,uk​)) evaluates at some random point to the expected value for the given challenges u1​,⋯,uk​. Recall from the previous section that this check only requires logd work. At the end of checking π and G, the circuit is left with a new G′, along with the u1′​,⋯,uk′​ challenges sampled for the check. To fully accept π as valid, we should perform a linear-time computation of G′=⟨G,s′⟩. Once again, we delay this computation by witnessing G′ and bringing G,u1​,⋯,uk​ out as public inputs to the proof π. This goes on from one proof instance to the next, until we are satisfied with the size of our batch of proofs. We finally perform a single linear-time computation, thus deciding the validity of the whole batch. We recall from the section Cycles of curves that we can instantiate this protocol over a two-cycle, where a proof produced by one curve is efficiently verified in the circuit of the other curve. However, some of these verifier checks can actually be efficiently performed in the native circuit; these are \"deferred\" to the next native circuit (see diagram below) instead of being immediately passed over to the other curve.","breadcrumbs":"Background Material » Recursion » Recursion","id":"197","title":"Recursion"},"2":{"body":"Requires Rust 1.51 or higher. Minimum supported Rust version can be changed in the future, but it will be done with a minor version bump.","breadcrumbs":"halo2 » Minimum Supported Rust Version","id":"2","title":"Minimum Supported Rust Version"},"20":{"body":"For our circuit, we will build a chip that provides the above numeric instructions for a finite field. /// The chip that will implement our instructions! Chips store their own\n/// config, as well as type markers if necessary.\nstruct FieldChip { config: FieldConfig, _marker: PhantomData,\n} Every chip needs to implement the Chip trait. This defines the properties of the chip that a Layouter may rely on when synthesizing a circuit, as well as enabling any initial state that the chip requires to be loaded into the circuit. impl Chip for FieldChip { type Config = FieldConfig; type Loaded = (); fn config(&self) -> &Self::Config { &self.config } fn loaded(&self) -> &Self::Loaded { &() }\n}","breadcrumbs":"User Documentation » A simple example » Define a chip implementation","id":"20","title":"Define a chip implementation"},"21":{"body":"The chip needs to be configured with the columns, permutations, and gates that will be required to implement all of the desired instructions. /// Chip state is stored in a config struct. This is generated by the chip\n/// during configuration, and then stored inside the chip.\n#[derive(Clone, Debug)]\nstruct FieldConfig { /// For this chip, we will use two advice columns to implement our instructions. /// These are also the columns through which we communicate with other parts of /// the circuit. advice: [Column; 2], /// This is the public input (instance) column. instance: Column, // We need a selector to enable the multiplication gate, so that we aren't placing // any constraints on cells where `NumericInstructions::mul` is not being used. // This is important when building larger circuits, where columns are used by // multiple sets of instructions. s_mul: Selector,\n} impl FieldChip { fn construct(config: >::Config) -> Self { Self { config, _marker: PhantomData, } } fn configure( meta: &mut ConstraintSystem, advice: [Column; 2], instance: Column, constant: Column, ) -> >::Config { meta.enable_equality(instance); meta.enable_constant(constant); for column in &advice { meta.enable_equality(*column); } let s_mul = meta.selector(); // Define our multiplication gate! meta.create_gate(\"mul\", |meta| { // To implement multiplication, we need three advice cells and a selector // cell. We arrange them like so: // // | a0 | a1 | s_mul | // |-----|-----|-------| // | lhs | rhs | s_mul | // | out | | | // // Gates may refer to any relative offsets we want, but each distinct // offset adds a cost to the proof. The most common offsets are 0 (the // current row), 1 (the next row), and -1 (the previous row), for which // `Rotation` has specific constructors. let lhs = meta.query_advice(advice[0], Rotation::cur()); let rhs = meta.query_advice(advice[1], Rotation::cur()); let out = meta.query_advice(advice[0], Rotation::next()); let s_mul = meta.query_selector(s_mul); // Finally, we return the polynomial expressions that constrain this gate. // For our multiplication gate, we only need a single polynomial constraint. // // The polynomial expressions returned from `create_gate` will be // constrained by the proving system to equal zero. Our expression // has the following properties: // - When s_mul = 0, any value is allowed in lhs, rhs, and out. // - When s_mul != 0, this constrains lhs * rhs = out. vec![s_mul * (lhs * rhs - out)] }); FieldConfig { advice, instance, s_mul, } }\n}","breadcrumbs":"User Documentation » A simple example » Configure the chip","id":"21","title":"Configure the chip"},"22":{"body":"/// A variable representing a number.\n#[derive(Clone)]\nstruct Number(AssignedCell); impl NumericInstructions for FieldChip { type Num = Number; fn load_private( &self, mut layouter: impl Layouter, value: Option, ) -> Result { let config = self.config(); layouter.assign_region( || \"load private\", |mut region| { region .assign_advice( || \"private input\", config.advice[0], 0, || value.ok_or(Error::Synthesis), ) .map(Number) }, ) } fn load_constant( &self, mut layouter: impl Layouter, constant: F, ) -> Result { let config = self.config(); layouter.assign_region( || \"load constant\", |mut region| { region .assign_advice_from_constant(|| \"constant value\", config.advice[0], 0, constant) .map(Number) }, ) } fn mul( &self, mut layouter: impl Layouter, a: Self::Num, b: Self::Num, ) -> Result { let config = self.config(); layouter.assign_region( || \"mul\", |mut region: Region<'_, F>| { // We only want to use a single multiplication gate in this region, // so we enable it at region offset 0; this means it will constrain // cells at offsets 0 and 1. config.s_mul.enable(&mut region, 0)?; // The inputs we've been given could be located anywhere in the circuit, // but we can only rely on relative offsets inside this region. So we // assign new cells inside the region and constrain them to have the // same values as the inputs. a.0.copy_advice(|| \"lhs\", &mut region, config.advice[0], 0)?; b.0.copy_advice(|| \"rhs\", &mut region, config.advice[1], 0)?; // Now we can assign the multiplication result, which is to be assigned // into the output position. let value = a.0.value().and_then(|a| b.0.value().map(|b| *a * *b)); // Finally, we do the assignment to the output, returning a // variable to be used in another part of the circuit. region .assign_advice( || \"lhs * rhs\", config.advice[0], 1, || value.ok_or(Error::Synthesis), ) .map(Number) }, ) } fn expose_public( &self, mut layouter: impl Layouter, num: Self::Num, row: usize, ) -> Result<(), Error> { let config = self.config(); layouter.constrain_instance(num.0.cell(), config.instance, row) }\n}","breadcrumbs":"User Documentation » A simple example » Implement chip traits","id":"22","title":"Implement chip traits"},"23":{"body":"Now that we have the instructions we need, and a chip that implements them, we can finally build our circuit! /// The full circuit implementation.\n///\n/// In this struct we store the private input variables. We use `Option` because\n/// they won't have any value during key generation. During proving, if any of these\n/// were `None` we would get an error.\n#[derive(Default)]\nstruct MyCircuit { constant: F, a: Option, b: Option,\n} impl Circuit for MyCircuit { // Since we are using a single chip for everything, we can just reuse its config. type Config = FieldConfig; type FloorPlanner = SimpleFloorPlanner; fn without_witnesses(&self) -> Self { Self::default() } fn configure(meta: &mut ConstraintSystem) -> Self::Config { // We create the two advice columns that FieldChip uses for I/O. let advice = [meta.advice_column(), meta.advice_column()]; // We also need an instance column to store public inputs. let instance = meta.instance_column(); // Create a fixed column to load constants. let constant = meta.fixed_column(); FieldChip::configure(meta, advice, instance, constant) } fn synthesize( &self, config: Self::Config, mut layouter: impl Layouter, ) -> Result<(), Error> { let field_chip = FieldChip::::construct(config); // Load our private values into the circuit. let a = field_chip.load_private(layouter.namespace(|| \"load a\"), self.a)?; let b = field_chip.load_private(layouter.namespace(|| \"load b\"), self.b)?; // Load the constant factor into the circuit. let constant = field_chip.load_constant(layouter.namespace(|| \"load constant\"), self.constant)?; // We only have access to plain multiplication. // We could implement our circuit as: // asq = a*a // bsq = b*b // absq = asq*bsq // c = constant*asq*bsq // // but it's more efficient to implement it as: // ab = a*b // absq = ab^2 // c = constant*absq let ab = field_chip.mul(layouter.namespace(|| \"a * b\"), a, b)?; let absq = field_chip.mul(layouter.namespace(|| \"ab * ab\"), ab.clone(), ab)?; let c = field_chip.mul(layouter.namespace(|| \"constant * absq\"), constant, absq)?; // Expose the result as a public input to the circuit. field_chip.expose_public(layouter.namespace(|| \"expose c\"), c, 0) }\n}","breadcrumbs":"User Documentation » A simple example » Build the circuit","id":"23","title":"Build the circuit"},"24":{"body":"halo2_proofs::dev::MockProver can be used to test that the circuit is working correctly. The private and public inputs to the circuit are constructed as we will do to create a proof, but by passing them to MockProver::run we get an object that can test every constraint in the circuit, and tell us exactly what is failing (if anything). // The number of rows in our circuit cannot exceed 2^k. Since our example // circuit is very small, we can pick a very small value here. let k = 4; // Prepare the private and public inputs to the circuit! let constant = Fp::from(7); let a = Fp::from(2); let b = Fp::from(3); let c = constant * a.square() * b.square(); // Instantiate the circuit with the private inputs. let circuit = MyCircuit { constant, a: Some(a), b: Some(b), }; // Arrange the public input. We expose the multiplication result in row 0 // of the instance column, so we position it there in our public inputs. let mut public_inputs = vec![c]; // Given the correct public input, our circuit will verify. let prover = MockProver::run(k, &circuit, vec![public_inputs.clone()]).unwrap(); assert_eq!(prover.verify(), Ok(())); // If we try some other public input, the proof will fail! public_inputs[0] += Fp::one(); let prover = MockProver::run(k, &circuit, vec![public_inputs]).unwrap(); assert!(prover.verify().is_err());","breadcrumbs":"User Documentation » A simple example » Testing the circuit","id":"24","title":"Testing the circuit"},"25":{"body":"You can find the source code for this example here .","breadcrumbs":"User Documentation » A simple example » Full example","id":"25","title":"Full example"},"26":{"body":"In normal programs, you can trade memory for CPU to improve performance, by pre-computing and storing lookup tables for some part of the computation. We can do the same thing in halo2 circuits! A lookup table can be thought of as enforcing a relation between variables, where the relation is expressed as a table. Assuming we have only one lookup argument in our constraint system, the total size of tables is constrained by the size of the circuit: each table entry costs one row, and it also costs one row to do each lookup. TODO","breadcrumbs":"User Documentation » Lookup tables » Lookup tables","id":"26","title":"Lookup tables"},"27":{"body":"","breadcrumbs":"User Documentation » Gadgets » Gadgets","id":"27","title":"Gadgets"},"28":{"body":"This section contains various ideas and snippets that you might find useful while writing halo2 circuits.","breadcrumbs":"User Documentation » Tips and tricks » Tips and tricks","id":"28","title":"Tips and tricks"},"29":{"body":"A common constraint used in R1CS circuits is the boolean constraint: b∗(1−b)=0. This constraint can only be satisfied by b=0 or b=1. In halo2 circuits, you can similarly constrain a cell to have one of a small set of values. For example, to constrain a to the range [0..5], you would create a gate of the form: a⋅(1−a)⋅(2−a)⋅(3−a)⋅(4−a)=0 while to constraint c to be either 7 or 13, you would use: (7−c)⋅(13−c)=0 The underlying principle here is that we create a polynomial constraint with roots at each value in the set of possible values we want to allow. In R1CS circuits, the maximum supported polynomial degree is 2 (due to all constraints being of the form a∗b=c). In halo2 circuits, you can use arbitrary-degree polynomials - with the proviso that higher-degree constraints are more expensive to use. Note that the roots don't have to be constants; for example (a−x)⋅(a−y)⋅(a−z)=0 will constrain a to be equal to one of {x,y,z} where the latter can be arbitrary polynomials, as long as the whole expression stays within the maximum degree bound.","breadcrumbs":"User Documentation » Tips and tricks » Small range constraints","id":"29","title":"Small range constraints"},"3":{"body":"halo2 currently uses rayon for parallel computation. The RAYON_NUM_THREADS environment variable can be used to set the number of threads.","breadcrumbs":"halo2 » Controlling parallelism","id":"3","title":"Controlling parallelism"},"30":{"body":"We can use Lagrange interpolation to create a polynomial constraint that maps f(X)=Y for small sets of X∈{xi​},Y∈{yi​}. For instance, say we want to map a 2-bit value to a \"spread\" version interleaved with zeros. We first precompute the evaluations at each point: 00→000001→000110→010011→0101​⟹⟹⟹⟹​0→01→12→43→5​ Then, we construct the Lagrange basis polynomial for each point using the identity: lj​(X)=0≤m B\n^ +\n| |\n+ v\nD <---+ C E +---> F ^ + | | + v H <---+ G After adding constraint B≡E the above algorithm produces the cycle: A +---> B +-------------+\n^ |\n| |\n+ v\nD <---+ C <---+ E F ^ + | | + v H <---+ G","breadcrumbs":"Design » Proving system » Permutation argument » Algorithm","id":"46","title":"Algorithm"},"47":{"body":"If we did not check whether left and right were already in the same cycle, then we could end up undoing an equality constraint. For example, if we have the following constraints: a≡b b≡c c≡d b≡d and we tried to implement adding an equality constraint just using step 5 of the above algorithm, then we would end up constructing the cycle (a b) (c d), rather than the correct (a b c d).","breadcrumbs":"Design » Proving system » Permutation argument » Broken alternatives","id":"47","title":"Broken alternatives"},"48":{"body":"We need to check a permutation of cells in m columns, represented in Lagrange basis by polynomials v0​,…,vm−1​. We will label each cell in those m columns with a unique element of F×. Suppose that we have a permutation on these labels, σ(column:i,row:j)=(column:i′,row:j′). in which the cycles correspond to equality-constraint sets. If we consider the set of pairs {(label,value)}, then the values within each cycle are equal if and only if permuting the label in each pair by σ yields the same set: An example for a cycle (A B C D). The set before permuting the labels is {(A, 7), (B, 7), (C, 7), (D, 7)}, and the set after is {(D, 7), (A, 7), (B, 7), (C, 7)} which is the same. If one of the 7s is replaced by 3, then the set after permuting labels is not the same. Since the labels are distinct, set equality is the same as multiset equality, which we can check using a product argument. Let ω be a 2k root of unity and let δ be a T root of unity, where T⋅2S+1=p with T odd and k≤S. We will use δi⋅ωj∈F× as the label for the cell in the jth row of the ith column of the permutation argument. We represent σ by a vector of m polynomials si​(X) such that si​(ωj)=δi′⋅ωj′. Notice that the identity permutation can be represented by the vector of m polynomials IDi​(ωj) such that IDi​(ωj)=δi⋅ωj. We will use a challenge β to compress each (label,value) pair to value+β⋅label. Just as in the product argument we used for lookups , we also use a challenge γ to randomize each term of the product. Now given our permutation represented by s0​,…,sm−1​ over columns represented by v0​,…,vm−1​, we want to ensure that: i=0∏m−1​j=0∏n−1​(vi​(ωj)+β⋅si​(ωj)+γvi​(ωj)+β⋅δi⋅ωj+γ​)=1 Here vi​(ωj)+β⋅δi⋅ωj represents the unpermuted (label,value) pair, and vi​(ωj)+β⋅si​(ωj) represents the permuted (σ(label),value) pair. Let ZP​ be such that ZP​(ω0)=ZP​(ωn)=1 and for 0≤j=r(λ) verifier challenges and let the ith challenge be sampled from Chi​. The advantage metric AdvIPSRS​(P,λ)=Pr[SRSPIP​(λ)] of a state restoration prover P is defined with respect to the following game. ​GameSRSIPP​(λ):​win←false;tr←ϵpp←IP.Setup(1λ)(x,stP​)←Pλ​(pp)RunPλOSRS​​(stP​)Return win​​​OracleOSRS​(τ=(a1​,c1​,...,ai−1​,ci−1​),ai​):​Ifτ∈trthenIfi≤rthenci​←Chi​;tr←tr∣∣(τ,ai​,ci​);Returnci​Else ifi=r+1thend←IP.V(pp,x,(τ,ai​));tr←(τ,ai​)Ifd=1then win←trueReturndReturn⊥​​ As shown in [GT20] (Theorem 1) state restoration soundness is tightly related to soundness after applying the Fiat-Shamir transformation. Knowledge Soundness We will show that our protocol satisfies a strengthened notion of knowledge soundness known as witness extended emulation . Informally, this notion states that for any successful prover algorithm there exists an efficient emulator that can extract a witness from it by rewinding it and supplying it with fresh randomness. However, we must slightly adjust our definition of witness extended emulation to account for the fact that our provers are state restoration provers and can rewind the verifier. Further, to avoid the need for rewinding the state restoration prover during witness extraction we study our protocol in the algebraic group model. Algebraic Group Model (AGM). An adversary Palg​ is said to be algebraic if whenever it outputs a group element X it also outputs a representation x∈Fn such that ⟨x,G⟩=X where G∈Gn is the vector of group elements that Palg​ has seen so far. Notationally, we write {X} to describe a group element X enhanced with this representation. We also write {X}iG​ to identify the component of the representation of X that corresponds with Gi​. In other words, X=i=0∑n−1​[{X}iG​]Gi​ The algebraic group model allows us to perform so-called \"online\" extraction for some protocols: the extractor can obtain the witness from the representations themselves for a single (accepting) transcript. State Restoration Witness Extended Emulation Let IP be an interactive argument for relation R with r=r(λ) challenges. We define for all non-uniform algebraic provers Palg​, extractors E, and computationally unbounded distinguishers D the advantage metric AdvIP,Rsr-wee​(Palg​,D,E,λ)=Pr[WEE-realIP,RP,D​(λ)]−Pr[WEE-idealIP,RE,P,D​(λ)] is defined with the respect to the following games. ​GameWEE-realIP,RPalg​,D​(λ):​tr←ϵpp←IP.Setup(1λ)(x,stP​)←Palg​(pp)RunPalg​Oreal​(stP​)b←D(tr)Returnb=1GameWEE-idealIP,RE,Palg​,D​(λ):​tr←ϵpp←IP.Setup(1λ)(x,stP​)←Palg​(pp)stE​←(1λ,pp,x)RunPalg​Oideal​(stP​)w←E(stE​,⊥)b←D(tr)Return(b=1)∧(Acc(tr)⟹(x,w)∈R)​​​OracleOreal​(τ=(a1​,c1​,...,ai−1​,ci−1​),ai​):​Ifτ∈trthenIfi≤rthenci​←Chi​;tr←tr∣∣(τ,ai​,ci​);Returnci​Else ifi=r+1thend←IP.V(pp,x,(τ,ai​));tr←(τ,ai​)Ifd=1then win←trueReturndReturn⊥OracleOideal​(τ,a):​Ifτ∈trthen(r,stE​)←E(stE​,[(τ,a)])tr←tr∣∣(τ,a,r)ReturnrReturn⊥​​ Zero Knowledge We say that an argument of knowledge is zero knowledge if the verifier also does not learn anything from their interaction besides that which can be learned from the existence of a valid w. More formally, Perfect Special Honest-Verifier Zero Knowledge. A public coin interactive argument (Setup,P,V) has perfect special honest-verifier zero knowledge (PSHVZK) if for all polynomial-time decidable relations R and for all (x,w)∈R and for all non-uniform polynomial-time adversaries A1​,A2​ there exists a probabilistic polynomial-time simulator S such that =​Pr⎣⎡​A1​(σ,x,tr)=1∣∣​​pp←Setup(1λ);(x,w,ρ)←A2​(pp);tr←⟨P(pp,x,w),V(pp,x,ρ)⟩​⎦⎤​Pr⎣⎡​A1​(σ,x,tr)=1∣∣​​pp←Setup(1λ);(x,w,ρ)←A2​(pp);tr←S(pp,x,ρ)​⎦⎤​​ where ρ is the internal randomness of the verifier. In this (common) definition of zero-knowledge the verifier is expected to act \"honestly\" and send challenges that correspond only with their internal randomness; they cannot adaptively respond to the prover based on the prover's messages. We use a strengthened form of this definition that forces the simulator to output a transcript with the same (adversarially provided) challenges that the verifier algorithm sends to the prover.","breadcrumbs":"Design » Protocol Description » Zero knowledge Arguments of Knowledge","id":"68","title":"Zero knowledge Arguments of Knowledge"},"69":{"body":"Let ω∈F be a n=2k primitive root of unity forming the domain D=(ω0,ω1,...,ωn−1) with t(X)=Xn−1 the vanishing polynomial over this domain. Let ng​,na​,ne​ be positive integers with na​,ne​p. That is, we must check that 0≤αq. Thus, given a scalar α, we witness the boolean decomposition of k=α+tq​. 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Typically, though, we will use them via gadgets . This indirection is useful because, for reasons of efficiency and limitations imposed by PLONKish circuits, the chip interfaces will often be dependent on low-level implementation details. The gadget interface can provide a more convenient and stable API that abstracts away from extraneous detail. For example, consider a hash function such as SHA-256. The interface of a chip supporting SHA-256 might be dependent on internals of the hash function design such as the separation between message schedule and compression function. The corresponding gadget interface can provide a more convenient and familiar update/finalize API, and can also handle parts of the hash function that do not need chip support, such as padding. This is similar to how accelerated instructions for cryptographic primitives on CPUs are typically accessed via software libraries, rather than directly. Gadgets can also provide modular and reusable abstractions for circuit programming at a higher level, similar to their use in libraries such as libsnark and bellman . As well as abstracting functions , they can also abstract types , such as elliptic curve points or integers of specific sizes.","breadcrumbs":"Concepts » Gadgets » Gadgets","id":"10","title":"Gadgets"},"100":{"body":"We use an optimized double-and-add algorithm, copied from \"Faster variable-base scalar multiplication in zk-SNARK circuits\" with some variable name changes: Acc := [2] T\nfor i from n-1 down to 0 { P := k_{i+1} ? T : −T Acc := (Acc + P) + Acc\n}\nreturn (k_0 = 0) ? (Acc - T) : Acc It remains to check that the x-coordinates of each pair of points to be added are distinct. When adding points in a prime-order group, we can rely on Theorem 3 from Appendix C of the Halo paper , which says that if we have two such points with nonzero indices wrt a given odd-prime order base, where the indices taken in the range −(q−1)/2..(q−1)/2 are distinct disregarding sign, then they have different x-coordinates. This is helpful, because it is easier to reason about the indices of points occurring in the scalar multiplication algorithm than it is to reason about their x-coordinates directly. So, the required check is equivalent to saying that the following \"indexed version\" of the above algorithm never asserts: acc := 2\nfor i from n-1 down to 0 { p = k_{i+1} ? 1 : −1 assert acc ≠ ± p assert (acc + p) ≠ acc // X acc := (acc + p) + acc assert 0 < acc ≤ (q-1)/2\n}\nif k_0 = 0 { assert acc ≠ 1 acc := acc - 1\n} The maximum value of acc is: <--- n 1s ---> 1011111...111111\n= 1100000...000000 - 1 = 2n+1+2n−1 The assertion labelled X obviously cannot fail because p=0. It is possible to see that acc is monotonically increasing except in the last conditional. It reaches its largest value when k is maximal, i.e. 2n+1+2n−1. So to entirely avoid exceptional cases, we would need 2n+1+2n−1<(q−1)/2. But we can use n larger by c if the last c iterations use complete addition . The first i for which the algorithm using only incomplete addition fails is going to be 252, since 2252+1+2252−1>(q−1)/2. We need n=254 to make the wraparound technique above work. sage: q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001\nsage: 2^253 + 2^252 - 1 < (q-1)//2\nFalse\nsage: 2^252 + 2^251 - 1 < (q-1)//2\nTrue So the last three iterations of the loop (i=2..0) need to use complete addition , as does the conditional subtraction at the end. Writing this out using ⸭ for incomplete addition (as we do in the spec), we have: Acc := [2] T\nfor i from 253 down to 3 { P := k_{i+1} ? T : −T Acc := (Acc ⸭ P) ⸭ Acc\n}\nfor i from 2 down to 0 { P := k_{i+1} ? T : −T Acc := (Acc + P) + Acc // complete addition\n}\nreturn (k_0 = 0) ? (Acc + (-T)) : Acc // complete addition","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Variable-base scalar multiplication","id":"100","title":"Variable-base scalar multiplication"},"101":{"body":"Define a running sum zj​=∑i=jn​(ki​⋅2i−j), where n=254 and: ​zn+1​=0,zn​=kn​,(most significant bit)z0​=k.​ Initialize A254​=[2]T.for i from 254 down to 4:bool_check(ki​)=0zi​=2zi+1​+ki​xP,i​=xT​yP,i​=(2ki​−1)⋅yT​(conditionally negate)λ1,i​⋅(xA,i​−xP,i​)=yA,i​−yP,i​λ1,i2​=xR,i​+xA,i​+xP,i​(λ1,i​+λ2,i​)⋅(xA,i​−xR,i​)=2yA,i​λ2,i2​=xA,i−1​+xR,i​+xA,i​λ2,i​⋅(xA,i​−xA,i−1​)=yA,i​+yA,i−1​,​ where xR,i​=(λ1,i2​−xA,i​−xT​). The helper bool_check(x)=x⋅(1−x). After substitution of xP,i​,yP,i​,xR,i​,yA,i​, and yA,i−1​, this becomes: Initialize A254​=[2]T.for i from 254 down to 4:// let ki​=zi​−2zi+1​// let yA,i​=2(λ1,i​+λ2,i​)⋅(xA,i​−(λ1,i2​−xA,i​−xT​))​bool_check(ki​)=0λ1,i​⋅(xA,i​−xT​)=yA,i​−(2ki​−1)⋅yT​λ2,i2​=xA,i−1​+λ1,i2​−xT​{λ2,i​⋅(xA,i​−xA,i−1​)=yA,i​+yA,i−1​,λ2,4​⋅(xA,4​−xA,3​)=yA,4​+yA,3witnessed​,​if i>4if i=4.​​ Here, yA,3witnessed​ is assigned to a cell. This is unlike previous yA,i​'s, which were implicitly derived from λ1,i​,λ2,i​,xA,i​,xT​, but never actually assigned. The bits k3…1​ are used in three further steps, using complete addition : for i from 3 down to 1:// let ki​=zi​−2zi+1​bool_check(ki​)=0(xA,i−1​,yA,i−1​)=((xA,i​,yA,i​)+(xT​,yT​))+(xA,i​,yA,i​)​ If the least significant bit k0​=1, we set B=O, otherwise we set B=−T. Then we return A+B using complete addition. Let B={(0,0),(xT​,−yT​),​ if k0​=1, otherwise.​ Output (xA,0​,yA,0​)+B. (Note that (0,0) represents O.)","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Constraint program for optimized double-and-add (incomplete addition)","id":"101","title":"Constraint program for optimized double-and-add (incomplete addition)"},"102":{"body":"We need six advice columns to witness (xT​,yT​,λ1​,λ2​,xA,i​,zi​). However, since (xT​,yT​) are the same, we can perform two incomplete additions in a single row, reusing the same (xT​,yT​). We split the scalar bits used in incomplete addition into hi and lo halves and process them in parallel. This means that we effectively have two for loops: the first, covering the hi half for i from 254 down to 130, with a special case at i=130; and the second, covering the lo half for the remaining i from 129 down to 4, with a special case at i=4. xT​xT​xT​⋮xT​​yT​yT​yT​⋮yT​​zhiz255​=0z254​z253​⋮z130​​xAhi​xA,254​=2[T]x​xA,253​⋮xA,130​xA,129​​λ1hi​yA,254​=2[T]y​λ1,254​λ1,253​⋮λ1,130​yA,129​​λ2hi​λ2,254​λ2,253​⋮λ2,130​​q1hi​100⋮0​q2hi​011⋮0​q3hi​000⋮1​zloz130​z129​z128​⋮z5​z4​​xAlo​xA,129​xA,128​⋮xA,5​xA,4​xA,3​​λ1lo​yA,129​λ1,129​λ1,128​⋮λ1,5​λ1,4​yA,3​​λ2lo​λ2,129​λ2,128​⋮λ2,5​λ2,4​​q1lo​100⋮00​q2lo​011⋮10​q3lo​000⋮01​​ For each hi and lo half, we have three sets of gates. Note that i is going from 255..=3; i is NOT indexing the rows. q1​=1 This gate is only used on the first row (before the for loop). We check that λ1​,λ2​ are initialized to values consistent with the initial yA​. Degree3​Constraintq1​⋅(yA,nwitnessed​−yA,n​)=0​​ where yA,n​yA,nwitnessed​​=2(λ1,n​+λ2,n​)⋅(xA,n​−(λ1,n2​−xA,n​−xT​))​, is witnessed.​ q2​=1 This gate is used on all rows corresponding to the for loop except the last. Degree223433​Constraintq2​⋅(xT,cur​−xT,next​)=0q2​⋅(yT,cur​−yT,next​)=0q2​⋅bool_check(ki​)=0, where ki​=zi​−2zi+1​q2​⋅(λ1,i​⋅(xA,i​−xT,i​)−yA,i​+(2ki​−1)⋅yT,i​)=0q2​⋅(λ2,i2​−xA,i−1​−λ1,i2​+xT,i​)=0q2​⋅(λ2,i​⋅(xA,i​−xA,i−1​)−yA,i​−yA,i−1​)=0​​ where yA,i​yA,i−1​​=2(λ1,i​+λ2,i​)⋅(xA,i​−(λ1,i2​−xA,i​−xT​))​,=2(λ1,i−1​+λ2,i−1​)⋅(xA,i−1​−(λ1,i−12​−xA,i−1​−xT​))​,​ q3​=1 This gate is used on the final iteration of the for loop, handling the special case where we check that the output yA​ has been witnessed correctly. Degree3433​Constraintq3​⋅bool_check(ki​)=0, where ki​=zi​−2zi+1​q3​⋅(λ1,i​⋅(xA,i​−xT,i​)−yA,i​+(2ki​−1)⋅yT,i​)=0q3​⋅(λ2,i2​−xA,i−1​−λ1,i2​+xT,i​)=0q3​⋅(λ2,i​⋅(xA,i​−xA,i−1​)−yA,i​−yA,i−1witnessed​)=0​​ where yA,i​yA,i−1witnessed​​=2(λ1,i​+λ2,i​)⋅(xA,i​−(λ1,i2​−xA,i​−xT​))​, is witnessed.​","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Circuit design","id":"102","title":"Circuit design"},"103":{"body":"zi​ cannot overflow for any i≥1, because it is a weighted sum of bits only up to 2n−1=2253, which is smaller than p (and also q). However, z0​=α+tq​ can overflow [0,p). Since overflow can only occur in the final step that constrains z0​=2⋅z1​+k0​, we have z0​=k(modp). It is then sufficient to also check that z0​=α+tq​(modp) (so that k=α+tq​(modp)) and that k∈[tq​,p+tq​). These conditions together imply that k=α+tq​ as an integer, and so 2254+k=α(modq) as required. Note: the bits k254..0​ do not represent a value reduced modulo q, but rather a representation of the unreduced α+tq​.","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Overflow check","id":"103","title":"Overflow check"},"104":{"body":"Since tp​+tq​<2130, we have [tq​,p+tq​)=[tq​,tq​+2130)∪[2130,2254)∪([2254,2254+2130)∩[p+tq​−2130,p+tq​)). We may assume that k=α+tq​(modp). Therefore, k∈[tq​,p+tq​)​⇔⇔⇔​(k∈[tq​,tq​+2130)∨k∈[2130,2254))∨(k∈[2254,2254+2130)∧k∈[p+tq​−2130,p+tq​))(k254​=0⟹(k∈[tq​,tq​+2130)∨k∈[2130,2254)))∧(k254​=1⟹(k∈[2254,2254+2130)∧k∈[p+tq​−2130,p+tq​))(k254​=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254​=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))Ⓐ​ Given k∈[2254,2254+2130), we prove equivalence of k∈[p+tq​−2130,p+tq​) and (α+2130)modp∈[0,2130) as follows: shift the range by 2130−p−tq​ to give k+2130−p−tq​∈[0,2130); observe that k+2130−p−tq​ is guaranteed to be in [2130−tp​−tq​,2131−tp​−tq​) and therefore cannot overflow or underflow modulo p; using the fact that k=α+tq​(modp), observe that (k+2130−p−tq​)modp=(α+tq​+2130−p−tq​)modp=(α+2130)modp. (We can see in a different way that this is correct by observing that it checks whether αmodp∈[p−2130,p), so the upper bound is aligned as we would expect.) Now, we can continue optimizing from Ⓐ: k∈[tq​,p+tq​)​⇔⇔​(k254​=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254​=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))(k254​=0⟹(α∈[0,2130)∨k253..130​ are not all 0))∧(k254​=1⟹(k253..130​ are all 0∧(α+2130)modp∈[0,2130)))​ Constraining k253..130​ to be all-0 or not-all-0 can be implemented almost \"for free\", as follows. Recall that zi​=∑h=in​(kh​⋅2h−i), so we have: z130​z130​z130​−k254​⋅2124​===​∑h=130254​(kh​⋅2h−130)k254​⋅2254−130+∑h=130253​(kh​⋅2h−130)∑h=130253​(kh​⋅2h−130)​ So k253..130​ are all 0 exactly when z130​=k254​⋅2124. Finally, we can merge the 130-bit decompositions for the k254​=0 and k254​=1 cases by checking that (α+k254​⋅2130)modp∈[0,2130).","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Optimized check for k∈[tq​,p+tq​)","id":"104","title":"Optimized check for k∈[tq​,p+tq​)"},"105":{"body":"Let s=α+k254​⋅2130. The constraints for the overflow check are: z0​k254​=1⟹(z130​k254​=0⟹(z130​​=α+tq​(modp)=2124∧smodp∈[0,2130))=0∨smodp∈[0,2130))​ Define inv0(x)={0,1/x,​if x=0otherwise.​ Witness η=inv0(z130​), and decompose smodp as s129..0​. Then the needed gates are: Degree22335​Constraintq_muloverflow⋅(s−(α+k254​⋅2130))=0q_muloverflow⋅(z0​−α−tq​)=0q_muloverflow⋅(k254​⋅(z130​−2124))=0q_muloverflow⋅(k254​⋅(s−i=0∑129​2i⋅si​)/2130)=0q_muloverflow⋅((1−k254​)⋅(1−z130​⋅η)⋅(s−i=0∑129​2i⋅si​)/2130)=0​​ where (s−i=0∑129​2i⋅si​)/2130 can be computed by another running sum. Note that the factor of 1/2130 has no effect on the constraint, since the RHS is zero. Running sum range check We make use of a 10-bit lookup range check in the circuit to subtract the low 130 bits of s. The range check subtracts the first 13⋅10 bits of s, and right-shifts the result to give (s−i=0∑129​2i⋅si​)/2130.","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Overflow check constraints","id":"105","title":"Overflow check constraints"},"106":{"body":"","breadcrumbs":"Design » Gadgets » Sinsemilla » Sinsemilla","id":"106","title":"Sinsemilla"},"107":{"body":"Sinsemilla is a collision-resistant hash function and commitment scheme designed to be efficient in algebraic circuit models that support lookups , such as PLONK or Halo 2. The security properties of Sinsemilla are similar to Pedersen hashes; it is not designed to be used where a random oracle, PRF, or preimage-resistant hash is required. The only claimed security property of the hash function is collision-resistance for fixed-length inputs. Sinsemilla is roughly 4 times less efficient than the algebraic hashes Rescue and Poseidon inside a circuit, but around 19 times more efficient than Rescue outside a circuit. Unlike either of these hashes, the collision resistance property of Sinsemilla can be proven based on cryptographic assumptions that have been well-established for at least 20 years. Sinsemilla can also be used as a computationally binding and perfectly hiding commitment scheme. The general approach is to split the message into k-bit pieces, and for each piece, select from a table of 2k bases in our cryptographic group. We combine the selected bases using a double-and-add algorithm. This ends up being provably as secure as a vector Pedersen hash, and makes advantageous use of the lookup facility supported by Halo 2.","breadcrumbs":"Design » Gadgets » Sinsemilla » Overview","id":"107","title":"Overview"},"108":{"body":"This section is an outline of how Sinsemilla works: for the normative specification, refer to §5.4.1.9 Sinsemilla Hash Function in the protocol spec. The incomplete point addition operator, ⸭, that we use below is also defined there. Let G be a cryptographic group of prime order q. We write G additively, with identity O, and using [m]P for scalar multiplication of P by m. Let k≥1 be an integer chosen based on efficiency considerations (the table size will be 2k). Let n be an integer, fixed for each instantiation, such that messages are kn bits, where 2n≤2q−1​. We use zero-padding to the next multiple of k bits if necessary. Setup: Choose Q and P[0..2k−1] as 2k+1 independent, verifiably random generators of G, using a suitable hash into G, such that none of Q or P[0..2k−1] are O. In Orchard, we define Q to be dependent on a domain separator D. The protocol specification uses Q(D) in place of Q and S(m) in place of P[m]. Hash(M): Split M into n groups of k bits. Interpret each group as a k-bit little-endian integer mi​. let Acc0​:=Q for i from 0 up to n−1: let Acci+1​:=(Acci​⸭P[mi+1​])⸭Acci​ return Accn​ Let ShortHash(M) be the x-coordinate of Hash(M). (This assumes that G is a prime-order elliptic curve in short Weierstrass form, as is the case for Pallas and Vesta.) It is slightly more efficient to express a double-and-add [2]A+R as (A+R)+A. We also use incomplete additions: it is shown in the Sinsemilla security argument that in the case where G is a prime-order short Weierstrass elliptic curve, an exceptional case for addition would lead to finding a discrete logarithm, which can be assumed to occur with negligible probability even for adversarial input.","breadcrumbs":"Design » Gadgets » Sinsemilla » Description","id":"108","title":"Description"},"109":{"body":"Choose another generator H independently of Q and P[0..2k−1]. The randomness r for a commitment is chosen uniformly on [0,q). Let Commitr​(M)=Hash(M)⸭[r]H. Let ShortCommitr​(M) be the x-coordinate of Commitr​(M). (This again assumes that G is a prime-order elliptic curve in short Weierstrass form.) Note that unlike a simple Pedersen commitment, this commitment scheme (Commit or ShortCommit) is not additively homomorphic.","breadcrumbs":"Design » Gadgets » Sinsemilla » Use as a commitment scheme","id":"109","title":"Use as a commitment scheme"},"11":{"body":"You're probably here because you want to write circuits? Excellent! This section will guide you through the process of creating circuits with halo2.","breadcrumbs":"User Documentation » User Documentation","id":"11","title":"User Documentation"},"110":{"body":"The aim of the design is to optimize the number of bits that can be processed for each step of the algorithm (which requires a doubling and addition in G) for a given table size. Using a single table of size 2k group elements, we can process k bits at a time.","breadcrumbs":"Design » Gadgets » Sinsemilla » Efficient implementation","id":"110","title":"Efficient implementation"},"111":{"body":"Let P={(j,xP[j]​,yP[j]​) for j∈{0..2k−1}}. Input: m1..=n​. (The message words are 1-indexed here, as in the protocol spec , but we start the loop from i=0 so that (xA,i​,yA,i​) corresponds to Acci​ in the protocol spec.) Output: (xA,n​,yA,n​). (xA,0​,yA,0​)=Q for i from 0 up to n−1: yP,i​=yA,i​−λ1,i​⋅(xA,i​−xP,i​) xR,i​=λ1,i2​−xA,i​−xP,i​ 2⋅yA,i​=(λ1,i​+λ2,i​)⋅(xA,i​−xR,i​) (mi+1​,xP,i​,yP,i​)∈P λ2,i2​=xA,i+1​+xR,i​+xA,i​ λ2,i​⋅(xA,i​−xA,i+1​)=yA,i​+yA,i+1​","breadcrumbs":"Design » Gadgets » Sinsemilla » Constraint program","id":"111","title":"Constraint program"},"112":{"body":"","breadcrumbs":"Design » Gadgets » Sinsemilla » PLONK / Halo 2 constraints","id":"112","title":"PLONK / Halo 2 constraints"},"113":{"body":"We have an n-bit message m=m1​+2km2​+...+2k⋅(n−1)mn​. (Note that the message words are 1-indexed as in the protocol spec .) Initialise the running sum z0​=α and define zi+1​:=2Kzi​−mi+1​​. We will end up with zn​=0. Rearranging gives us an expression for each word of the original message mi+1​=zi​−2k⋅zi+1​, which we can look up in the table. In other words, zn−i​=h=0∑i−1​2kh⋅mh+1​. For a little-endian decomposition as used here, the running sum is initialized to the scalar and ends at 0. For a big-endian decomposition as used in variable-base scalar multiplication , the running sum would start at 0 and end with recovering the original scalar. The running sum only applies to message words within a single field element, i.e. if n≥PrimeField::NUM_BITS then we will have several disjoint running sums. A longer message can be constructed by splitting the message words across several field elements, and then running several instances of the constraints below. An additional qS2​ selector is set to 0 for the last step of each element, except for the last element where it is set to 2. In order to support chaining multiple field elements without a gap, we will use a slightly more complicated expression for mi+1​ that effectively forces zn​ to zero for the last step of each element, as indicated by qS2​. This allows the cell that would have been zn​ to be used to reinitialize the running sum for the next element.","breadcrumbs":"Design » Gadgets » Sinsemilla » Message decomposition","id":"113","title":"Message decomposition"},"114":{"body":"The Sinsemilla circuit makes use of 210 pre-computed random generators. These are loaded into a lookup table: tableidx​012⋮210−1​tablex​xP[0]​xP[1]​xP[2]​⋮xP[210−1]​​tabley​yP[0]​yP[1]​yP[2]​⋮yP[210−1]​​​","breadcrumbs":"Design » Gadgets » Sinsemilla » Generator lookup table","id":"114","title":"Generator lookup table"},"115":{"body":"Note: qS3​ is synthesized from qS1​ and qS2​; it is shown here only for clarity. Step012⋮n−10′1′2′⋮n−1′n′​xA​xQ​xA,1​xA,2​⋮xA,n−1​xA,0′​xA,1′​xA,2′​⋮xA,n−1′​xA,n′​​xP​xP[m1​]​xP[m2​]​xP[m3​]​⋮xP[mn​]​xP[m1′​]​xP[m2′​]​xP[m3′​]​⋮xP[mn′​]​​bitsz0​z1​z2​⋮zn−1​z0′​z1′​z2′​⋮zn−1′​​λ1​λ1,0​λ1,1​λ1,2​⋮λ1,n−1​λ1,0′​λ1,1′​λ1,2′​⋮λ1,n−1′​yA,n​​λ2​λ2,0​λ2,1​λ2,2​⋮λ2,n−1​λ2,0′​λ2,1′​λ2,2′​⋮λ2,n−1′​​qS1​11111111110​qS2​11110111120​qS3​00000000020​qS4​10000000000​fixed_y_QyQ​0000000000​​ xQ​, z0​, z0′​, etc. would be copied in using equality constraints.","breadcrumbs":"Design » Gadgets » Sinsemilla » Layout","id":"115","title":"Layout"},"116":{"body":"For i∈[0,n), let​xR,i​YA,i​yP,i​mi+1​qS3​​=====​λ1,i2​−xA,i​−xP,i​(λ1,i​+λ2,i​)⋅(xA,i​−xR,i​)YA,i​/2−λ1,i​⋅(xA,i​−xP,i​)zi​−2k⋅(qS2,i​−qS3,i​)⋅zi+1​qS2​⋅(qS2​−1)​ The Halo 2 circuit API can automatically substitute yP,i​, xR,i​, yA,i​, and yA,i+1​, so we don't need to do that manually. xA,0​=xQ​ 2⋅yQ​=YA,0​ for i from 0 up to n−1: (mi+1​,xP,i​,yP,i​)∈P λ2,i2​=xA,i+1​+xR,i​+xA,i​ 4⋅λ2,i​⋅(xA,i​−xA,i+1​)=2⋅YA,i​+(2−qS3​)⋅YA,i+1​+2qS3​⋅yA,n​ Note that each term of the last constraint is multiplied by 4 relative to the constraint program given earlier. This is a small optimization that avoids divisions by 2. By gating the lookup expression on qS1​, we avoid the need to fill in unused cells with dummy values to pass the lookup argument. The optimized lookup value (using a default index of 0) is: (​qS1​⋅mi+1​,qS1​⋅xP,i​+(1−qS1​)⋅xP,0​,qS1​⋅yP,i​+(1−qS1​)⋅yP,0​)​ This increases the degree of the lookup argument to 6. Degree4635​ConstraintqS4​⋅(2⋅yQ​−YA,0​)=0qS1,i​⇒(mi+1​,xP,i​,yP,i​)∈PqS1,i​⋅(λ2,i2​−(xA,i+1​+xR,i​+xA,i​))qS1,i​⋅(4⋅λ2,i​⋅(xA,i​−xA,i+1​)−(2⋅YA,i​+(2−qS3,i​)⋅YA,i+1​+2⋅qS3,i​⋅yA,n​))=0​​","breadcrumbs":"Design » Gadgets » Sinsemilla » Optimized Sinsemilla gate","id":"116","title":"Optimized Sinsemilla gate"},"117":{"body":"","breadcrumbs":"Design » Gadgets » Sinsemilla » MerkleCRH » MerkleCRH","id":"117","title":"MerkleCRH"},"118":{"body":"SinsemillaHash is used in the MerkleCRHOrchard hash function . The input to SinsemillaHash is: l⋆∣∣left⋆∣∣right⋆, where: l⋆=I2LEBSP10​(l)=I2LEBSP10​(MerkleDepthOrchard−1−layer), left⋆=I2LEBSPℓMerkleOrchard​​(left), right⋆=I2LEBSPℓMerkleOrchard​​(right), with ℓMerkleOrchard​=255. left and right are allowed to be non-canonical 255-bit encodings. We break these inputs into the following MessagePieces: a (250 bits)b (20 bits)c (250 bits)​=a0​∣∣a1​=l⋆∣∣(bits 0..=239 of left )=b0​∣∣b1​∣∣b2​=(bits 240..=249 of left)∣∣(bits 250..=254 of left)∣∣(bits 0..=4 of right)=bits 5..=254 of right​ a,b,c are constrained by the SinsemillaHash to be 250 bits, 20 bits, and 250 bits respectively. In a custom gate, we check this message decomposition by enforcing the following constraints: a0​=l z1,a​, the index-1 running sum output of SinsemillaHash(a), is copied into the gate. z1,a​ has been constrained by the SinsemillaHash to be 240 bits. We recover the subpieces a0​,a1​ using a,z1,a​: z1,a​⟹a0​​=210a−a0​​=a1​=a−z1,a​⋅210.​ l+1 is loaded into a fixed column at each layer of the hash. It is used both as a gate selector, and to fix the value of l. We check that a0​=(l+1)−1. Note: The reason for using l+1 instead of l is that l=0 when layer=31 (hashing two leaves). We cannot have a zero-valued selector, since a constraint gated by a zero-valued selector is never checked. b1​+25⋅b2​=z1,b​ z1,b​, the index-1 running sum output of SinsemillaHash(b), is copied into the gate. z1,b​ has been constrained by the SinsemillaHash to be 10 bits. We witness the subpieces b1​,b2​ outside this gate, and constrain them each to be 5 bits. Inside the gate, we check that b1​+25⋅b2​=z1,b​. We also recover the subpiece b0​ using (b,z1,b​): z1,b​⟹b0​​=210b−b0..=10​​=b−(z1,b​⋅210).​ We have now derived or witnessed every subpiece, and range-constrained every subpiece: a0​ (10 bits), derived as a0​=a−210⋅z1,a​; a1​ (240 bits), equal to z1,a​; b0​ (10 bits), derived as b0​=b−210⋅z1,b​; b1​ (5 bits) is witnessed and constrained outside the gate; b2​ (5 bits) is witnessed and constrained outside the gate; b1​+25⋅b2​ is constrained to equal z1,b​, and we use them to reconstruct the original field element inputs: left=a1​+2240⋅b0​+2254⋅b1​ right=b2​+25⋅c","breadcrumbs":"Design » Gadgets » Sinsemilla » MerkleCRH » Message decomposition","id":"118","title":"Message decomposition"},"119":{"body":"The Orchard circuit spans 10 advice columns while the Sinsemilla chip only uses 5 advice columns. We distribute the path hashing evenly across two Sinsemilla chips to make better use of the available circuit area. Since the output from the previous layer hash is copied into the next layer hash, we maintain continuity even when moving from one chip to the other.","breadcrumbs":"Design » Gadgets » Sinsemilla » MerkleCRH » Circuit components","id":"119","title":"Circuit components"},"12":{"body":"The halo2 crate includes several utilities to help you design and implement your circuits.","breadcrumbs":"User Documentation » Developer tools » Developer tools","id":"12","title":"Developer tools"},"120":{"body":"Given a field element α, these gadgets decompose it into W K-bit windows α=k0​+2K⋅k1​+22K⋅k2​+⋯+2(W−1)K⋅kW−1​ where each ki​ a K-bit value. This is done using a running sum zi​,i∈[0..W). We initialize the running sum z0​=α, and compute subsequent terms zi+1​=2Kzi​−ki​​. This gives us: z0​z1​z2​↓zW​​=α=k0​+2K⋅k1​+22K⋅k2​+23K⋅k3​+⋯,=(z0​−k0​)/2K=k1​+2K⋅k2​+22K⋅k3​+⋯,=(z1​−k1​)/2K=k2​+2K⋅k3​+⋯,⋮ (in strict mode)=(zW−1​−kW−1​)/2K=0 (because zW−1​=kW−1​)​","breadcrumbs":"Design » Gadgets » Decomposition » Decomposition","id":"120","title":"Decomposition"},"121":{"body":"Strict mode constrains the running sum output zW​ to be zero, thus range-constraining the field element to be within W⋅K bits. In strict mode, we are also assured that zW−1​=kW−1​ gives us the last window in the decomposition.","breadcrumbs":"Design » Gadgets » Decomposition » Strict mode","id":"121","title":"Strict mode"},"122":{"body":"This gadget makes use of a K-bit lookup table to decompose a field element α into K-bit words. Each K-bit word ki​=zi​−2K⋅zi+1​ is range-constrained by a lookup in the K-bit table. The region layout for the lookup decomposition uses a single advice column z, and two selectors qlookup​ and qrunning​. zz0​z1​⋮zn−1​zn​​qlookup​11⋮10​qrunning​11⋮10​​","breadcrumbs":"Design » Gadgets » Decomposition » Lookup decomposition","id":"122","title":"Lookup decomposition"},"123":{"body":"Using two K-bit lookups, we can range-constrain a field element α to be n bits, where n≤K. To do this: Constrain 0≤α<2K to be within K bits using a K-bit lookup. Constrain 0≤α⋅2K−n<2K to be within K bits using a K-bit lookup. The short variant of the lookup decomposition introduces a qbitshift​ selector. The same advice column z has here been renamed to word for clarity: wordαα′2K−n​qlookup​110​qrunning​000​qbitshift​010​​ where α′=α⋅2K−n. Note that 2K−n is assigned to a fixed column at keygen, and copied in at proving time. This is used in the gate enabled by the qbitshift​ selector to check that α was shifted correctly: Degree2​Constraintqbitshift​⋅(α′−(α⋅2K−n))​​","breadcrumbs":"Design » Gadgets » Decomposition » Short range check","id":"123","title":"Short range check"},"124":{"body":"Since the lookup decomposition and its short variant both make use of the same lookup table, we combine their lookup input expressions into a single one: qlookup​⋅(qrunning​⋅(zi​−2K⋅zi+1​)+(1−qrunning​)⋅word) where zi​ and word are the same cell (but distinguished here for clarity of usage).","breadcrumbs":"Design » Gadgets » Decomposition » Combined lookup expression","id":"124","title":"Combined lookup expression"},"125":{"body":"For a short range (for instance, [0,range) where range≤8), we can range-constrain each word using a degree-range polynomial constraint instead of a lookup: range_check(word,range)=word⋅(1−word)⋯(range−1−word).","breadcrumbs":"Design » Gadgets » Decomposition » Short range decomposition","id":"125","title":"Short range decomposition"},"126":{"body":"","breadcrumbs":"Design » Gadgets » SHA-256 » SHA-256","id":"126","title":"SHA-256"},"127":{"body":"SHA-256 is specified in NIST FIPS PUB 180-4 . Unlike the specification, we use ⊞ for addition modulo 232, and + for field addition. ⊕ is used for XOR.","breadcrumbs":"Design » Gadgets » SHA-256 » Specification","id":"127","title":"Specification"},"128":{"body":"SHA-256 maintains state in eight 32-bit variables. It processes input as 512-bit blocks, but internally splits these blocks into 32-bit chunks. We therefore designed the SHA-256 gadget to consume input in 32-bit chunks.","breadcrumbs":"Design » Gadgets » SHA-256 » Gadget interface","id":"128","title":"Gadget interface"},"129":{"body":"The SHA-256 gadget requires a chip with the following instructions: # extern crate halo2_proofs;\n# use halo2_proofs::plonk::Error;\n# use std::fmt;\n#\n# trait Chip: Sized {}\n# trait Layouter {}\nconst BLOCK_SIZE: usize = 16;\nconst DIGEST_SIZE: usize = 8; pub trait Sha256Instructions: Chip { /// Variable representing the SHA-256 internal state. type State: Clone + fmt::Debug; /// Variable representing a 32-bit word of the input block to the SHA-256 compression /// function. type BlockWord: Copy + fmt::Debug; /// Places the SHA-256 IV in the circuit, returning the initial state variable. fn initialization_vector(layouter: &mut impl Layouter) -> Result; /// Starting from the given initial state, processes a block of input and returns the /// final state. fn compress( layouter: &mut impl Layouter, initial_state: &Self::State, input: [Self::BlockWord; BLOCK_SIZE], ) -> Result; /// Converts the given state into a message digest. fn digest( layouter: &mut impl Layouter, state: &Self::State, ) -> Result<[Self::BlockWord; DIGEST_SIZE], Error>;\n} TODO: Add instruction for computing padding. This set of instructions was chosen to strike a balance between the reusability of the instructions, and the scope for chips to internally optimise them. In particular, we considered splitting the compression function into its constituent parts (Ch, Maj etc), and providing a compression function gadget that implemented the round logic. However, this would prevent chips from using relative references between the various parts of a compression round. Having an instruction that implements all compression rounds is also similar to the Intel SHA extensions, which provide an instruction that performs multiple compression rounds.","breadcrumbs":"Design » Gadgets » SHA-256 » Chip instructions","id":"129","title":"Chip instructions"},"13":{"body":"halo2_proofs::dev::MockProver is a tool for debugging circuits, as well as cheaply verifying their correctness in unit tests. The private and public inputs to the circuit are constructed as would normally be done to create a proof, but MockProver::run instead creates an object that will test every constraint in the circuit directly. It returns granular error messages that indicate which specific constraint (if any) is not satisfied.","breadcrumbs":"User Documentation » Developer tools » Mock prover","id":"13","title":"Mock prover"},"130":{"body":"This chip implementation is based around a single 16-bit lookup table. It requires a minimum of 216 circuit rows, and is therefore suitable for use in larger circuits. We target a maximum constraint degree of 9. That will allow us to handle constraining carries and \"small pieces\" to a range of up to {0..7} in one row.","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » 16-bit table chip for SHA-256","id":"130","title":"16-bit table chip for SHA-256"},"131":{"body":"There are 64 compression rounds. Each round takes 32-bit values A,B,C,D,E,F,G,H as input, and performs the following operations: Ch(E,F,G)Maj(A,B,C)Σ0​(A)Σ1​(E)H′Enew​Anew​​========​(E∧F)⊕(¬E∧G)(A∧B)⊕(A∧C)⊕(B∧C)count(A,B,C)≥2(A⋙2)⊕(A⋙13)⊕(A⋙22)(E⋙6)⊕(E⋙11)⊕(E⋙25)H+Ch(E,F,G)+Σ1​(E)+Kt​+Wt​reduce6​(H′+D)reduce7​(H′+Maj(A,B,C)+Σ0​(A))​ where reducei​ must handle a carry 0≤carry = MyCircuit { a: None, lookup_table, }; // Create the area you want to draw on. // Use SVGBackend if you want to render to .svg instead. use plotters::prelude::*; let root = BitMapBackend::new(\"layout.png\", (1024, 768)).into_drawing_area(); root.fill(&WHITE).unwrap(); let root = root .titled(\"Example Circuit Layout\", (\"sans-serif\", 60)) .unwrap(); halo2_proofs::dev::CircuitLayout::default() // You can optionally render only a section of the circuit. .view_width(0..2) .view_height(0..16) // You can hide labels, which can be useful with smaller areas. .show_labels(false) // Render the circuit onto your area! // The first argument is the size parameter for the circuit. .render(5, &circuit, &root) .unwrap();\n} Columns are laid out from left to right as instance, advice and fixed. The order of columns is otherwise without meaning. Instance columns have a white background. Advice columns have a red background. Fixed columns have a blue background. Regions are shown as labelled green boxes (overlaying the background colour). A region may appear as multiple boxes if some of its columns happen to not be adjacent. Cells that have been assigned to by the circuit will be shaded in grey. If any cells are assigned to more than once (which is usually a mistake), they will be shaded darker than the surrounding cells.","breadcrumbs":"User Documentation » Developer tools » Circuit layout","id":"15","title":"Circuit layout"},"150":{"body":"v1 v1 of the σ1​ gate takes in a word that's split into (10,7,2,13)-bit chunks (already constrained by message scheduling). We refer to these chunks respectively as (a(10),b(7),c(2),d(13)). b(7) is further split into (2,2,3)-bit chunks b(7)lo,b(7)mid,b(7)hi. We witness the spread versions of the small chunks. We already have spread(a(10)) and spread(d(13)) from the message scheduling. (X⋙17)⊕(X⋙19)⊕(X≫10) is equivalent to (X⋘15)⊕(X⋘13)⊕(X≫10). s_low_sigma_1 a0​ a1​ a2​ a3​ a4​ a5​ a6​ 0 {0,1,2,3,4,5} R0even​ spread(R0even​) b(7)lo spread(b(7)lo) b(7)mid spread(b(7)mid) 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(a(10)) spread(d(13)) b(7) 0 {0,1,2,3,4,5} R1even​ spread(R1even​) c(2) spread(c(2)) b(7)hi spread(b(7)hi) 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) Constraints: s_low_sigma_1 (σ1​ v1 constraint): LHS−RHS=0 LHS​=​spread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​)​ RHS=​429b(7)hi430c(2)​++​49d(13)427b(7)mid427b(7)hi​+++​47c(2)425b(7)lo425b(7)mid​+++​44b(7)hi415a(10)423b(7)lo​+++​42b(7)mid42d(13)413a(10)​+++​b(7)loc(2)d(13)​++​ check that b was properly split into subsections for 7-bit pieces. Wb(7)lo+22Wb(7)mid+24Wb(7)hi−W=0 2-bit range check and 2-bit spread check on b(7)lo,b(7)mid,c(2) 3-bit range check and 3-bit spread check on b(7)hi v2 v2 of the σ1​ gate takes in a word that's split into (3,4,3,7,1,1,13)-bit chunks (already constrained by message scheduling). We refer to these chunks respectively as (a(3),b(4),c(3),d(7),e(1),f(1),g(13)). We already have spread(d(7)),spread(g(13)) from the message scheduling. The 1-bit e(1),f(1) remain unchanged by the spread operation and can be used directly. We further split b(4) into two 2-bit chunks b(4)lo,b(4)hi. We witness the spread versions of the small chunks. (X⋙17)⊕(X⋙19)⊕(X≫10) is equivalent to (X⋘15)⊕(X⋘13)⊕(X≫10). s_low_sigma_1_v2 a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ 0 {0,1,2,3,4,5} R0even​ spread(R0even​) b(4)lo spread(b(4)lo) b(4)hi spread(b(4)hi) 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(7)) spread(g(13)) b(4) e(1) 0 {0,1,2,3,4,5} R1even​ spread(R1even​) a(3) spread(a(3)) c(3) spread(c(3)) f(1) 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) Constraints: s_low_sigma_1_v2 (σ1​ v2 constraint): LHS−RHS=0 LHS​=​spread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​)​ RHS=​425d(7)431f(1)​++​422c(3)430e(1)​++​420b(4)hi423d(7)​++​49g(13)418b(4)lo420c(3)​+++​48f(1)415a418b(4)hi​+++​47e(1)42g(13)416b(4)lo​+++​d(7)41f(1)413a​+++​e(1)g(13)​+​ check that b was properly split into subsections for 4-bit pieces. Wb(4)lo+22Wb(4)hi−W=0 2-bit range check and 2-bit spread check on b(4)lo,b(4)hi 3-bit range check and 3-bit spread check on a(3),c(3)","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » σ_1 gate","id":"150","title":"σ_1 gate"},"151":{"body":"Small range constraints Let constrainn​(x)=∏i=0n​(x−i). Constraining this expression to equal zero enforces that x is in [0..n]. 2-bit range check (a−3)(a−2)(a−1)(a)=0 sr2 a0​ 1 a 2-bit spread l1​(a)+4∗l2​(a)+5∗l3​(a)−a′=0 ss2 a0​ a1​ 1 a a' with interpolation polynomials: l0​(a)=(−3)(−2)(−1)(a−3)(a−2)(a−1)​ (spread(00)=0000) l1​(a)=(−2)(−1)(1)(a−3)(a−2)(a)​ (spread(01)=0001) l2​(a)=(−1)(1)(2)(a−3)(a−1)(a)​ (spread(10)=0100) l3​(a)=(1)(2)(3)(a−2)(a−1)(a)​ (spread(11)=0101) 3-bit range check (a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)(a)=0 sr3 a0​ 1 a 3-bit spread l1​(a)+4∗l2​(a)+5∗l3​(a)+16∗l4​(a)+17∗l5​(a)+20∗l6​(a)+21∗l7​(a)−a′=0 ss3 a0​ a1​ 1 a a' with interpolation polynomials: l0​(a)=(−7)(−6)(−5)(−4)(−3)(−2)(−1)(a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)​ (spread(000)=000000) l1​(a)=(−6)(−5)(−4)(−3)(−2)(−1)(1)(a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a)​ (spread(001)=000001) l2​(a)=(−5)(−4)(−3)(−2)(−1)(1)(2)(a−7)(a−6)(a−5)(a−4)(a−3)(a−1)(a)​ (spread(010)=000100) l3​(a)=(−4)(−3)(−2)(−1)(1)(2)(3)(a−7)(a−6)(a−5)(a−4)(a−2)(a−1)(a)​ (spread(011)=000101) l4​(a)=(−3)(−2)(−1)(1)(2)(3)(4)(a−7)(a−6)(a−5)(a−3)(a−2)(a−1)(a)​ (spread(100)=010000) l5​(a)=(−2)(−1)(1)(2)(3)(4)(5)(a−7)(a−6)(a−4)(a−3)(a−2)(a−1)(a)​ (spread(101)=010001) l6​(a)=(−1)(1)(2)(3)(4)(5)(6)(a−7)(a−5)(a−4)(a−3)(a−2)(a−1)(a)​ (spread(110)=010100) l7​(a)=(1)(2)(3)(4)(5)(6)(7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)(a)​ (spread(111)=010101) reduce_6 gate Addition (mod232) of 6 elements Input: E {eilo​,eihi​}i=05​ carry Check: E=e0​+e1​+e2​+e3​+e4​+e5​(mod32) Assume inputs are constrained to 16 bits. Addition gate (sa): a0​+a1​+a2​+a3​+a4​+a5​+a6​−a7​=0 Carry gate (sc): 216a6​ω−1+a6​+[(a6​−5)(a6​−4)(a6​−3)(a6​−2)(a6​−1)(a6​)]=0 sa sc a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ 1 0 e0lo​ e1lo​ e2lo​ e3lo​ e4lo​ e5lo​ −carry∗216 Elo 1 1 e0hi​ e1hi​ e2hi​ e3hi​ e4hi​ e5hi​ carry Ehi Assume inputs are constrained to 16 bits. Addition gate (sa): a0​ω−1+a1​ω−1+a2​ω−1+a0​+a1​+a2​+a3​ω−1−a3​=0 Carry gate (sc): 216a3​ω+a3​ω−1=0 sa sc a0​ a1​ a2​ a3​ 0 0 e0lo​ e1lo​ e2lo​ −carry∗216 1 1 e3lo​ e4lo​ e5lo​ Elo 0 0 e0hi​ e1hi​ e2hi​ carry 1 0 e3hi​ e4hi​ e5hi​ Ehi reduce_7 gate Addition (mod232) of 7 elements Input: E {eilo​,eihi​}i=06​ carry Check: E=e0​+e1​+e2​+e3​+e4​+e5​+e6​(mod32) Assume inputs are constrained to 16 bits. Addition gate (sa): a0​+a1​+a2​+a3​+a4​+a5​+a6​+a7​−a8​=0 Carry gate (sc): 216a7​ω−1+a7​+[(a7​−6)(a7​−5)(a7​−4)(a7​−3)(a7​−2)(a7​−1)(a7​)]=0 sa sc a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ 1 0 e0lo​ e1lo​ e2lo​ e3lo​ e4lo​ e5lo​ e6lo​ −carry∗216 Elo 1 1 e0hi​ e1hi​ e2hi​ e3hi​ e4hi​ e5hi​ e6hi​ carry Ehi","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » Helper gates","id":"151","title":"Helper gates"},"152":{"body":"For each block M∈{0,1}512 of the padded message, 64 words of 32 bits each are constructed as follows: the first 16 are obtained by splitting M into 32-bit blocks M=W0​∣∣W1​∣∣⋯∣∣W14​∣∣W15​; the remaining 48 words are constructed using the formula: Wi​=σ1​(Wi−2​)⊞Wi−7​⊞σ0​(Wi−15​)⊞Wi−16​, for 16≤i<64. sw sd0 sd1 sd2 sd3 ss0 ss0_v2 ss1 ss1_v2 a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W0lo​ spread(W0lo​) W0lo​ W0hi​ W0​ σ0​(W1​)lo σ1​(W14​)lo W9lo​ 1 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W0hi​ spread(W0hi​) W16​ σ0​(W1​)hi σ1​(W14​)hi W9hi​ carry16​ 0 1 1 0 0 0 0 0 0 {0,1,2,3,4} W1d(14)​ spread(W1d(14)​) W1lo​ W1hi​ W1​ σ0​(W2​)lo σ1​(W15​)lo W10lo​ 1 0 0 0 0 0 0 0 0 {0,1,2} W1c(11)​ spread(W1c(11)​) W1a(3)​ W1b(4)​ W17​ σ0​(W2​)hi σ1​(W15​)hi W10hi​ carry17​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W1b(4)lo​ spread(W1b(4)lo​) W1b(4)hi​ spread(W1b(4)hi​) 0 0 0 0 0 1 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R0odd​) spread(R1odd​) spread(W1c(11)​) spread(W1d(14)​) W1b(4)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R1even​) 0 0 W1a(3)​ spread(W1a(3)​) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1odd​) σ0​v1R0​ σ0​v1R1​ σ0​v1R0even​ σ0​v1R0odd​ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 0 0 0 0 0 0 0 0 {0,1,2,3} W14g(13)​ spread(W14g(13)​) W14a(3)​ W14c(3)​ 0 1 0 1 0 0 0 0 0 0 W14d(7)​ spread(W14d(7)​) W14lo​ W14hi​ W14​ σ0​(W15​)lo σ1​(W28​)lo W23lo​ 1 0 0 0 0 0 0 0 0 0 W14b(4)​ spread(W14b(4)​) W14e(1)​ W14f(1)​ W30​ σ0​(W15​)hi σ1​(W28​)hi W23hi​ carry30​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W14b(4)lo​ spread(W14b(4)lo​) W14b(4)hi​ spread(W14b(4)hi​) 0 0 0 0 0 0 1 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(W14d(7)​) spread(W14g(13)​) W1b(14)​ W14e(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) W14a(3)​ spread(W14a(3)​) W14c(3)​ spread(W14c(3)​) W14f(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) σ0​v2R0​ σ0​v2R1​ σ0​v2R0even​ σ0​v2R0odd​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W14b(4)lo​ spread(W14b(4)lo​) W14b(4)hi​ spread(W14b(4)hi​) 0 0 0 0 0 0 0 0 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d) spread(g) W14e(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) W14a(3)​ spread(W14a(3)​) W14c(3)​ spread(W14c(3)​) W14f(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) σ1​v2R0​ σ1​v2R1​ σ1​v2R0even​ σ1​v2R0odd​ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 1 0 0 1 0 0 0 0 {0,1,2,3} W49d(13)​ spread(W49d(13)​) W49lo​ W49hi​ W49​ 0 0 0 0 0 0 0 0 0 {0,1} W49a(10)​ spread(W49a(10)​) W49c(2)​ W49b(7)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W49b(7)lo​ spread(W49b(7)lo​) W49b(7)mid​ spread(W49b(7)mid​) 0 0 0 0 0 0 0 0 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(a) spread(d) W1b(49)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) W49c(2)​ spread(W49c(2)​) W49b(7)hi​ spread(W49b(7)hi​) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) σ1​v1R0​ σ1​v1R1​ σ1​v1R0even​ σ1​v1R0odd​ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W62lo​ spread(W62lo​) W62lo​ W62hi​ W62​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W62hi​ spread(W62hi​) 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W63lo​ spread(W63lo​) W63lo​ W63hi​ W63​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W63hi​ spread(W63hi​) Constraints: sw: construct word using reduce4​ sd0: decomposition gate for W0​,W62​,W63​ Wlo+216Whi−W=0 sd1: decomposition gate for W1..13​ (split into (3,4,11,14)-bit pieces) Wa(3)+23Wb(4)lo+25Wb(4)hi+27Wc(11)+218Wd(14)−W=0 sd2: decomposition gate for W14..48​ (split into (3,4,3,7,1,1,13)-bit pieces) Wa(3)+23Wb(4)lo+25Wb(4)hi+27Wc(11)+210Wd(14)+217We(1)+218Wf(1)+219Wg(13)−W=0 sd3: decomposition gate for W49..61​ (split into (10,7,2,13)-bit pieces) Wa(10)+210Wb(7)lo+212Wb(7)mid+215Wb(7)hi+217Wc(2)+219Wd(13)−W=0","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » Message scheduling region","id":"152","title":"Message scheduling region"},"153":{"body":"+----------------------------------------------------------+\n| |\n| decompose E, |\n| Σ_1(E) |\n| |\n| +---------------------------------------+\n| | |\n| | reduce_5() to get H' |\n| | |\n+----------------------------------------------------------+\n| decompose F, decompose G |\n| |\n| Ch(E,F,G) |\n| |\n+----------------------------------------------------------+\n| |\n| decompose A, |\n| Σ_0(A) |\n| |\n| |\n| +---------------------------------------+\n| | |\n| | reduce_7() to get A_new, |\n| | using H' |\n| | |\n+------------------+---------------------------------------+\n| decompose B, decompose C |\n| |\n| Maj(A,B,C) |\n| |\n| +---------------------------------------+\n| | reduce_6() to get E_new, |\n| | using H' |\n+------------------+---------------------------------------+ Initial round: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0​d(7) spread(E0​d(7)) E0​b(5)lo spread(E0​b(5)lo) E0​b(5)hi spread(E0​b(5)hi) E0lo​ spread(E0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} E0​c(14) spread(E0​c(14)) E0​a(6)lo spread(E0​a(6)lo) E0​a(6)hi spread(E0​a(6)hi) E0hi​ spread(E0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(E0​b(5)lo) spread(E0​b(5)hi) 0 0 0 0 1 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(E0​d(7)) spread(E0​c(14)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(E0​a(6)lo) spread(E0​a(6)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0​d(7) spread(F0​d(7)) F0​b(5)lo spread(F0​b(5)lo) F0​b(5)hi spread(F0​b(5)hi) F0lo​ spread(F0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} F0​c(14) spread(F0​c(14)) F0​a(6)lo spread(F0​a(6)lo) F0​a(6)hi spread(F0​a(6)hi) F0hi​ spread(F0hi​) 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} G0​d(7) spread(G0​d(7)) G0​b(5)lo spread(G0​b(5)lo) G0​b(5)hi spread(G0​b(5)hi) G0lo​ spread(G0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} G0​c(14) spread(G0​c(14)) G0​a(6)lo spread(G0​a(6)lo) G0​a(6)hi spread(G0​a(6)hi) G0hi​ spread(G0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P0even​ spread(P0even​) spread(Elo) spread(Ehi) Q0odd​ K0lo​ H0lo​ W0lo​ 0 0 0 0 0 0 0 1 0 0 1 {0,1,2,3,4,5} P0odd​ spread(P0odd​) spread(P1odd​) Σ1​(E0​)lo Σ1​(E0​)hi K0hi​ H0hi​ W0hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1even​ spread(P1even​) spread(Flo) spread(Fhi) Q1odd​ P1odd​ Hprime0lo​ Hprime0hi​ Hprime0​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1odd​ spread(P1odd​) D0lo​ E1lo​ 0 0 0 0 0 0 0 0 0 1 0 {0,1,2,3,4,5} Q0even​ spread(Q0even​) spread(Eneglo​) spread(Eneghi​) spread(Elo) D0hi​ E1hi​ E1​carry 0 0 0 0 0 0 1 0 0 0 0 {0,1,2,3,4,5} Q0odd​ spread(Q0odd​) spread(Q1odd​) spread(Ehi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1even​ spread(Q1even​) spread(Glo) spread(Ghi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1odd​ spread(Q1odd​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} A0​b(11) spread(A0​b(11)) A0​c(9)lo spread(A0​c(9)lo) A0​c(9)mid spread(A0​c(9)mid) A0lo​ spread(A0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} A0​d(10) spread(A0​d(10)) A0​a(2) spread(A0​a(2)) A0​c(9)hi spread(A0​c(9)hi) A0hi​ spread(A0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(c(9)lo) spread(c(9)mid) 0 0 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(10)) spread(b(11)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(a(2)) spread(c(9)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} B0​b(11) spread(B0​b(11)) B0​c(9)lo spread(B0​c(9)lo) B0​c(9)mid spread(B0​c(9)mid) B0lo​ spread(B0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} B0​d(10) spread(B0​d(10)) B0​a(2) spread(B0​a(2)) B0​c(9)hi spread(B0​c(9)hi) B0hi​ spread(B0hi​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} C0​b(11) spread(C0​b(11)) C0​c(9)lo spread(C0​c(9)lo) C0​c(9)mid spread(C0​c(9)mid) C0lo​ spread(C0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} C0​d(10) spread(C0​d(10)) C0​a(2) spread(C0​a(2)) C0​c(9)hi spread(C0​c(9)hi) C0hi​ spread(C0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M0even​ spread(M0even​) M1odd​ spread(A0lo​) spread(A0hi​) Hprime0lo​ Hprime0hi​ 0 0 0 0 0 1 0 0 1 0 0 {0,1,2,3,4,5} M0odd​ spread(M0odd​) spread(M1odd​) spread(B0lo​) spread(B0hi​) Σ0​(A0​)lo A1lo​ A1​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1even​ spread(M1even​) spread(C0lo​) spread(C0hi​) Σ0​(A0​)hi A1hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1odd​ spread(M1odd​) Steady-state: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0​d(7) spread(E0​d(7)) E0​b(5)lo spread(E0​b(5)lo) E0​b(5)hi spread(E0​b(5)hi) E0lo​ spread(E0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} E0​c(14) spread(E0​c(14)) E0​a(6)lo spread(E0​a(6)lo) E0​a(6)hi spread(E0​a(6)hi) E0hi​ spread(E0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(E0​b(5)lo) spread(E0​b(5)hi) 0 0 0 0 1 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(E0​d(7)) spread(E0​c(14)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(E0​a(6)lo) spread(E0​a(6)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P0even​ spread(P0even​) spread(Elo) spread(Ehi) Q0odd​ K0lo​ H0lo​ W0lo​ 0 0 0 0 0 0 0 1 0 0 1 {0,1,2,3,4,5} P0odd​ spread(P0odd​) spread(P1odd​) Σ1​(E0​)lo Σ1​(E0​)hi K0hi​ H0hi​ W0hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1even​ spread(P1even​) spread(Flo) spread(Fhi) Q1odd​ P1odd​ Hprime0lo​ Hprime0hi​ Hprime0​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1odd​ spread(P1odd​) D0lo​ E1lo​ 0 0 0 0 0 0 0 0 0 1 0 {0,1,2,3,4,5} Q0even​ spread(Q0even​) spread(Eneglo​) spread(Eneghi​) spread(Elo) D0hi​ E1hi​ E1​carry 0 0 0 0 0 0 1 0 0 0 0 {0,1,2,3,4,5} Q0odd​ spread(Q0odd​) spread(Q1odd​) spread(Ehi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1even​ spread(Q1even​) spread(Glo) spread(Ghi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1odd​ spread(Q1odd​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} A0​b(11) spread(A0​b(11)) A0​c(9)lo spread(A0​c(9)lo) A0​c(9)mid spread(A0​c(9)mid) A0lo​ spread(A0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} A0​d(10) spread(A0​d(10)) A0​a(2) spread(A0​a(2)) A0​c(9)hi spread(A0​c(9)hi) A0hi​ spread(A0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(c(9)lo) spread(c(9)mid) 0 0 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(10)) spread(b(11)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(a(2)) spread(c(9)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M0even​ spread(M0even​) M1odd​ spread(A0lo​) spread(A0hi​) Hprime0lo​ Hprime0hi​ 0 0 0 0 0 1 0 0 1 0 0 {0,1,2,3,4,5} M0odd​ spread(M0odd​) spread(M1odd​) spread(B0lo​) spread(B0hi​) Σ0​(A0​)lo A1lo​ A1​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1even​ spread(M1even​) spread(C0lo​) spread(C0hi​) Σ0​(A0​)hi A1hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1odd​ spread(M1odd​) Final digest: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 A63lo​ A63hi​ A63​ B63lo​ B63hi​ B63​ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C63lo​ C63hi​ C63​ C63lo​ C63hi​ C63​ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 E63lo​ E63hi​ E63​ G63lo​ G63hi​ G63​ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 F63lo​ F63hi​ F63​ H63lo​ H63hi​ H63​","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » Compression region","id":"153","title":"Compression region"},"154":{"body":"This section covers the background material required to understand the Halo 2 proving system. It is targeted at an ELI15 (Explain It Like I'm 15) level; if you think anything could do with additional explanation, let us know !","breadcrumbs":"Background Material » Background Material","id":"154","title":"Background Material"},"155":{"body":"A fundamental component of many cryptographic protocols is the algebraic structure known as a field . Fields are sets of objects (usually numbers) with two associated binary operators + and × such that various field axioms hold. The real numbers R are an example of a field with uncountably many elements. Halo makes use of finite fields which have a finite number of elements. Finite fields are fully classified as follows: if F is a finite field, it contains ∣F∣=pk elements for some integer k≥1 and some prime p; any two finite fields with the same number of elements are isomorphic. In particular, all of the arithmetic in a prime field Fp​ is isomorphic to addition and multiplication of integers modulo p, i.e. in Zp​. This is why we often refer to p as the modulus . We'll write a field as Fq​ where q=pk. The prime p is called its characteristic . In the cases where k>1 the field Fq​ is a k-degree extension of the field Fp​. (By analogy, the complex numbers C=R(i) are an extension of the real numbers.) However, in Halo we do not use extension fields. Whenever we write Fp​ we are referring to what we call a prime field which has a prime p number of elements, i.e. k=1. Important notes: There are two special elements in any field: 0, the additive identity, and 1, the multiplicative identity. The least significant bit of a field element, when represented as an integer in binary format, can be interpreted as its \"sign\" to help distinguish it from its additive inverse (negation). This is because for some nonzero element a which has a least significant bit 0 we have that −a=p−a has a least significant bit 1, and vice versa. We could also use whether or not an element is larger than (p−1)/2 to give it a \"sign.\" Finite fields will be useful later for constructing polynomials and elliptic curves . Elliptic curves are examples of groups, which we discuss next.","breadcrumbs":"Background Material » Fields » Fields","id":"155","title":"Fields"},"156":{"body":"Groups are simpler and more limited than fields; they have only one binary operator ⋅ and fewer axioms. They also have an identity, which we'll denote as 1. Any non-zero element a in a group has an inverse b=a−1, which is the unique element b such that a⋅b=1. For example, the set of nonzero elements of Fp​ forms a group, where the group operation is given by multiplication on the field. (aside) Additive vs multiplicative notation If ⋅ is written as × or omitted (i.e. a⋅b written as ab), the identity as 1, and inversion as a−1, as we did above, then we say that the group is \"written multiplicatively\". If ⋅ is written as +, the identity as 0 or O, and inversion as −a, then we say it is \"written additively\". It's conventional to use additive notation for elliptic curve groups, and multiplicative notation when the elements come from a finite field. When additive notation is used, we also write [k]A=k timesA+A+⋯+A​​ for nonnegative k and call this \"scalar multiplication\"; we also often use uppercase letters for variables denoting group elements. When multiplicative notation is used, we also write ak=k timesa×a×⋯×a​​ and call this \"exponentiation\". In either case we call the scalar k such that [k]g=a or gk=a the \"discrete logarithm\" of a to base g. We can extend scalars to negative integers by inversion, i.e. [−k]A+[k]A=O or a−k×ak=1. The order of an element a of a finite group is defined as the smallest positive integer k such that ak=1 (in multiplicative notation) or [k]a=O (in additive notation). The order of the group is the number of elements. Groups always have a generating set , which is a set of elements such that we can produce any element of the group as (in multiplicative terminology) a product of powers of those elements. So if the generating set is g1..k​, we can produce any element of the group as i=1∏k​giai​​. There can be many different generating sets for a given group. A group is called cyclic if it has a (not necessarily unique) generating set with only a single element — call it g. In that case we can say that g generates the group, and that the order of g is the order of the group. Any finite cyclic group G of order n is isomorphic to the integers modulo n (denoted Z/nZ), such that: the operation ⋅ in G corresponds to addition modulo n; the identity in G corresponds to 0; some generator g∈G corresponds to 1. Given a generator g, the isomorphism is always easy to compute in the Z/nZ→G direction; it is just a↦ga (or in additive notation, a↦[a]g). It may be difficult in general to compute in the G→Z/nZ direction; we'll discuss this further when we come to elliptic curves . If the order n of a finite group is prime, then the group is cyclic, and every non-identity element is a generator.","breadcrumbs":"Background Material » Fields » Groups","id":"156","title":"Groups"},"157":{"body":"We use the notation Fp×​ for the multiplicative group (i.e. the group operation is multiplication in Fp​) over the set Fp​−{0}. A quick way of obtaining the inverse in Fp×​ is a−1=ap−2. The reason for this stems from Fermat's little theorem , which states that ap=a(modp) for any integer a. If a is nonzero, we can divide by a twice to get ap−2=a−1. Let's assume that α is a generator of Fp×​, so it has order p−1 (equal to the number of elements in Fp×​). Therefore, for any element in a∈Fp×​ there is a unique integer i∈{0..p−2} such that a=αi. Notice that a×b where a,b∈Fp×​ can really be interpreted as αi×αj where a=αi and b=αj. Indeed, it holds that αi×αj=αi+j for all 0≤i,j = MyCircuit { a: None, lookup_table, }; // Generate the DOT graph string. let dot_string = halo2_proofs::dev::circuit_dot_graph(&circuit); // Now you can either handle it in Rust, or just // print it out to use with command-line tools. print!(\"{}\", dot_string);\n}","breadcrumbs":"User Documentation » Developer tools » Circuit structure","id":"16","title":"Circuit structure"},"160":{"body":"In a field Fp​ exactly half of all nonzero elements are squares; the remainder are non-squares or \"quadratic non-residues\". In order to see why, consider an α that generates the 2-order multiplicative subgroup of Fp×​ (this exists because p−1 is divisible by 2 since p is a prime greater than 2) and β that generates the t-order multiplicative subgroup of Fp×​ where p−1=2t. Then every element a∈Fp×​ can be written uniquely as αi⋅βj with i∈Z2​ and j∈Zt​. Half of all elements will have i=0 and the other half will have i=1. Let's consider the simple case where p≡3(mod4) and so t is odd (if t is even, then p−1 would be divisible by 4, which contradicts p being 3(mod4)). If a∈Fp×​ is a square, then there must exist b=αi⋅βj such that b2=a. But this means that a=(αi⋅βj)2=α2i⋅β2j=β2j. In other words, all squares in this particular field do not generate the 2-order multiplicative subgroup, and so since half of the elements generate the 2-order subgroup then at most half of the elements are square. In fact exactly half of the elements are square (since squaring each nonsquare element gives a unique square). This means we can assume all squares can be written as βm for some m, and therefore finding the square root is a matter of exponentiating by 2−1(modt). In the event that p≡1(mod4) then things get more complicated because 2−1(modt) does not exist. Let's write p−1 as 2k⋅t with t odd. The case k=0 is impossible, and the case k=1 is what we already described, so consider k≥2. α generates a 2k-order multiplicative subgroup and β generates the odd t-order multiplicative subgroup. Then every element a∈Fp×​ can be written as αi⋅βj for i∈Z2k​ and j∈Zt​. If the element is a square, then there exists some b=a​ which can be written b=αi′⋅βj′ for i′∈Z2k​ and j′∈Zt​. This means that a=b2=α2i′⋅β2j′, therefore we have i≡2i′(mod2k), and j≡2j′(modt). i would have to be even in this case because otherwise it would be impossible to have i≡2i′(mod2k) for any i′. In the case that a is not a square, then i is odd, and so half of all elements are squares. In order to compute the square root, we can first raise the element a=αi⋅βj to the power t to \"kill\" the t-order component, giving at=αit(mod2)k⋅βjt(modt)=αit(mod2)k and then raise this result to the power t−1(mod2k) to undo the effect of the original exponentiation on the 2k-order component: (αitmod2k)t−1(mod2k)=αi (since t is relatively prime to 2k). This leaves bare the αi value which we can trivially handle. We can similarly kill the 2k-order component to obtain βj⋅2−1(modt), and put the values together to obtain the square root. It turns out that in the cases k=2,3 there are simpler algorithms that merge several of these exponentiations together for efficiency. For other values of k, the only known way is to manually extract i by squaring until you obtain the identity for every single bit of i. This is the essence of the Tonelli-Shanks square root algorithm and describes the general strategy. (There is another square root algorithm that uses quadratic extension fields, but it doesn't pay off in efficiency until the prime becomes quite large.)","breadcrumbs":"Background Material » Fields » Square roots","id":"160","title":"Square roots"},"161":{"body":"In the previous sections we wrote p−1=2k⋅t with t odd, and stated that an element α∈Fp×​ generated the 2k-order subgroup. For convenience, let's denote n:=2k. The elements {1,α,…,αn−1} are known as the nth roots of unity . The primitive root of unity , ω, is an nth root of unity such that ωi=1 except when i≡0(modn). Important notes: If α is an nth root of unity, α satisfies αn−1=0. If α=1, then 1+α+α2+⋯+αn−1=0. Equivalently, the roots of unity are solutions to the equation Xn−1=(X−1)(X−α)(X−α2)⋯(X−αn−1). ω2n​+i=−ωi​ (\"Negation lemma\") . Proof: ωn=1​⟹ωn−1=0⟹(ωn/2+1)(ωn/2−1)=0.​ Since the order of ω is n, ωn/2=1. Therefore, ωn/2=−1. (ω2n​+i)2=(ωi)2​ (\"Halving lemma\") . Proof: (ω2n​+i)2=ωn+2i=ωn⋅ω2i=ω2i=(ωi)2. In other words, if we square each element in the nth roots of unity, we would get back only half the elements, {(ωni​)2}={ωn/2​} (i.e. the 2n​th roots of unity). There is a two-to-one mapping between the elements and their squares.","breadcrumbs":"Background Material » Fields » Roots of unity","id":"161","title":"Roots of unity"},"162":{"body":"Friedman, R. (n.d.) \"Cyclic Groups and Elementary Number Theory II\" (p. 5).","breadcrumbs":"Background Material » Fields » References","id":"162","title":"References"},"163":{"body":"Let A(X) be a polynomial over Fp​ with formal indeterminate X. As an example, A(X)=a0​+a1​X+a2​X2+a3​X3 defines a degree-3 polynomial. a0​ is referred to as the constant term. Polynomials of degree n−1 have n coefficients. We will often want to compute the result of replacing the formal indeterminate X with some concrete value x, which we denote by A(x). In mathematics this is commonly referred to as \"evaluating A(X) at a point x\". The word \"point\" here stems from the geometrical usage of polynomials in the form y=A(x), where (x,y) is the coordinate of a point in two-dimensional space. However, the polynomials we deal with are almost always constrained to equal zero, and x will be an element of some field . This should not be confused with points on an elliptic curve , which we also make use of, but never in the context of polynomial evaluation. Important notes: Multiplication of polynomials produces a product polynomial that is the sum of the degrees of its factors. Polynomial division subtracts from the degree. deg(A(X)B(X))=deg(A(X))+deg(B(X)), deg(A(X)/B(X))=deg(A(X))−deg(B(X)). Given a polynomial A(X) of degree n−1, if we obtain n evaluations of the polynomial at distinct points then these evaluations perfectly define the polynomial. In other words, given these evaluations we can obtain a unique polynomial A(X) of degree n−1 via polynomial interpolation. [a0​,a1​,⋯,an−1​] is the coefficient representation of the polynomial A(X). Equivalently, we could use its evaluation representation [(x0​,A(x0​)),(x1​,A(x1​)),⋯,(xn−1​,A(xn−1​))] at n distinct points. Either representation uniquely specifies the same polynomial. (aside) Horner's rule Horner's rule allows for efficient evaluation of a polynomial of degree n−1, using only n−1 multiplications and n−1 additions. It is the following identity: a0​​+a1​X+a2​X2+⋯+an−1​Xn−1=a0​+X(a1​+X(a2​+⋯+X(an−2​+Xan−1​))),​","breadcrumbs":"Background Material » Polynomials » Polynomials","id":"163","title":"Polynomials"},"164":{"body":"The FFT is an efficient way of converting between the coefficient and evaluation representations of a polynomial. It evaluates the polynomial at the nth roots of unity {ω0,ω1,⋯,ωn−1}, where ω is a primitive nth root of unity. By exploiting symmetries in the roots of unity, each round of the FFT reduces the evaluation into a problem only half the size. Most commonly we use polynomials of length some power of two, n=2k, and apply the halving reduction recursively.","breadcrumbs":"Background Material » Polynomials » Fast Fourier Transform (FFT)","id":"164","title":"Fast Fourier Transform (FFT)"},"165":{"body":"In the coefficient representation, it takes O(n2) operations to multiply two polynomials A(X)⋅B(X)=C(X): A(X)B(X)C(X)​=a0​+a1​X+a2​X2+⋯+an−1​Xn−1,=b0​+b1​X+b2​X2+⋯+bn−1​Xn−1,=a0​⋅(b0​+b1​X+b2​X2+⋯+bn−1​Xn−1)+a1​X⋅(b0​+b1​X+b2​X2+⋯+bn−1​Xn−1)+⋯+an−1​Xn−1⋅(b0​+b1​X+b2​X2+⋯+bn−1​Xn−1),​ where each of the n terms in the first polynomial has to be multiplied by the n terms of the second polynomial. In the evaluation representation, however, polynomial multiplication only requires O(n) operations: ABC​:{(x0​,A(x0​)),(x1​,A(x1​)),⋯,(xn−1​,A(xn−1​))},:{(x0​,B(x0​)),(x1​,B(x1​)),⋯,(xn−1​,B(xn−1​))},:{(x0​,A(x0​)B(x0​)),(x1​,A(x1​)B(x1​)),⋯,(xn−1​,A(xn−1​)B(xn−1​))},​ where each evaluation is multiplied pointwise. This suggests the following strategy for fast polynomial multiplication: Evaluate polynomials at all n points; Perform fast pointwise multiplication in the evaluation representation (O(n)); Convert back to the coefficient representation. The challenge now is how to evaluate and interpolate the polynomials efficiently. Naively, evaluating a polynomial at n points would require O(n2) operations (we use the O(n) Horner's rule at each point): ⎣⎡​A(1)A(ω)A(ω2)⋮A(ωn−1)​⎦⎤​=⎣⎡​111⋮1​1ωω2⋮ωn−1​1ω2ω2⋅2⋮ω2(n−1)​………⋯​1ωn−1ω2⋅(n−1)⋮ω(n−1)2​⎦⎤​⋅⎣⎡​a0​a1​a2​⋮an−1​​⎦⎤​. For convenience, we will denote the matrices above as: A^=Vω​⋅A. (A^ is known as the Discrete Fourier Transform of A; Vω​ is also called the Vandermonde matrix .)","breadcrumbs":"Background Material » Polynomials » Motivation: Fast polynomial multiplication","id":"165","title":"Motivation: Fast polynomial multiplication"},"166":{"body":"Our strategy is to divide a DFT of size n into two interleaved DFTs of size n/2. Given the polynomial A(X)=a0​+a1​X+a2​X2+⋯+an−1​Xn−1, we split it up into even and odd terms: Aeven​Aodd​​=a0​+a2​X+⋯+an−2​X2n​−1,=a1​+a3​X+⋯+an−1​X2n​−1.​ To recover the original polynomial, we do A(X)=Aeven​(X2)+XAodd​(X2). Trying this out on points ωni​ and ωn2n​+i​, i∈[0..2n​−1], we start to notice some symmetries: A(ωni​)A(ωn2n​+i​)​=Aeven​((ωni​)2)+ωni​Aodd​((ωni​)2),=Aeven​((ωn2n​+i​)2)+ωn2n​+i​Aodd​((ωn2n​+i​)2)=Aeven​((−ωni​)2)−ωni​Aodd​((−ωni​)2)←(negation lemma)=Aeven​((ωni​)2)−ωni​Aodd​((ωni​)2).​ Notice that we are only evaluating Aeven​(X) and Aodd​(X) over half the domain {(ωn0​)2,(ωn​)2,⋯,(ωn2n​−1​)2}={ωn/2i​},i=[0..2n​−1] (halving lemma). This gives us all the terms we need to reconstruct A(X) over the full domain {ω0,ω,⋯,ωn−1}: which means we have transformed a length-n DFT into two length-2n​ DFTs. We choose n=2k to be a power of two (by zero-padding if needed), and apply this divide-and-conquer strategy recursively. By the Master Theorem [1] , this gives us an evaluation algorithm with O(nlog2​n) operations, also known as the Fast Fourier Transform (FFT).","breadcrumbs":"Background Material » Polynomials » The (radix-2) Cooley-Tukey algorithm","id":"166","title":"The (radix-2) Cooley-Tukey algorithm"},"167":{"body":"So we've evaluated our polynomials and multiplied them pointwise. What remains is to convert the product from the evaluation representation back to coefficient representation. To do this, we simply call the FFT on the evaluation representation. However, this time we also: replace ωi by ω−i in the Vandermonde matrix, and multiply our final result by a factor of 1/n. In other words: A=n1​Vω−1​⋅A^. (To understand why the inverse FFT has a similar form to the FFT, refer to Slide 13-1 of [2] . The below image was also taken from [2] .)","breadcrumbs":"Background Material » Polynomials » Inverse FFT","id":"167","title":"Inverse FFT"},"168":{"body":"The Schwartz-Zippel lemma informally states that \"different polynomials are different at most points.\" Formally, it can be written as follows: Let p(x1​,x2​,⋯,xn​) be a nonzero polynomial of n variables with degree d. Let S be a finite set of numbers with at least d elements in it. If we choose random α1​,α1​,⋯,αn​ from S, Pr[p(α1​,α2​,⋯,αn​)=0]≤∣S∣d​. In the familiar univariate case p(X), this reduces to saying that a nonzero polynomial of degree d has at most d roots. The Schwartz-Zippel lemma is used in polynomial equality testing. Given two multi-variate polynomials p1​(x1​,⋯,xn​) and p2​(x1​,⋯,xn​) of degrees d1​,d2​ respectively, we can test if p1​(α1​,⋯,αn​)−p2​(α1​,⋯,αn​)=0 for random α1​,⋯,αn​←S, where the size of S is at least ∣S∣≥(d1​+d2​). If the two polynomials are identical, this will always be true, whereas if the two polynomials are different then the equality holds with probability at most ∣S∣max(d1​,d2​)​.","breadcrumbs":"Background Material » Polynomials » The Schwartz-Zippel lemma","id":"168","title":"The Schwartz-Zippel lemma"},"169":{"body":"Consider the order-n multiplicative subgroup H with primitive root of unity ω. For all ωi∈H,i∈[n−1], we have (ωi)n=(ωn)i=(ω0)i=1. In other words, every element of H fulfils the equation ZH​(X)​=Xn−1=(X−ω0)(X−ω1)(X−ω2)⋯(X−ωn−1),​ meaning every element is a root of ZH​(X). We call ZH​(X) the vanishing polynomial over H because it evaluates to zero on all elements of H. This comes in particularly handy when checking polynomial constraints. For instance, to check that A(X)+B(X)=C(X) over H, we simply have to check that A(X)+B(X)−C(X) is some multiple of ZH​(X). In other words, if dividing our constraint by the vanishing polynomial still yields some polynomial ZH​(X)A(X)+B(X)−C(X)​=H(X), we are satisfied that A(X)+B(X)−C(X)=0 over H.","breadcrumbs":"Background Material » Polynomials » Vanishing polynomial","id":"169","title":"Vanishing polynomial"},"17":{"body":"The cost-model binary takes high-level parameters for a circuit design, and estimates the verification cost, as well as resulting proof size. Usage: cargo run --example cost-model -- [OPTIONS] k Positional arguments: k 2^K bound on the number of rows. Optional arguments: -h, --help Print this message. -a, --advice R[,R..] An advice column with the given rotations. May be repeated. -i, --instance R[,R..] An instance column with the given rotations. May be repeated. -f, --fixed R[,R..] A fixed column with the given rotations. May be repeated. -g, --gate-degree D Maximum degree of the custom gates. -l, --lookup N,I,T A lookup over N columns with max input degree I and max table degree T. May be repeated. -p, --permutation N A permutation over N columns. May be repeated. For example, to estimate the cost of a circuit with three advice columns and one fixed column (with various rotations), and a maximum gate degree of 4: > cargo run --example cost-model -- -a 0,1 -a 0 -a-0,-1,1 -f 0 -g 4 11 Finished dev [unoptimized + debuginfo] target(s) in 0.03s Running `target/debug/examples/cost-model -a 0,1 -a 0 -a 0,-1,1 -f 0 -g 4 11`\nCircuit { k: 11, max_deg: 4, advice_columns: 3, lookups: 0, permutations: [], column_queries: 7, point_sets: 3, estimator: Estimator,\n}\nProof size: 1440 bytes\nVerification: at least 81.689ms","breadcrumbs":"User Documentation » Developer tools » Cost estimator","id":"17","title":"Cost estimator"},"170":{"body":"TODO: explain what a basis is in general (briefly). Polynomials are commonly written in the monomial basis (e.g. X,X2,...Xn). However, when working over a multiplicative subgroup of order n, we find a more natural expression in the Lagrange basis. Consider the order-n multiplicative subgroup H with primitive root of unity ω. The Lagrange basis corresponding to this subgroup is a set of functions {Li​}i=0n−1​, where Li​(ωj)={10​if i=j,otherwise.​ We can write this more compactly as Li​(ωj)=δij​, where δ is the Kronecker delta function. Now, we can write our polynomial as a linear combination of Lagrange basis functions, A(X)=i=0∑n−1​ai​Li​(X),X∈H, which is equivalent to saying that p(X) evaluates to a0​ at ω0, to a1​ at ω1, to a2​ at ω2,⋯, and so on. When working over a multiplicative subgroup, the Lagrange basis function has a convenient sparse representation of the form Li​(X)=X−ωici​⋅(Xn−1)​, where ci​ is the barycentric weight. (To understand how this form was derived, refer to [3] .) For i=0, we have c=1/n⟹L0​(X)=n1​X−1(Xn−1)​. Suppose we are given a set of evaluation points {x0​,x1​,⋯,xn−1​}. Since we cannot assume that the xi​'s form a multiplicative subgroup, we consider also the Lagrange polynomials Li​'s in the general case. Then we can construct: Li​(X)=j=i∏​xi​−xj​X−xj​​,i∈[0..n−1]. Here, every X=xj​=xi​ will produce a zero numerator term (xj​−xj​), causing the whole product to evaluate to zero. On the other hand, X=xi​ will evaluate to xi​−xj​xi​−xj​​ at every term, resulting in an overall product of one. This gives the desired Kronecker delta behaviour Li​(xj​)=δij​ on the set {x0​,x1​,⋯,xn−1​}.","breadcrumbs":"Background Material » Polynomials » Lagrange basis functions","id":"170","title":"Lagrange basis functions"},"171":{"body":"Given a polynomial in its evaluation representation A:{(x0​,A(x0​)),(x1​,A(x1​)),⋯,(xn−1​,A(xn−1​))}, we can reconstruct its coefficient form in the Lagrange basis: A(X)=i=0∑n−1​A(xi​)Li​(X), where X∈{x0​,x1​,⋯,x1−n​}.","breadcrumbs":"Background Material » Polynomials » Lagrange interpolation","id":"171","title":"Lagrange interpolation"},"172":{"body":"Dasgupta, S., Papadimitriou, C. H., & Vazirani, U. V. (2008). \"Algorithms\" (ch. 2). New York: McGraw-Hill Higher Education. Golin, M. (2016). \"The Fast Fourier Transform and Polynomial Multiplication\" [lecture notes], COMP 3711H Design and Analysis of Algorithms, Hong Kong University of Science and Technology. Berrut, J. and Trefethen, L. (2004). \"Barycentric Lagrange Interpolation.\"","breadcrumbs":"Background Material » Polynomials » References","id":"172","title":"References"},"173":{"body":"In the section Inverses and groups we introduced the concept of groups . A group has an identity and a group operation. In this section we will write groups additively, i.e. the identity is O and the group operation is +. Some groups can be used as cryptographic groups . At the risk of oversimplifying, this means that the problem of finding a discrete logarithm of a group element P to a given base G, i.e. finding x such that P=[x]G, is hard in general.","breadcrumbs":"Background Material » Cryptographic groups » Cryptographic groups","id":"173","title":"Cryptographic groups"},"174":{"body":"The Pedersen commitment [ P99 ] is a way to commit to a secret message in a verifiable way. It uses two random public generators G,H∈G, where G is a cryptographic group of order p. A random secret r is chosen in Zq​, and the message to commit to m is from any subset of Zq​. The commitment is c=Commit(m,r)=[m]G+[r]H. To open the commitment, the committer reveals m and r, thus allowing anyone to verify that c is indeed a commitment to m. Notice that the Pedersen commitment scheme is homomorphic: Commit(m,r)+Commit(m′,r′)​=[m]G+[r]H+[m′]G+[r′]H=[m+m′]G+[r+r′]H=Commit(m+m′,r+r′).​ Assuming the discrete log assumption holds, Pedersen commitments are also perfectly hiding and computationally binding: hiding : the adversary chooses messages m0​,m1​. The committer commits to one of these messages c=Commit(mb​;r),b∈{0,1}. Given c, the probability of the adversary guessing the correct b is no more than 21​. binding : the adversary cannot pick two different messages m0​=m1​, and randomness r0​,r1​, such that Commit(m0​,r0​)=Commit(m1​,r1​).","breadcrumbs":"Background Material » Cryptographic groups » Pedersen commitment","id":"174","title":"Pedersen commitment"},"175":{"body":"We can use a variant of the Pedersen commitment scheme to commit to multiple messages at once, m=(m1​,⋯,mn​). This time, we'll have to sample a corresponding number of random public generators G=(G0​,⋯,Gn−1​), along with a single random generator H as before (for use in hiding). Then, our commitment scheme is: Commit(m;r)​=Commit((m0​,⋯,mn−1​);r)=[r]H+[m0​]G0​+⋯+[mn−1​]Gn−1​=[r]H+i=0∑n−1​[mi​]Gi​.​ TODO: is this positionally binding?","breadcrumbs":"Background Material » Cryptographic groups » Vector Pedersen commitment","id":"175","title":"Vector Pedersen commitment"},"176":{"body":"An example of a protocol that uses cryptographic groups is Diffie--Hellman key agreement [ DH1976 ]. The Diffie--Hellman protocol is a method for two users, Alice and Bob, to generate a shared private key. It proceeds as follows: Alice and Bob publicly agree on two prime numbers, p and G, where p is large and G is a primitive root (modp). (Note that g is a generator of the group Fp×​.) Alice chooses a large random number a as her private key. She computes her public key A=[a]G(modp), and sends A to Bob. Similarly, Bob chooses a large random number b as his private key. He computes his public key B=[b]G(modp), and sends B to Alice. Now both Alice and Bob compute their shared key K=[ab]G(modp), which Alice computes as K=[a]B(modp)=[a]([b]G)(modp), and Bob computes as K=[b]A(modp)=[b]([a]G)(modp). A potential eavesdropper would need to derive K=[ab]g(modp) knowing only g,p,A=[a]G, and B=[b]G: in other words, they would need to either get the discrete logarithm a from A=[a]G or b from B=[b]G, which we assume to be computationally infeasible in Fp×​. More generally, protocols that use similar ideas to Diffie--Hellman are used throughout cryptography. One way of instantiating a cryptographic group is as an elliptic curve . Before we go into detail on elliptic curves, we'll describe some algorithms that can be used for any group.","breadcrumbs":"Background Material » Cryptographic groups » Diffie--Hellman","id":"176","title":"Diffie--Hellman"},"177":{"body":"","breadcrumbs":"Background Material » Cryptographic groups » Multiscalar multiplication","id":"177","title":"Multiscalar multiplication"},"178":{"body":"Reference: https://jbootle.github.io/Misc/pippenger.pdf","breadcrumbs":"Background Material » Cryptographic groups » TODO: Pippenger's algorithm","id":"178","title":"TODO: Pippenger's algorithm"},"179":{"body":"Elliptic curves constructed over finite fields are another important cryptographic tool. We use elliptic curves because they provide a cryptographic group , i.e. a group in which the discrete logarithm problem (discussed below) is hard. There are several ways to define the curve equation, but for our purposes, let Fp​ be a large (255-bit) field, and then let the set of solutions (x,y) to y2=x3+b for some constant b define the Fp​-rational points on an elliptic curve E(Fp​). These (x,y) coordinates are called \"affine coordinates\". Each of the Fp​-rational points, together with a \"point at infinity\" O that serves as the group identity, can be interpreted as an element of a group. By convention, elliptic curve groups are written additively. \"Three points on a line sum to zero, which is the point at infinity.\" The group addition law is simple: to add two points together, find the line that intersects both points and obtain the third point, and then negate its y-coordinate. The case that a point is being added to itself, called point doubling, requires special handling: we find the line tangent to the point, and then find the single other point that intersects this line and then negate. Otherwise, in the event that a point is being \"added\" to its negation, the result is the point at infinity. The ability to add and double points naturally gives us a way to scale them by integers, called scalars . The number of points on the curve is the group order. If this number is a prime q, then the scalars can be considered as elements of a scalar field , Fq​. Elliptic curves, when properly designed, have an important security property. Given two random elements G,H∈E(Fp​) finding a such that [a]G=H, otherwise known as the discrete log of H with respect to G, is considered computationally infeasible with classical computers. This is called the elliptic curve discrete log assumption. If an elliptic curve group G has prime order q (like the ones used in Halo 2), then it is a finite cyclic group. Recall from the section on groups that this implies it is isomorphic to Z/qZ, or equivalently, to the scalar field Fq​. Each possible generator G fixes the isomorphism; then an element on the scalar side is precisely the discrete log of the corresponding group element with respect to G. In the case of a cryptographically secure elliptic curve, the isomorphism is hard to compute in the G→Fq​ direction because the elliptic curve discrete log problem is hard. It is sometimes helpful to make use of this isomorphism by thinking of group-based cryptographic protocols and algorithms in terms of the scalars instead of in terms of the group elements. This can make proofs and notation simpler. For instance, it has become common in papers on proof systems to use the notation [x] to denote a group element with discrete log x, where the generator is implicit. We also used this idea in the \" distinct-x theorem \", in order to prove correctness of optimizations for elliptic curve scalar multiplication in Sapling, and an endomorphism-based optimization in Appendix C of the original Halo paper .","breadcrumbs":"Background Material » Elliptic curves » Elliptic curves","id":"179","title":"Elliptic curves"},"18":{"body":"Let's start with a simple circuit, to introduce you to the common APIs and how they are used. The circuit will take a public input c, and will prove knowledge of two private inputs a and b such that a2⋅b2=c.","breadcrumbs":"User Documentation » A simple example » A simple example","id":"18","title":"A simple example"},"180":{"body":"","breadcrumbs":"Background Material » Elliptic curves » Curve arithmetic","id":"180","title":"Curve arithmetic"},"181":{"body":"The simplest situation is doubling a point (x0​,y0​). Continuing with our example y2=x3+b, this is done first by computing the derivative λ=dxdy​=2y3x2​. To obtain expressions for (x1​,y1​)=(x0​,y0​)+(x0​,y0​), we consider x1​−x0​−y1​−y0​​=λ​⟹−y1​=λ(x1​−x0​)+y0​⟹y1​=λ(x0​−x1​)−y0​​.​ To get the expression for x1​, we substitute y=λ(x0​−x)−y0​ into the elliptic curve equation: y2=x3+b​⟹(λ(x0​−x)−y0​)2=x3+b⟹x3−λ2x2+⋯=0←(rearranging terms)=(x−x0​)(x−x0​)(x−x1​)←(known roots x0​,x0​,x1​)=x3−(x0​+x0​+x1​)x2+⋯.​ Comparing coefficients for the x2 term gives us λ2=x0​+x0​+x1​⟹x1​=λ2−2x0​​.","breadcrumbs":"Background Material » Elliptic curves » Point doubling","id":"181","title":"Point doubling"},"182":{"body":"This unfortunately requires an expensive inversion of 2y. We can avoid this by arranging our equations to \"defer\" the computation of the inverse, since we often do not need the actual affine (x′,y′) coordinate of the resulting point immediately after an individual curve operation. Let's introduce a third coordinate Z and scale our curve equation by Z3 like so: Z3y2=Z3x3+Z3b Our original curve is just this curve at the restriction Z=1. If we allow the affine point (x,y) to be represented by X=xZ, Y=yZ and Z=0 then we have the homogenous projective curve Y2Z=X3+Z3b. Obtaining (x,y) from (X,Y,Z) is as simple as computing (X/Z,Y/Z) when Z=0. (When Z=0, we are dealing with the point at infinity O:=(0:1:0).) In this form, we now have a convenient way to defer the inversion required by doubling a point. The general strategy is to express x′,y′ as rational functions using x=X/Z and y=Y/Z, rearrange to make their denominators the same, and then take the resulting point (X,Y,Z) to have Z be the shared denominator and X=x′Z,Y=y′Z. Projective coordinates are often, but not always, more efficient than affine coordinates. There may be exceptions to this when either we have a different way to apply Montgomery's trick, or when we're in the circuit setting where multiplications and inversions are about equally as expensive (at least in terms of circuit size). The following shows an example of doubling a point (X,Y,Z)=(xZ,yZ,Z) without an inversion. Substituting with X,Y,Z gives us λ=2y3x2​=2(Y/Z)3(X/Z)2​=2YZ3X2​ and gives us x′y′​=λ2−2x=λ2−Z2X​=4Y2Z29X4​−Z2X​=4Y2Z29X4−8XY2Z​=8Y3Z318X4YZ−16XY3Z2​=λ(x−x′)−y=λ(ZX​−4Y2Z29X4−8XY2Z​)−ZY​=2YZ3X2​(ZX​−4Y2Z29X4−8XY2Z​)−ZY​=2YZ23X3​−8Y3Z327X6−24X3Y2Z​−ZY​=8Y3Z312X3Y2Z−8Y4Z2−27X6+24X3Y2Z​​ Notice how the denominators of x′ and y′ are the same. Thus, instead of computing (x′,y′) we can compute (X,Y,Z) with Z=8Y3Z3 and X,Y set to the corresponding numerators such that X/Z=x′ and Y/Z=y′. This completely avoids the need to perform an inversion when doubling, and something analogous to this can be done when adding two distinct points.","breadcrumbs":"Background Material » Elliptic curves » Projective coordinates","id":"182","title":"Projective coordinates"},"183":{"body":"We now add two points with distinct x-coordinates, P=(x0​,y0​) and Q=(x1​,y1​), where x0​=x1​, to obtain R=P+Q=(x2​,y2​). The line PQ​ has slope λ=fracy1​−y0​x1​−x0​⟹y−y0​=λ⋅(x−x0​). Using the expression for PQ​, we compute y-coordinate −y2​ of −R as: −y2​−y0​=λ⋅(x2​−x0​)⟹y2​=(x0​−x2​)−y0​​. Plugging the expression for PQ​ into the curve equation y2=x3+b yields y2=x3+b​⟹(λ⋅(x−x0​)+y0​)2=x3+b⟹x3−λ2x2+⋯=0←(rearranging terms)=(x−x0​)(x−x1​)(x−x2​)←(known roots x0​,x1​,x2​)=x3−(x0​+x1​+x2​)x2+⋯.​ Comparing coefficients for the x2 term gives us λ2=x0​+x1​+x2​⟹x2​=λ2−x0​−x1​​. Important notes: There exist efficient formulae [1] for point addition that do not have edge cases (so-called \"complete\" formulae) and that unify the addition and doubling cases together. The result of adding a point to its negation using those formulae produces Z=0, which represents the point at infinity. In addition, there are other models like the Jacobian representation where (x,y)=(xZ2,yZ3,Z) where the curve is rescaled by Z6 instead of Z3, and this representation has even more efficient arithmetic but no unified/complete formulae. We can easily compare two curve points (X1​,Y1​,Z1​) and (X2​,Y2​,Z2​) for equality in the homogenous projective coordinate space by \"homogenizing\" their Z-coordinates; the checks become X1​Z2​=X2​Z1​ and Y1​Z2​=Y2​Z1​.","breadcrumbs":"Background Material » Elliptic curves » Point addition","id":"183","title":"Point addition"},"184":{"body":"Imagine that Fp​ has a primitive cube root of unity, or in other words that 3∣p−1 and so an element ζp​ generates a 3-order multiplicative subgroup. Notice that a point (x,y) on our example elliptic curve y2=x3+b has two cousin points: (ζp​x,ζp2​x), because the computation x3 effectively kills the ζ component of the x-coordinate. Applying the map (x,y)↦(ζp​x,y) is an application of an endomorphism over the curve. The exact mechanics involved are complicated, but when the curve has a prime q number of points (and thus a prime \"order\") the effect of the endomorphism is to multiply the point by a scalar in Fq​ which is also a primitive cube root ζq​ in the scalar field.","breadcrumbs":"Background Material » Elliptic curves » Curve endomorphisms","id":"184","title":"Curve endomorphisms"},"185":{"body":"Given a point on the curve P=(x,y), we know that its negation −P=(x,−y) is also on the curve. To uniquely specify a point, we need only encode its x-coordinate along with the sign of its y-coordinate.","breadcrumbs":"Background Material » Elliptic curves » Curve point compression","id":"185","title":"Curve point compression"},"186":{"body":"As mentioned in the Fields section, we can interpret the least significant bit of a field element as its \"sign\", since its additive inverse will always have the opposite LSB. So we record the LSB of the y-coordinate as sign. Pallas and Vesta are defined over the Fp​ and Fq​ fields, which elements can be expressed in 255 bits. This conveniently leaves one unused bit in a 32-byte representation. We pack the y-coordinate sign bit into the highest bit in the representation of the x-coordinate: <----------------------------------- x --------------------------------->\nEnc(P) = [_ _ _ _ _ _ _ _] [_ _ _ _ _ _ _ _] ... [_ _ _ _ _ _ _ _] [_ _ _ _ _ _ _ sign] ^ <-------------------------------------> ^ LSB 30 bytes MSB The \"point at infinity\" O that serves as the group identity, does not have an affine (x,y) representation. However, it turns out that there are no points on either the Pallas or Vesta curve with x=0 or y=0. We therefore use the \"fake\" affine coordinates (0,0) to encode O, which results in the all-zeroes 32-byte array.","breadcrumbs":"Background Material » Elliptic curves » Serialization","id":"186","title":"Serialization"},"187":{"body":"When deserializing a compressed curve point, we first read the most significant bit as ysign, the sign of the y-coordinate. Then, we set this bit to zero to recover the original x-coordinate. If x=0,y=0, we return the \"point at infinity\" O. Otherwise, we proceed to compute y=x3+b​. Here, we read the least significant bit of y as sign. If sign == ysign, we already have the correct sign and simply return the curve point (x,y). Otherwise, we negate y and return (x,−y).","breadcrumbs":"Background Material » Elliptic curves » Deserialization","id":"187","title":"Deserialization"},"188":{"body":"Let Ep​ be an elliptic curve over a finite field Fp​, where p is a prime. We denote this by Ep​/Fp​. and we denote the group of points of Ep​ over Fp​, with order q=#E(Fp​). For this curve, we call Fp​ the \"base field\" and Fq​ the \"scalar field\". We instantiate our proof system over the elliptic curve Ep​/Fp​. This allows us to prove statements about Fq​-arithmetic circuit satisfiability. (aside) If our curve Ep​ is over Fp​, why is the arithmetic circuit instead in Fq​? The proof system is basically working on encodings of the scalars in the circuit (or more precisely, commitments to polynomials whose coefficients are scalars). The scalars are in Fq​ when their encodings/commitments are elliptic curve points in Ep​/Fp​. However, most of the verifier's arithmetic computations are over the base field Fp​, and are thus efficiently expressed as an Fp​-arithmetic circuit. (aside) Why are the verifier's computations (mainly) over Fp​? The Halo 2 verifier actually has to perform group operations using information output by the circuit. Group operations like point doubling and addition use arithmetic in Fp​, because the coordinates of points are in Fp​. This motivates us to construct another curve with scalar field Fp​, which has an Fp​-arithmetic circuit that can efficiently verify proofs from the first curve. As a bonus, if this second curve had base field Eq​/Fq​, it would generate proofs that could be efficiently verified in the first curve's Fq​-arithmetic circuit. In other words, we instantiate a second proof system over Eq​/Fq​, forming a 2-cycle with the first:","breadcrumbs":"Background Material » Elliptic curves » Cycles of curves","id":"188","title":"Cycles of curves"},"189":{"body":"Reference: https://github.com/zcash/pasta","breadcrumbs":"Background Material » Elliptic curves » TODO: Pallas-Vesta curves","id":"189","title":"TODO: Pallas-Vesta curves"},"19":{"body":"Firstly, we need to define the instructions that our circuit will rely on. Instructions are the boundary between high-level gadgets and the low-level circuit operations. Instructions may be as coarse or as granular as desired, but in practice you want to strike a balance between an instruction being large enough to effectively optimize its implementation, and small enough that it is meaningfully reusable. For our circuit, we will use three instructions: Load a private number into the circuit. Multiply two numbers. Expose a number as a public input to the circuit. We also need a type for a variable representing a number. Instruction interfaces provide associated types for their inputs and outputs, to allow the implementations to represent these in a way that makes the most sense for their optimization goals. trait NumericInstructions: Chip { /// Variable representing a number. type Num; /// Loads a number into the circuit as a private input. fn load_private(&self, layouter: impl Layouter, a: Option) -> Result; /// Loads a number into the circuit as a fixed constant. fn load_constant(&self, layouter: impl Layouter, constant: F) -> Result; /// Returns `c = a * b`. fn mul( &self, layouter: impl Layouter, a: Self::Num, b: Self::Num, ) -> Result; /// Exposes a number as a public input to the circuit. fn expose_public( &self, layouter: impl Layouter, num: Self::Num, row: usize, ) -> Result<(), Error>;\n}","breadcrumbs":"User Documentation » A simple example » Define instructions","id":"19","title":"Define instructions"},"190":{"body":"Sometimes it is useful to be able to produce a random point on an elliptic curve Ep​/Fp​ corresponding to some input, in such a way that no-one will know its discrete logarithm (to any other base). This is described in detail in the Internet draft on Hashing to Elliptic Curves . Several algorithms can be used depending on efficiency and security requirements. The framework used in the Internet Draft makes use of several functions: hash_to_field: takes a byte sequence input and maps it to a element in the base field Fp​ map_to_curve: takes an Fp​ element and maps it to Ep​.","breadcrumbs":"Background Material » Elliptic curves » Hashing to curves","id":"190","title":"Hashing to curves"},"191":{"body":"Reference: https://eprint.iacr.org/2019/403.pdf","breadcrumbs":"Background Material » Elliptic curves » TODO: Simplified SWU","id":"191","title":"TODO: Simplified SWU"},"192":{"body":"Renes, J., Costello, C., & Batina, L. (2016, May). \"Complete addition formulas for prime order elliptic curves.\" In Annual International Conference on the Theory and Applications of Cryptographic Techniques (pp. 403-428). Springer, Berlin, Heidelberg.","breadcrumbs":"Background Material » Elliptic curves » References","id":"192","title":"References"},"193":{"body":"We want to commit to some polynomial p(X)∈Fp​[X], and be able to provably evaluate the committed polynomial at arbitrary points. The naive solution would be for the prover to simply send the polynomial's coefficients to the verifier: however, this requires O(n) communication. Our polynomial commitment scheme gets the job done using O(logn) communication.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Polynomial commitment using inner product argument","id":"193","title":"Polynomial commitment using inner product argument"},"194":{"body":"Given a parameter d=2k, we generate the common reference string σ=(G,G,H,Fp​) defining certain constants for this scheme: G is a group of prime order p; G∈Gd is a vector of d random group elements; H∈G is a random group element; and Fp​ is the finite field of order p.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Setup","id":"194","title":"Setup"},"195":{"body":"The Pedersen vector commitment Commit is defined as Commit(σ,p(X);r)=⟨a,G⟩+[r]H, for some polynomial p(X)∈Fp​[X] and some blinding factor r∈Fp​. Here, each element of the vector ai​∈Fp​ is the coefficient for the ith degree term of p(X), and p(X) is of maximal degree d−1.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Commit","id":"195","title":"Commit"},"196":{"body":"The modified inner product argument is an argument of knowledge for the relation {((P,x,v);(a,r)):P=⟨a,G⟩+[r]H,v=⟨a,b⟩}​, where b=(1,x,x2,⋯,xd−1) is composed of increasing powers of the evaluation point x. This allows a prover to demonstrate to a verifier that the polynomial contained “inside” the commitment P evaluates to v at x, and moreover, that the committed polynomial has maximum degree d−1. The inner product argument proceeds in k=log2​d rounds. For our purposes, it is sufficient to know about its final outputs, while merely providing intuition about the intermediate rounds. (Refer to Section 3 in the Halo paper for a full explanation.) Before beginning the argument, the verifier selects a random group element U and sends it to the prover. We initialize the argument at round k, with the vectors a(k):=a, G(k):=G and b(k):=b. In each round j=k,k−1,⋯,1: the prover computes two values Lj​ and Rj​ by taking some inner product of a(j) with G(j) and b(j). Note that are in some sense \"cross-terms\": the lower half of a is used with the higher half of G and b, and vice versa: Lj​Rj​​=⟨alo(j)​,Ghi(j)​⟩+[lj​]H+[⟨alo(j)​,bhi(j)​⟩]U=⟨ahi(j)​,Glo(j)​⟩+[lj​]H+[⟨ahi(j)​,blo(j)​⟩]U​ the verifier issues a random challenge uj​; the prover uses uj​ to compress the lower and higher halves of a(j), thus producing a new vector of half the original length a(j−1)=ahi(j)​⋅uj−1​+alo(j)​⋅uj​. The vectors G(j) and b(j) are similarly compressed to give G(j−1) and b(j−1). a(j−1), G(j−1) and b(j−1) are input to the next round j−1. Note that at the end of the last round j=1, we are left with a:=a(0), G:=G(0), b:=b(0), each of length 1. The intuition is that these final scalars, together with the challenges {uj​} and \"cross-terms\" {Lj​,Rj​} from each round, encode the compression in each round. Since the prover did not know the challenges U,{uj​} in advance, they would have been unable to manipulate the round compressions. Thus, checking a constraint on these final terms should enforce that the compression had been performed correctly, and that the original a satisfied the relation before undergoing compression. Note that G,b are simply rearrangements of the publicly known G,b, with the round challenges {uj​} mixed in: this means the verifier can compute G,b independently and verify that the prover had provided those same values.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Open (prover) and OpenVerify (verifier)","id":"196","title":"Open (prover) and OpenVerify (verifier)"},"197":{"body":"Alternative terms: Induction; Accumulation scheme; Proof-carrying data However, the computation of G requires a length-2k multiexponentiation ⟨G,s⟩, where s is composed of the round challenges u1​,⋯,uk​ arranged in a binary counting structure. This is the linear-time computation that we want to amortise across a batch of proof instances. Instead of computing G, notice that we can express G as a commitment to a polynomial G=Commit(σ,g(X,u1​,⋯,uk​)), where g(X,u1​,⋯,uk​):=∏i=1k​(ui​+ui−1​X2i−1) is a polynomial with degree 2k−1. Since G is a commitment, it can be checked in an inner product argument. The verifier circuit witnesses G and brings G,u1​,⋯,uk​ out as public inputs to the proof π. The next verifier instance checks π using the inner product argument; this includes checking that G=Commit(g(X,u1​,⋯,uk​)) evaluates at some random point to the expected value for the given challenges u1​,⋯,uk​. Recall from the previous section that this check only requires logd work. At the end of checking π and G, the circuit is left with a new G′, along with the u1′​,⋯,uk′​ challenges sampled for the check. To fully accept π as valid, we should perform a linear-time computation of G′=⟨G,s′⟩. Once again, we delay this computation by witnessing G′ and bringing G,u1​,⋯,uk​ out as public inputs to the proof π. This goes on from one proof instance to the next, until we are satisfied with the size of our batch of proofs. We finally perform a single linear-time computation, thus deciding the validity of the whole batch. We recall from the section Cycles of curves that we can instantiate this protocol over a two-cycle, where a proof produced by one curve is efficiently verified in the circuit of the other curve. However, some of these verifier checks can actually be efficiently performed in the native circuit; these are \"deferred\" to the next native circuit (see diagram below) instead of being immediately passed over to the other curve.","breadcrumbs":"Background Material » Recursion » Recursion","id":"197","title":"Recursion"},"2":{"body":"Requires Rust 1.56.1 or higher. Minimum supported Rust version can be changed in the future, but it will be done with a minor version bump.","breadcrumbs":"halo2 » Minimum Supported Rust Version","id":"2","title":"Minimum Supported Rust Version"},"20":{"body":"For our circuit, we will build a chip that provides the above numeric instructions for a finite field. /// The chip that will implement our instructions! Chips store their own\n/// config, as well as type markers if necessary.\nstruct FieldChip { config: FieldConfig, _marker: PhantomData,\n} Every chip needs to implement the Chip trait. This defines the properties of the chip that a Layouter may rely on when synthesizing a circuit, as well as enabling any initial state that the chip requires to be loaded into the circuit. impl Chip for FieldChip { type Config = FieldConfig; type Loaded = (); fn config(&self) -> &Self::Config { &self.config } fn loaded(&self) -> &Self::Loaded { &() }\n}","breadcrumbs":"User Documentation » A simple example » Define a chip implementation","id":"20","title":"Define a chip implementation"},"21":{"body":"The chip needs to be configured with the columns, permutations, and gates that will be required to implement all of the desired instructions. /// Chip state is stored in a config struct. This is generated by the chip\n/// during configuration, and then stored inside the chip.\n#[derive(Clone, Debug)]\nstruct FieldConfig { /// For this chip, we will use two advice columns to implement our instructions. /// These are also the columns through which we communicate with other parts of /// the circuit. advice: [Column; 2], /// This is the public input (instance) column. instance: Column, // We need a selector to enable the multiplication gate, so that we aren't placing // any constraints on cells where `NumericInstructions::mul` is not being used. // This is important when building larger circuits, where columns are used by // multiple sets of instructions. s_mul: Selector,\n} impl FieldChip { fn construct(config: >::Config) -> Self { Self { config, _marker: PhantomData, } } fn configure( meta: &mut ConstraintSystem, advice: [Column; 2], instance: Column, constant: Column, ) -> >::Config { meta.enable_equality(instance); meta.enable_constant(constant); for column in &advice { meta.enable_equality(*column); } let s_mul = meta.selector(); // Define our multiplication gate! meta.create_gate(\"mul\", |meta| { // To implement multiplication, we need three advice cells and a selector // cell. We arrange them like so: // // | a0 | a1 | s_mul | // |-----|-----|-------| // | lhs | rhs | s_mul | // | out | | | // // Gates may refer to any relative offsets we want, but each distinct // offset adds a cost to the proof. The most common offsets are 0 (the // current row), 1 (the next row), and -1 (the previous row), for which // `Rotation` has specific constructors. let lhs = meta.query_advice(advice[0], Rotation::cur()); let rhs = meta.query_advice(advice[1], Rotation::cur()); let out = meta.query_advice(advice[0], Rotation::next()); let s_mul = meta.query_selector(s_mul); // Finally, we return the polynomial expressions that constrain this gate. // For our multiplication gate, we only need a single polynomial constraint. // // The polynomial expressions returned from `create_gate` will be // constrained by the proving system to equal zero. Our expression // has the following properties: // - When s_mul = 0, any value is allowed in lhs, rhs, and out. // - When s_mul != 0, this constrains lhs * rhs = out. vec![s_mul * (lhs * rhs - out)] }); FieldConfig { advice, instance, s_mul, } }\n}","breadcrumbs":"User Documentation » A simple example » Configure the chip","id":"21","title":"Configure the chip"},"22":{"body":"/// A variable representing a number.\n#[derive(Clone)]\nstruct Number(AssignedCell); impl NumericInstructions for FieldChip { type Num = Number; fn load_private( &self, mut layouter: impl Layouter, value: Option, ) -> Result { let config = self.config(); layouter.assign_region( || \"load private\", |mut region| { region .assign_advice( || \"private input\", config.advice[0], 0, || value.ok_or(Error::Synthesis), ) .map(Number) }, ) } fn load_constant( &self, mut layouter: impl Layouter, constant: F, ) -> Result { let config = self.config(); layouter.assign_region( || \"load constant\", |mut region| { region .assign_advice_from_constant(|| \"constant value\", config.advice[0], 0, constant) .map(Number) }, ) } fn mul( &self, mut layouter: impl Layouter, a: Self::Num, b: Self::Num, ) -> Result { let config = self.config(); layouter.assign_region( || \"mul\", |mut region: Region<'_, F>| { // We only want to use a single multiplication gate in this region, // so we enable it at region offset 0; this means it will constrain // cells at offsets 0 and 1. config.s_mul.enable(&mut region, 0)?; // The inputs we've been given could be located anywhere in the circuit, // but we can only rely on relative offsets inside this region. So we // assign new cells inside the region and constrain them to have the // same values as the inputs. a.0.copy_advice(|| \"lhs\", &mut region, config.advice[0], 0)?; b.0.copy_advice(|| \"rhs\", &mut region, config.advice[1], 0)?; // Now we can assign the multiplication result, which is to be assigned // into the output position. let value = a.0.value().and_then(|a| b.0.value().map(|b| *a * *b)); // Finally, we do the assignment to the output, returning a // variable to be used in another part of the circuit. region .assign_advice( || \"lhs * rhs\", config.advice[0], 1, || value.ok_or(Error::Synthesis), ) .map(Number) }, ) } fn expose_public( &self, mut layouter: impl Layouter, num: Self::Num, row: usize, ) -> Result<(), Error> { let config = self.config(); layouter.constrain_instance(num.0.cell(), config.instance, row) }\n}","breadcrumbs":"User Documentation » A simple example » Implement chip traits","id":"22","title":"Implement chip traits"},"23":{"body":"Now that we have the instructions we need, and a chip that implements them, we can finally build our circuit! /// The full circuit implementation.\n///\n/// In this struct we store the private input variables. We use `Option` because\n/// they won't have any value during key generation. During proving, if any of these\n/// were `None` we would get an error.\n#[derive(Default)]\nstruct MyCircuit { constant: F, a: Option, b: Option,\n} impl Circuit for MyCircuit { // Since we are using a single chip for everything, we can just reuse its config. type Config = FieldConfig; type FloorPlanner = SimpleFloorPlanner; fn without_witnesses(&self) -> Self { Self::default() } fn configure(meta: &mut ConstraintSystem) -> Self::Config { // We create the two advice columns that FieldChip uses for I/O. let advice = [meta.advice_column(), meta.advice_column()]; // We also need an instance column to store public inputs. let instance = meta.instance_column(); // Create a fixed column to load constants. let constant = meta.fixed_column(); FieldChip::configure(meta, advice, instance, constant) } fn synthesize( &self, config: Self::Config, mut layouter: impl Layouter, ) -> Result<(), Error> { let field_chip = FieldChip::::construct(config); // Load our private values into the circuit. let a = field_chip.load_private(layouter.namespace(|| \"load a\"), self.a)?; let b = field_chip.load_private(layouter.namespace(|| \"load b\"), self.b)?; // Load the constant factor into the circuit. let constant = field_chip.load_constant(layouter.namespace(|| \"load constant\"), self.constant)?; // We only have access to plain multiplication. // We could implement our circuit as: // asq = a*a // bsq = b*b // absq = asq*bsq // c = constant*asq*bsq // // but it's more efficient to implement it as: // ab = a*b // absq = ab^2 // c = constant*absq let ab = field_chip.mul(layouter.namespace(|| \"a * b\"), a, b)?; let absq = field_chip.mul(layouter.namespace(|| \"ab * ab\"), ab.clone(), ab)?; let c = field_chip.mul(layouter.namespace(|| \"constant * absq\"), constant, absq)?; // Expose the result as a public input to the circuit. field_chip.expose_public(layouter.namespace(|| \"expose c\"), c, 0) }\n}","breadcrumbs":"User Documentation » A simple example » Build the circuit","id":"23","title":"Build the circuit"},"24":{"body":"halo2_proofs::dev::MockProver can be used to test that the circuit is working correctly. The private and public inputs to the circuit are constructed as we will do to create a proof, but by passing them to MockProver::run we get an object that can test every constraint in the circuit, and tell us exactly what is failing (if anything). // The number of rows in our circuit cannot exceed 2^k. Since our example // circuit is very small, we can pick a very small value here. let k = 4; // Prepare the private and public inputs to the circuit! let constant = Fp::from(7); let a = Fp::from(2); let b = Fp::from(3); let c = constant * a.square() * b.square(); // Instantiate the circuit with the private inputs. let circuit = MyCircuit { constant, a: Some(a), b: Some(b), }; // Arrange the public input. We expose the multiplication result in row 0 // of the instance column, so we position it there in our public inputs. let mut public_inputs = vec![c]; // Given the correct public input, our circuit will verify. let prover = MockProver::run(k, &circuit, vec![public_inputs.clone()]).unwrap(); assert_eq!(prover.verify(), Ok(())); // If we try some other public input, the proof will fail! public_inputs[0] += Fp::one(); let prover = MockProver::run(k, &circuit, vec![public_inputs]).unwrap(); assert!(prover.verify().is_err());","breadcrumbs":"User Documentation » A simple example » Testing the circuit","id":"24","title":"Testing the circuit"},"25":{"body":"You can find the source code for this example here .","breadcrumbs":"User Documentation » A simple example » Full example","id":"25","title":"Full example"},"26":{"body":"In normal programs, you can trade memory for CPU to improve performance, by pre-computing and storing lookup tables for some part of the computation. We can do the same thing in halo2 circuits! A lookup table can be thought of as enforcing a relation between variables, where the relation is expressed as a table. Assuming we have only one lookup argument in our constraint system, the total size of tables is constrained by the size of the circuit: each table entry costs one row, and it also costs one row to do each lookup. TODO","breadcrumbs":"User Documentation » Lookup tables » Lookup tables","id":"26","title":"Lookup tables"},"27":{"body":"","breadcrumbs":"User Documentation » Gadgets » Gadgets","id":"27","title":"Gadgets"},"28":{"body":"This section contains various ideas and snippets that you might find useful while writing halo2 circuits.","breadcrumbs":"User Documentation » Tips and tricks » Tips and tricks","id":"28","title":"Tips and tricks"},"29":{"body":"A common constraint used in R1CS circuits is the boolean constraint: b∗(1−b)=0. This constraint can only be satisfied by b=0 or b=1. In halo2 circuits, you can similarly constrain a cell to have one of a small set of values. For example, to constrain a to the range [0..5], you would create a gate of the form: a⋅(1−a)⋅(2−a)⋅(3−a)⋅(4−a)=0 while to constraint c to be either 7 or 13, you would use: (7−c)⋅(13−c)=0 The underlying principle here is that we create a polynomial constraint with roots at each value in the set of possible values we want to allow. In R1CS circuits, the maximum supported polynomial degree is 2 (due to all constraints being of the form a∗b=c). In halo2 circuits, you can use arbitrary-degree polynomials - with the proviso that higher-degree constraints are more expensive to use. Note that the roots don't have to be constants; for example (a−x)⋅(a−y)⋅(a−z)=0 will constrain a to be equal to one of {x,y,z} where the latter can be arbitrary polynomials, as long as the whole expression stays within the maximum degree bound.","breadcrumbs":"User Documentation » Tips and tricks » Small range constraints","id":"29","title":"Small range constraints"},"3":{"body":"halo2 currently uses rayon for parallel computation. The RAYON_NUM_THREADS environment variable can be used to set the number of threads.","breadcrumbs":"halo2 » Controlling parallelism","id":"3","title":"Controlling parallelism"},"30":{"body":"We can use Lagrange interpolation to create a polynomial constraint that maps f(X)=Y for small sets of X∈{xi​},Y∈{yi​}. For instance, say we want to map a 2-bit value to a \"spread\" version interleaved with zeros. We first precompute the evaluations at each point: 00→000001→000110→010011→0101​⟹⟹⟹⟹​0→01→12→43→5​ Then, we construct the Lagrange basis polynomial for each point using the identity: lj​(X)=0≤m B\n^ +\n| |\n+ v\nD <---+ C E +---> F ^ + | | + v H <---+ G After adding constraint B≡E the above algorithm produces the cycle: A +---> B +-------------+\n^ |\n| |\n+ v\nD <---+ C <---+ E F ^ + | | + v H <---+ G","breadcrumbs":"Design » Proving system » Permutation argument » Algorithm","id":"46","title":"Algorithm"},"47":{"body":"If we did not check whether left and right were already in the same cycle, then we could end up undoing an equality constraint. For example, if we have the following constraints: a≡b b≡c c≡d b≡d and we tried to implement adding an equality constraint just using step 5 of the above algorithm, then we would end up constructing the cycle (a b) (c d), rather than the correct (a b c d).","breadcrumbs":"Design » Proving system » Permutation argument » Broken alternatives","id":"47","title":"Broken alternatives"},"48":{"body":"We need to check a permutation of cells in m columns, represented in Lagrange basis by polynomials v0​,…,vm−1​. We will label each cell in those m columns with a unique element of F×. Suppose that we have a permutation on these labels, σ(column:i,row:j)=(column:i′,row:j′). in which the cycles correspond to equality-constraint sets. If we consider the set of pairs {(label,value)}, then the values within each cycle are equal if and only if permuting the label in each pair by σ yields the same set: An example for a cycle (A B C D). The set before permuting the labels is {(A, 7), (B, 7), (C, 7), (D, 7)}, and the set after is {(D, 7), (A, 7), (B, 7), (C, 7)} which is the same. If one of the 7s is replaced by 3, then the set after permuting labels is not the same. Since the labels are distinct, set equality is the same as multiset equality, which we can check using a product argument. Let ω be a 2k root of unity and let δ be a T root of unity, where T⋅2S+1=p with T odd and k≤S. We will use δi⋅ωj∈F× as the label for the cell in the jth row of the ith column of the permutation argument. We represent σ by a vector of m polynomials si​(X) such that si​(ωj)=δi′⋅ωj′. Notice that the identity permutation can be represented by the vector of m polynomials IDi​(ωj) such that IDi​(ωj)=δi⋅ωj. We will use a challenge β to compress each (label,value) pair to value+β⋅label. Just as in the product argument we used for lookups , we also use a challenge γ to randomize each term of the product. Now given our permutation represented by s0​,…,sm−1​ over columns represented by v0​,…,vm−1​, we want to ensure that: i=0∏m−1​j=0∏n−1​(vi​(ωj)+β⋅si​(ωj)+γvi​(ωj)+β⋅δi⋅ωj+γ​)=1 Here vi​(ωj)+β⋅δi⋅ωj represents the unpermuted (label,value) pair, and vi​(ωj)+β⋅si​(ωj) represents the permuted (σ(label),value) pair. Let ZP​ be such that ZP​(ω0)=ZP​(ωn)=1 and for 0≤j=r(λ) verifier challenges and let the ith challenge be sampled from Chi​. The advantage metric AdvIPSRS​(P,λ)=Pr[SRSPIP​(λ)] of a state restoration prover P is defined with respect to the following game. ​GameSRSIPP​(λ):​win←false;tr←ϵpp←IP.Setup(1λ)(x,stP​)←Pλ​(pp)RunPλOSRS​​(stP​)Return win​​​OracleOSRS​(τ=(a1​,c1​,...,ai−1​,ci−1​),ai​):​Ifτ∈trthenIfi≤rthenci​←Chi​;tr←tr∣∣(τ,ai​,ci​);Returnci​Else ifi=r+1thend←IP.V(pp,x,(τ,ai​));tr←(τ,ai​)Ifd=1then win←trueReturndReturn⊥​​ As shown in [GT20] (Theorem 1) state restoration soundness is tightly related to soundness after applying the Fiat-Shamir transformation. Knowledge Soundness We will show that our protocol satisfies a strengthened notion of knowledge soundness known as witness extended emulation . Informally, this notion states that for any successful prover algorithm there exists an efficient emulator that can extract a witness from it by rewinding it and supplying it with fresh randomness. However, we must slightly adjust our definition of witness extended emulation to account for the fact that our provers are state restoration provers and can rewind the verifier. Further, to avoid the need for rewinding the state restoration prover during witness extraction we study our protocol in the algebraic group model. Algebraic Group Model (AGM). An adversary Palg​ is said to be algebraic if whenever it outputs a group element X it also outputs a representation x∈Fn such that ⟨x,G⟩=X where G∈Gn is the vector of group elements that Palg​ has seen so far. Notationally, we write {X} to describe a group element X enhanced with this representation. We also write {X}iG​ to identify the component of the representation of X that corresponds with Gi​. In other words, X=i=0∑n−1​[{X}iG​]Gi​ The algebraic group model allows us to perform so-called \"online\" extraction for some protocols: the extractor can obtain the witness from the representations themselves for a single (accepting) transcript. State Restoration Witness Extended Emulation Let IP be an interactive argument for relation R with r=r(λ) challenges. We define for all non-uniform algebraic provers Palg​, extractors E, and computationally unbounded distinguishers D the advantage metric AdvIP,Rsr-wee​(Palg​,D,E,λ)=Pr[WEE-realIP,RP,D​(λ)]−Pr[WEE-idealIP,RE,P,D​(λ)] is defined with the respect to the following games. ​GameWEE-realIP,RPalg​,D​(λ):​tr←ϵpp←IP.Setup(1λ)(x,stP​)←Palg​(pp)RunPalg​Oreal​(stP​)b←D(tr)Returnb=1GameWEE-idealIP,RE,Palg​,D​(λ):​tr←ϵpp←IP.Setup(1λ)(x,stP​)←Palg​(pp)stE​←(1λ,pp,x)RunPalg​Oideal​(stP​)w←E(stE​,⊥)b←D(tr)Return(b=1)∧(Acc(tr)⟹(x,w)∈R)​​​OracleOreal​(τ=(a1​,c1​,...,ai−1​,ci−1​),ai​):​Ifτ∈trthenIfi≤rthenci​←Chi​;tr←tr∣∣(τ,ai​,ci​);Returnci​Else ifi=r+1thend←IP.V(pp,x,(τ,ai​));tr←(τ,ai​)Ifd=1then win←trueReturndReturn⊥OracleOideal​(τ,a):​Ifτ∈trthen(r,stE​)←E(stE​,[(τ,a)])tr←tr∣∣(τ,a,r)ReturnrReturn⊥​​ Zero Knowledge We say that an argument of knowledge is zero knowledge if the verifier also does not learn anything from their interaction besides that which can be learned from the existence of a valid w. More formally, Perfect Special Honest-Verifier Zero Knowledge. A public coin interactive argument (Setup,P,V) has perfect special honest-verifier zero knowledge (PSHVZK) if for all polynomial-time decidable relations R and for all (x,w)∈R and for all non-uniform polynomial-time adversaries A1​,A2​ there exists a probabilistic polynomial-time simulator S such that =​Pr⎣⎡​A1​(σ,x,tr)=1∣∣​​pp←Setup(1λ);(x,w,ρ)←A2​(pp);tr←⟨P(pp,x,w),V(pp,x,ρ)⟩​⎦⎤​Pr⎣⎡​A1​(σ,x,tr)=1∣∣​​pp←Setup(1λ);(x,w,ρ)←A2​(pp);tr←S(pp,x,ρ)​⎦⎤​​ where ρ is the internal randomness of the verifier. In this (common) definition of zero-knowledge the verifier is expected to act \"honestly\" and send challenges that correspond only with their internal randomness; they cannot adaptively respond to the prover based on the prover's messages. We use a strengthened form of this definition that forces the simulator to output a transcript with the same (adversarially provided) challenges that the verifier algorithm sends to the prover.","breadcrumbs":"Design » Protocol Description » Zero knowledge Arguments of Knowledge","id":"68","title":"Zero knowledge Arguments of Knowledge"},"69":{"body":"Let ω∈F be a n=2k primitive root of unity forming the domain D=(ω0,ω1,...,ωn−1) with t(X)=Xn−1 the vanishing polynomial over this domain. Let ng​,na​,ne​ be positive integers with na​,ne​p. That is, we must check that 0≤αq. Thus, given a scalar α, we witness the boolean decomposition of k=α+tq​. 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Typically, though, we will use them via gadgets . This indirection is useful because, for reasons of efficiency and limitations imposed by PLONKish circuits, the chip interfaces will often be dependent on low-level implementation details. The gadget interface can provide a more convenient and stable API that abstracts away from extraneous detail. For example, consider a hash function such as SHA-256. The interface of a chip supporting SHA-256 might be dependent on internals of the hash function design such as the separation between message schedule and compression function. The corresponding gadget interface can provide a more convenient and familiar update/finalize API, and can also handle parts of the hash function that do not need chip support, such as padding. This is similar to how accelerated instructions for cryptographic primitives on CPUs are typically accessed via software libraries, rather than directly. Gadgets can also provide modular and reusable abstractions for circuit programming at a higher level, similar to their use in libraries such as libsnark and bellman . As well as abstracting functions , they can also abstract types , such as elliptic curve points or integers of specific sizes.","breadcrumbs":"Concepts » Gadgets » Gadgets","id":"10","title":"Gadgets"},"100":{"body":"We use an optimized double-and-add algorithm, copied from \"Faster variable-base scalar multiplication in zk-SNARK circuits\" with some variable name changes: Acc := [2] T\nfor i from n-1 down to 0 { P := k_{i+1} ? T : −T Acc := (Acc + P) + Acc\n}\nreturn (k_0 = 0) ? (Acc - T) : Acc It remains to check that the x-coordinates of each pair of points to be added are distinct. When adding points in a prime-order group, we can rely on Theorem 3 from Appendix C of the Halo paper , which says that if we have two such points with nonzero indices wrt a given odd-prime order base, where the indices taken in the range −(q−1)/2..(q−1)/2 are distinct disregarding sign, then they have different x-coordinates. This is helpful, because it is easier to reason about the indices of points occurring in the scalar multiplication algorithm than it is to reason about their x-coordinates directly. So, the required check is equivalent to saying that the following \"indexed version\" of the above algorithm never asserts: acc := 2\nfor i from n-1 down to 0 { p = k_{i+1} ? 1 : −1 assert acc ≠ ± p assert (acc + p) ≠ acc // X acc := (acc + p) + acc assert 0 < acc ≤ (q-1)/2\n}\nif k_0 = 0 { assert acc ≠ 1 acc := acc - 1\n} The maximum value of acc is: <--- n 1s ---> 1011111...111111\n= 1100000...000000 - 1 = 2n+1+2n−1 The assertion labelled X obviously cannot fail because p=0. It is possible to see that acc is monotonically increasing except in the last conditional. It reaches its largest value when k is maximal, i.e. 2n+1+2n−1. So to entirely avoid exceptional cases, we would need 2n+1+2n−1<(q−1)/2. But we can use n larger by c if the last c iterations use complete addition . The first i for which the algorithm using only incomplete addition fails is going to be 252, since 2252+1+2252−1>(q−1)/2. We need n=254 to make the wraparound technique above work. sage: q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001\nsage: 2^253 + 2^252 - 1 < (q-1)//2\nFalse\nsage: 2^252 + 2^251 - 1 < (q-1)//2\nTrue So the last three iterations of the loop (i=2..0) need to use complete addition , as does the conditional subtraction at the end. Writing this out using ⸭ for incomplete addition (as we do in the spec), we have: Acc := [2] T\nfor i from 253 down to 3 { P := k_{i+1} ? T : −T Acc := (Acc ⸭ P) ⸭ Acc\n}\nfor i from 2 down to 0 { P := k_{i+1} ? T : −T Acc := (Acc + P) + Acc // complete addition\n}\nreturn (k_0 = 0) ? (Acc + (-T)) : Acc // complete addition","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Variable-base scalar multiplication","id":"100","title":"Variable-base scalar multiplication"},"101":{"body":"Define a running sum zj​=∑i=jn​(ki​⋅2i−j), where n=254 and: ​zn+1​=0,zn​=kn​,(most significant bit)z0​=k.​ Initialize A254​=[2]T.for i from 254 down to 4:bool_check(ki​)=0zi​=2zi+1​+ki​xP,i​=xT​yP,i​=(2ki​−1)⋅yT​(conditionally negate)λ1,i​⋅(xA,i​−xP,i​)=yA,i​−yP,i​λ1,i2​=xR,i​+xA,i​+xP,i​(λ1,i​+λ2,i​)⋅(xA,i​−xR,i​)=2yA,i​λ2,i2​=xA,i−1​+xR,i​+xA,i​λ2,i​⋅(xA,i​−xA,i−1​)=yA,i​+yA,i−1​,​ where xR,i​=(λ1,i2​−xA,i​−xT​). The helper bool_check(x)=x⋅(1−x). After substitution of xP,i​,yP,i​,xR,i​,yA,i​, and yA,i−1​, this becomes: Initialize A254​=[2]T.for i from 254 down to 4:// let ki​=zi​−2zi+1​// let yA,i​=2(λ1,i​+λ2,i​)⋅(xA,i​−(λ1,i2​−xA,i​−xT​))​bool_check(ki​)=0λ1,i​⋅(xA,i​−xT​)=yA,i​−(2ki​−1)⋅yT​λ2,i2​=xA,i−1​+λ1,i2​−xT​{λ2,i​⋅(xA,i​−xA,i−1​)=yA,i​+yA,i−1​,λ2,4​⋅(xA,4​−xA,3​)=yA,4​+yA,3witnessed​,​if i>4if i=4.​​ Here, yA,3witnessed​ is assigned to a cell. This is unlike previous yA,i​'s, which were implicitly derived from λ1,i​,λ2,i​,xA,i​,xT​, but never actually assigned. The bits k3…1​ are used in three further steps, using complete addition : for i from 3 down to 1:// let ki​=zi​−2zi+1​bool_check(ki​)=0(xA,i−1​,yA,i−1​)=((xA,i​,yA,i​)+(xT​,yT​))+(xA,i​,yA,i​)​ If the least significant bit k0​=1, we set B=O, otherwise we set B=−T. Then we return A+B using complete addition. Let B={(0,0),(xT​,−yT​),​ if k0​=1, otherwise.​ Output (xA,0​,yA,0​)+B. (Note that (0,0) represents O.)","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Constraint program for optimized double-and-add (incomplete addition)","id":"101","title":"Constraint program for optimized double-and-add (incomplete addition)"},"102":{"body":"We need six advice columns to witness (xT​,yT​,λ1​,λ2​,xA,i​,zi​). However, since (xT​,yT​) are the same, we can perform two incomplete additions in a single row, reusing the same (xT​,yT​). We split the scalar bits used in incomplete addition into hi and lo halves and process them in parallel. This means that we effectively have two for loops: the first, covering the hi half for i from 254 down to 130, with a special case at i=130; and the second, covering the lo half for the remaining i from 129 down to 4, with a special case at i=4. xT​xT​xT​⋮xT​​yT​yT​yT​⋮yT​​zhiz255​=0z254​z253​⋮z130​​xAhi​xA,254​=2[T]x​xA,253​⋮xA,130​xA,129​​λ1hi​yA,254​=2[T]y​λ1,254​λ1,253​⋮λ1,130​yA,129​​λ2hi​λ2,254​λ2,253​⋮λ2,130​​q1hi​100⋮0​q2hi​011⋮0​q3hi​000⋮1​zloz130​z129​z128​⋮z5​z4​​xAlo​xA,129​xA,128​⋮xA,5​xA,4​xA,3​​λ1lo​yA,129​λ1,129​λ1,128​⋮λ1,5​λ1,4​yA,3​​λ2lo​λ2,129​λ2,128​⋮λ2,5​λ2,4​​q1lo​100⋮00​q2lo​011⋮10​q3lo​000⋮01​​ For each hi and lo half, we have three sets of gates. Note that i is going from 255..=3; i is NOT indexing the rows. q1​=1 This gate is only used on the first row (before the for loop). We check that λ1​,λ2​ are initialized to values consistent with the initial yA​. Degree3​Constraintq1​⋅(yA,nwitnessed​−yA,n​)=0​​ where yA,n​yA,nwitnessed​​=2(λ1,n​+λ2,n​)⋅(xA,n​−(λ1,n2​−xA,n​−xT​))​, is witnessed.​ q2​=1 This gate is used on all rows corresponding to the for loop except the last. Degree223433​Constraintq2​⋅(xT,cur​−xT,next​)=0q2​⋅(yT,cur​−yT,next​)=0q2​⋅bool_check(ki​)=0, where ki​=zi​−2zi+1​q2​⋅(λ1,i​⋅(xA,i​−xT,i​)−yA,i​+(2ki​−1)⋅yT,i​)=0q2​⋅(λ2,i2​−xA,i−1​−λ1,i2​+xT,i​)=0q2​⋅(λ2,i​⋅(xA,i​−xA,i−1​)−yA,i​−yA,i−1​)=0​​ where yA,i​yA,i−1​​=2(λ1,i​+λ2,i​)⋅(xA,i​−(λ1,i2​−xA,i​−xT​))​,=2(λ1,i−1​+λ2,i−1​)⋅(xA,i−1​−(λ1,i−12​−xA,i−1​−xT​))​,​ q3​=1 This gate is used on the final iteration of the for loop, handling the special case where we check that the output yA​ has been witnessed correctly. Degree3433​Constraintq3​⋅bool_check(ki​)=0, where ki​=zi​−2zi+1​q3​⋅(λ1,i​⋅(xA,i​−xT,i​)−yA,i​+(2ki​−1)⋅yT,i​)=0q3​⋅(λ2,i2​−xA,i−1​−λ1,i2​+xT,i​)=0q3​⋅(λ2,i​⋅(xA,i​−xA,i−1​)−yA,i​−yA,i−1witnessed​)=0​​ where yA,i​yA,i−1witnessed​​=2(λ1,i​+λ2,i​)⋅(xA,i​−(λ1,i2​−xA,i​−xT​))​, is witnessed.​","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Circuit design","id":"102","title":"Circuit design"},"103":{"body":"zi​ cannot overflow for any i≥1, because it is a weighted sum of bits only up to 2n−1=2253, which is smaller than p (and also q). However, z0​=α+tq​ can overflow [0,p). Since overflow can only occur in the final step that constrains z0​=2⋅z1​+k0​, we have z0​=k(modp). It is then sufficient to also check that z0​=α+tq​(modp) (so that k=α+tq​(modp)) and that k∈[tq​,p+tq​). These conditions together imply that k=α+tq​ as an integer, and so 2254+k=α(modq) as required. Note: the bits k254..0​ do not represent a value reduced modulo q, but rather a representation of the unreduced α+tq​.","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Overflow check","id":"103","title":"Overflow check"},"104":{"body":"Since tp​+tq​<2130, we have [tq​,p+tq​)=[tq​,tq​+2130)∪[2130,2254)∪([2254,2254+2130)∩[p+tq​−2130,p+tq​)). We may assume that k=α+tq​(modp). Therefore, k∈[tq​,p+tq​)​⇔⇔⇔​(k∈[tq​,tq​+2130)∨k∈[2130,2254))∨(k∈[2254,2254+2130)∧k∈[p+tq​−2130,p+tq​))(k254​=0⟹(k∈[tq​,tq​+2130)∨k∈[2130,2254)))∧(k254​=1⟹(k∈[2254,2254+2130)∧k∈[p+tq​−2130,p+tq​))(k254​=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254​=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))Ⓐ​ Given k∈[2254,2254+2130), we prove equivalence of k∈[p+tq​−2130,p+tq​) and (α+2130)modp∈[0,2130) as follows: shift the range by 2130−p−tq​ to give k+2130−p−tq​∈[0,2130); observe that k+2130−p−tq​ is guaranteed to be in [2130−tp​−tq​,2131−tp​−tq​) and therefore cannot overflow or underflow modulo p; using the fact that k=α+tq​(modp), observe that (k+2130−p−tq​)modp=(α+tq​+2130−p−tq​)modp=(α+2130)modp. (We can see in a different way that this is correct by observing that it checks whether αmodp∈[p−2130,p), so the upper bound is aligned as we would expect.) Now, we can continue optimizing from Ⓐ: k∈[tq​,p+tq​)​⇔⇔​(k254​=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254​=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))(k254​=0⟹(α∈[0,2130)∨k253..130​ are not all 0))∧(k254​=1⟹(k253..130​ are all 0∧(α+2130)modp∈[0,2130)))​ Constraining k253..130​ to be all-0 or not-all-0 can be implemented almost \"for free\", as follows. Recall that zi​=∑h=in​(kh​⋅2h−i), so we have: z130​z130​z130​−k254​⋅2124​===​∑h=130254​(kh​⋅2h−130)k254​⋅2254−130+∑h=130253​(kh​⋅2h−130)∑h=130253​(kh​⋅2h−130)​ So k253..130​ are all 0 exactly when z130​=k254​⋅2124. Finally, we can merge the 130-bit decompositions for the k254​=0 and k254​=1 cases by checking that (α+k254​⋅2130)modp∈[0,2130).","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Optimized check for k∈[tq​,p+tq​)","id":"104","title":"Optimized check for k∈[tq​,p+tq​)"},"105":{"body":"Let s=α+k254​⋅2130. The constraints for the overflow check are: z0​k254​=1⟹(z130​k254​=0⟹(z130​​=α+tq​(modp)=2124∧smodp∈[0,2130))=0∨smodp∈[0,2130))​ Define inv0(x)={0,1/x,​if x=0otherwise.​ Witness η=inv0(z130​), and decompose smodp as s129..0​. Then the needed gates are: Degree22335​Constraintq_muloverflow⋅(s−(α+k254​⋅2130))=0q_muloverflow⋅(z0​−α−tq​)=0q_muloverflow⋅(k254​⋅(z130​−2124))=0q_muloverflow⋅(k254​⋅(s−i=0∑129​2i⋅si​)/2130)=0q_muloverflow⋅((1−k254​)⋅(1−z130​⋅η)⋅(s−i=0∑129​2i⋅si​)/2130)=0​​ where (s−i=0∑129​2i⋅si​)/2130 can be computed by another running sum. Note that the factor of 1/2130 has no effect on the constraint, since the RHS is zero. Running sum range check We make use of a 10-bit lookup range check in the circuit to subtract the low 130 bits of s. The range check subtracts the first 13⋅10 bits of s, and right-shifts the result to give (s−i=0∑129​2i⋅si​)/2130.","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Overflow check constraints","id":"105","title":"Overflow check constraints"},"106":{"body":"","breadcrumbs":"Design » Gadgets » Sinsemilla » Sinsemilla","id":"106","title":"Sinsemilla"},"107":{"body":"Sinsemilla is a collision-resistant hash function and commitment scheme designed to be efficient in algebraic circuit models that support lookups , such as PLONK or Halo 2. The security properties of Sinsemilla are similar to Pedersen hashes; it is not designed to be used where a random oracle, PRF, or preimage-resistant hash is required. The only claimed security property of the hash function is collision-resistance for fixed-length inputs. Sinsemilla is roughly 4 times less efficient than the algebraic hashes Rescue and Poseidon inside a circuit, but around 19 times more efficient than Rescue outside a circuit. Unlike either of these hashes, the collision resistance property of Sinsemilla can be proven based on cryptographic assumptions that have been well-established for at least 20 years. Sinsemilla can also be used as a computationally binding and perfectly hiding commitment scheme. The general approach is to split the message into k-bit pieces, and for each piece, select from a table of 2k bases in our cryptographic group. We combine the selected bases using a double-and-add algorithm. This ends up being provably as secure as a vector Pedersen hash, and makes advantageous use of the lookup facility supported by Halo 2.","breadcrumbs":"Design » Gadgets » Sinsemilla » Overview","id":"107","title":"Overview"},"108":{"body":"This section is an outline of how Sinsemilla works: for the normative specification, refer to §5.4.1.9 Sinsemilla Hash Function in the protocol spec. The incomplete point addition operator, ⸭, that we use below is also defined there. Let G be a cryptographic group of prime order q. We write G additively, with identity O, and using [m]P for scalar multiplication of P by m. Let k≥1 be an integer chosen based on efficiency considerations (the table size will be 2k). Let n be an integer, fixed for each instantiation, such that messages are kn bits, where 2n≤2q−1​. We use zero-padding to the next multiple of k bits if necessary. Setup: Choose Q and P[0..2k−1] as 2k+1 independent, verifiably random generators of G, using a suitable hash into G, such that none of Q or P[0..2k−1] are O. In Orchard, we define Q to be dependent on a domain separator D. The protocol specification uses Q(D) in place of Q and S(m) in place of P[m]. Hash(M): Split M into n groups of k bits. Interpret each group as a k-bit little-endian integer mi​. let Acc0​:=Q for i from 0 up to n−1: let Acci+1​:=(Acci​⸭P[mi+1​])⸭Acci​ return Accn​ Let ShortHash(M) be the x-coordinate of Hash(M). (This assumes that G is a prime-order elliptic curve in short Weierstrass form, as is the case for Pallas and Vesta.) It is slightly more efficient to express a double-and-add [2]A+R as (A+R)+A. We also use incomplete additions: it is shown in the Sinsemilla security argument that in the case where G is a prime-order short Weierstrass elliptic curve, an exceptional case for addition would lead to finding a discrete logarithm, which can be assumed to occur with negligible probability even for adversarial input.","breadcrumbs":"Design » Gadgets » Sinsemilla » Description","id":"108","title":"Description"},"109":{"body":"Choose another generator H independently of Q and P[0..2k−1]. The randomness r for a commitment is chosen uniformly on [0,q). Let Commitr​(M)=Hash(M)⸭[r]H. Let ShortCommitr​(M) be the x-coordinate of Commitr​(M). (This again assumes that G is a prime-order elliptic curve in short Weierstrass form.) Note that unlike a simple Pedersen commitment, this commitment scheme (Commit or ShortCommit) is not additively homomorphic.","breadcrumbs":"Design » Gadgets » Sinsemilla » Use as a commitment scheme","id":"109","title":"Use as a commitment scheme"},"11":{"body":"You're probably here because you want to write circuits? Excellent! This section will guide you through the process of creating circuits with halo2.","breadcrumbs":"User Documentation » User Documentation","id":"11","title":"User Documentation"},"110":{"body":"The aim of the design is to optimize the number of bits that can be processed for each step of the algorithm (which requires a doubling and addition in G) for a given table size. Using a single table of size 2k group elements, we can process k bits at a time.","breadcrumbs":"Design » Gadgets » Sinsemilla » Efficient implementation","id":"110","title":"Efficient implementation"},"111":{"body":"Let P={(j,xP[j]​,yP[j]​) for j∈{0..2k−1}}. Input: m1..=n​. (The message words are 1-indexed here, as in the protocol spec , but we start the loop from i=0 so that (xA,i​,yA,i​) corresponds to Acci​ in the protocol spec.) Output: (xA,n​,yA,n​). (xA,0​,yA,0​)=Q for i from 0 up to n−1: yP,i​=yA,i​−λ1,i​⋅(xA,i​−xP,i​) xR,i​=λ1,i2​−xA,i​−xP,i​ 2⋅yA,i​=(λ1,i​+λ2,i​)⋅(xA,i​−xR,i​) (mi+1​,xP,i​,yP,i​)∈P λ2,i2​=xA,i+1​+xR,i​+xA,i​ λ2,i​⋅(xA,i​−xA,i+1​)=yA,i​+yA,i+1​","breadcrumbs":"Design » Gadgets » Sinsemilla » Constraint program","id":"111","title":"Constraint program"},"112":{"body":"","breadcrumbs":"Design » Gadgets » Sinsemilla » PLONK / Halo 2 constraints","id":"112","title":"PLONK / Halo 2 constraints"},"113":{"body":"We have an n-bit message m=m1​+2km2​+...+2k⋅(n−1)mn​. (Note that the message words are 1-indexed as in the protocol spec .) Initialise the running sum z0​=α and define zi+1​:=2Kzi​−mi+1​​. We will end up with zn​=0. Rearranging gives us an expression for each word of the original message mi+1​=zi​−2k⋅zi+1​, which we can look up in the table. In other words, zn−i​=h=0∑i−1​2kh⋅mh+1​. For a little-endian decomposition as used here, the running sum is initialized to the scalar and ends at 0. For a big-endian decomposition as used in variable-base scalar multiplication , the running sum would start at 0 and end with recovering the original scalar. The running sum only applies to message words within a single field element, i.e. if n≥PrimeField::NUM_BITS then we will have several disjoint running sums. A longer message can be constructed by splitting the message words across several field elements, and then running several instances of the constraints below. An additional qS2​ selector is set to 0 for the last step of each element, except for the last element where it is set to 2. In order to support chaining multiple field elements without a gap, we will use a slightly more complicated expression for mi+1​ that effectively forces zn​ to zero for the last step of each element, as indicated by qS2​. This allows the cell that would have been zn​ to be used to reinitialize the running sum for the next element.","breadcrumbs":"Design » Gadgets » Sinsemilla » Message decomposition","id":"113","title":"Message decomposition"},"114":{"body":"The Sinsemilla circuit makes use of 210 pre-computed random generators. These are loaded into a lookup table: tableidx​012⋮210−1​tablex​xP[0]​xP[1]​xP[2]​⋮xP[210−1]​​tabley​yP[0]​yP[1]​yP[2]​⋮yP[210−1]​​​","breadcrumbs":"Design » Gadgets » Sinsemilla » Generator lookup table","id":"114","title":"Generator lookup table"},"115":{"body":"Note: qS3​ is synthesized from qS1​ and qS2​; it is shown here only for clarity. Step012⋮n−10′1′2′⋮n−1′n′​xA​xQ​xA,1​xA,2​⋮xA,n−1​xA,0′​xA,1′​xA,2′​⋮xA,n−1′​xA,n′​​xP​xP[m1​]​xP[m2​]​xP[m3​]​⋮xP[mn​]​xP[m1′​]​xP[m2′​]​xP[m3′​]​⋮xP[mn′​]​​bitsz0​z1​z2​⋮zn−1​z0′​z1′​z2′​⋮zn−1′​​λ1​λ1,0​λ1,1​λ1,2​⋮λ1,n−1​λ1,0′​λ1,1′​λ1,2′​⋮λ1,n−1′​yA,n​​λ2​λ2,0​λ2,1​λ2,2​⋮λ2,n−1​λ2,0′​λ2,1′​λ2,2′​⋮λ2,n−1′​​qS1​11111111110​qS2​11110111120​qS3​00000000020​qS4​10000000000​fixed_y_QyQ​0000000000​​ xQ​, z0​, z0′​, etc. would be copied in using equality constraints.","breadcrumbs":"Design » Gadgets » Sinsemilla » Layout","id":"115","title":"Layout"},"116":{"body":"For i∈[0,n), let​xR,i​YA,i​yP,i​mi+1​qS3​​=====​λ1,i2​−xA,i​−xP,i​(λ1,i​+λ2,i​)⋅(xA,i​−xR,i​)YA,i​/2−λ1,i​⋅(xA,i​−xP,i​)zi​−2k⋅(qS2,i​−qS3,i​)⋅zi+1​qS2​⋅(qS2​−1)​ The Halo 2 circuit API can automatically substitute yP,i​, xR,i​, yA,i​, and yA,i+1​, so we don't need to do that manually. xA,0​=xQ​ 2⋅yQ​=YA,0​ for i from 0 up to n−1: (mi+1​,xP,i​,yP,i​)∈P λ2,i2​=xA,i+1​+xR,i​+xA,i​ 4⋅λ2,i​⋅(xA,i​−xA,i+1​)=2⋅YA,i​+(2−qS3​)⋅YA,i+1​+2qS3​⋅yA,n​ Note that each term of the last constraint is multiplied by 4 relative to the constraint program given earlier. This is a small optimization that avoids divisions by 2. By gating the lookup expression on qS1​, we avoid the need to fill in unused cells with dummy values to pass the lookup argument. The optimized lookup value (using a default index of 0) is: (​qS1​⋅mi+1​,qS1​⋅xP,i​+(1−qS1​)⋅xP,0​,qS1​⋅yP,i​+(1−qS1​)⋅yP,0​)​ This increases the degree of the lookup argument to 6. Degree4635​ConstraintqS4​⋅(2⋅yQ​−YA,0​)=0qS1,i​⇒(mi+1​,xP,i​,yP,i​)∈PqS1,i​⋅(λ2,i2​−(xA,i+1​+xR,i​+xA,i​))qS1,i​⋅(4⋅λ2,i​⋅(xA,i​−xA,i+1​)−(2⋅YA,i​+(2−qS3,i​)⋅YA,i+1​+2⋅qS3,i​⋅yA,n​))=0​​","breadcrumbs":"Design » Gadgets » Sinsemilla » Optimized Sinsemilla gate","id":"116","title":"Optimized Sinsemilla gate"},"117":{"body":"","breadcrumbs":"Design » Gadgets » Sinsemilla » MerkleCRH » MerkleCRH","id":"117","title":"MerkleCRH"},"118":{"body":"SinsemillaHash is used in the MerkleCRHOrchard hash function . The input to SinsemillaHash is: l⋆∣∣left⋆∣∣right⋆, where: l⋆=I2LEBSP10​(l)=I2LEBSP10​(MerkleDepthOrchard−1−layer), left⋆=I2LEBSPℓMerkleOrchard​​(left), right⋆=I2LEBSPℓMerkleOrchard​​(right), with ℓMerkleOrchard​=255. left and right are allowed to be non-canonical 255-bit encodings. We break these inputs into the following MessagePieces: a (250 bits)b (20 bits)c (250 bits)​=a0​∣∣a1​=l⋆∣∣(bits 0..=239 of left )=b0​∣∣b1​∣∣b2​=(bits 240..=249 of left)∣∣(bits 250..=254 of left)∣∣(bits 0..=4 of right)=bits 5..=254 of right​ a,b,c are constrained by the SinsemillaHash to be 250 bits, 20 bits, and 250 bits respectively. In a custom gate, we check this message decomposition by enforcing the following constraints: a0​=l z1,a​, the index-1 running sum output of SinsemillaHash(a), is copied into the gate. z1,a​ has been constrained by the SinsemillaHash to be 240 bits. We recover the subpieces a0​,a1​ using a,z1,a​: z1,a​⟹a0​​=210a−a0​​=a1​=a−z1,a​⋅210.​ l+1 is loaded into a fixed column at each layer of the hash. It is used both as a gate selector, and to fix the value of l. We check that a0​=(l+1)−1. Note: The reason for using l+1 instead of l is that l=0 when layer=31 (hashing two leaves). We cannot have a zero-valued selector, since a constraint gated by a zero-valued selector is never checked. b1​+25⋅b2​=z1,b​ z1,b​, the index-1 running sum output of SinsemillaHash(b), is copied into the gate. z1,b​ has been constrained by the SinsemillaHash to be 10 bits. We witness the subpieces b1​,b2​ outside this gate, and constrain them each to be 5 bits. Inside the gate, we check that b1​+25⋅b2​=z1,b​. We also recover the subpiece b0​ using (b,z1,b​): z1,b​⟹b0​​=210b−b0..=10​​=b−(z1,b​⋅210).​ We have now derived or witnessed every subpiece, and range-constrained every subpiece: a0​ (10 bits), derived as a0​=a−210⋅z1,a​; a1​ (240 bits), equal to z1,a​; b0​ (10 bits), derived as b0​=b−210⋅z1,b​; b1​ (5 bits) is witnessed and constrained outside the gate; b2​ (5 bits) is witnessed and constrained outside the gate; b1​+25⋅b2​ is constrained to equal z1,b​, and we use them to reconstruct the original field element inputs: left=a1​+2240⋅b0​+2254⋅b1​ right=b2​+25⋅c","breadcrumbs":"Design » Gadgets » Sinsemilla » MerkleCRH » Message decomposition","id":"118","title":"Message decomposition"},"119":{"body":"The Orchard circuit spans 10 advice columns while the Sinsemilla chip only uses 5 advice columns. We distribute the path hashing evenly across two Sinsemilla chips to make better use of the available circuit area. Since the output from the previous layer hash is copied into the next layer hash, we maintain continuity even when moving from one chip to the other.","breadcrumbs":"Design » Gadgets » Sinsemilla » MerkleCRH » Circuit components","id":"119","title":"Circuit components"},"12":{"body":"The halo2 crate includes several utilities to help you design and implement your circuits.","breadcrumbs":"User Documentation » Developer tools » Developer tools","id":"12","title":"Developer tools"},"120":{"body":"Given a field element α, these gadgets decompose it into W K-bit windows α=k0​+2K⋅k1​+22K⋅k2​+⋯+2(W−1)K⋅kW−1​ where each ki​ a K-bit value. This is done using a running sum zi​,i∈[0..W). We initialize the running sum z0​=α, and compute subsequent terms zi+1​=2Kzi​−ki​​. This gives us: z0​z1​z2​↓zW​​=α=k0​+2K⋅k1​+22K⋅k2​+23K⋅k3​+⋯,=(z0​−k0​)/2K=k1​+2K⋅k2​+22K⋅k3​+⋯,=(z1​−k1​)/2K=k2​+2K⋅k3​+⋯,⋮ (in strict mode)=(zW−1​−kW−1​)/2K=0 (because zW−1​=kW−1​)​","breadcrumbs":"Design » Gadgets » Decomposition » Decomposition","id":"120","title":"Decomposition"},"121":{"body":"Strict mode constrains the running sum output zW​ to be zero, thus range-constraining the field element to be within W⋅K bits. In strict mode, we are also assured that zW−1​=kW−1​ gives us the last window in the decomposition.","breadcrumbs":"Design » Gadgets » Decomposition » Strict mode","id":"121","title":"Strict mode"},"122":{"body":"This gadget makes use of a K-bit lookup table to decompose a field element α into K-bit words. Each K-bit word ki​=zi​−2K⋅zi+1​ is range-constrained by a lookup in the K-bit table. The region layout for the lookup decomposition uses a single advice column z, and two selectors qlookup​ and qrunning​. zz0​z1​⋮zn−1​zn​​qlookup​11⋮10​qrunning​11⋮10​​","breadcrumbs":"Design » Gadgets » Decomposition » Lookup decomposition","id":"122","title":"Lookup decomposition"},"123":{"body":"Using two K-bit lookups, we can range-constrain a field element α to be n bits, where n≤K. To do this: Constrain 0≤α<2K to be within K bits using a K-bit lookup. Constrain 0≤α⋅2K−n<2K to be within K bits using a K-bit lookup. The short variant of the lookup decomposition introduces a qbitshift​ selector. The same advice column z has here been renamed to word for clarity: wordαα′2K−n​qlookup​110​qrunning​000​qbitshift​010​​ where α′=α⋅2K−n. Note that 2K−n is assigned to a fixed column at keygen, and copied in at proving time. This is used in the gate enabled by the qbitshift​ selector to check that α was shifted correctly: Degree2​Constraintqbitshift​⋅(α′−(α⋅2K−n))​​","breadcrumbs":"Design » Gadgets » Decomposition » Short range check","id":"123","title":"Short range check"},"124":{"body":"Since the lookup decomposition and its short variant both make use of the same lookup table, we combine their lookup input expressions into a single one: qlookup​⋅(qrunning​⋅(zi​−2K⋅zi+1​)+(1−qrunning​)⋅word) where zi​ and word are the same cell (but distinguished here for clarity of usage).","breadcrumbs":"Design » Gadgets » Decomposition » Combined lookup expression","id":"124","title":"Combined lookup expression"},"125":{"body":"For a short range (for instance, [0,range) where range≤8), we can range-constrain each word using a degree-range polynomial constraint instead of a lookup: range_check(word,range)=word⋅(1−word)⋯(range−1−word).","breadcrumbs":"Design » Gadgets » Decomposition » Short range decomposition","id":"125","title":"Short range decomposition"},"126":{"body":"","breadcrumbs":"Design » Gadgets » SHA-256 » SHA-256","id":"126","title":"SHA-256"},"127":{"body":"SHA-256 is specified in NIST FIPS PUB 180-4 . Unlike the specification, we use ⊞ for addition modulo 232, and + for field addition. ⊕ is used for XOR.","breadcrumbs":"Design » Gadgets » SHA-256 » Specification","id":"127","title":"Specification"},"128":{"body":"SHA-256 maintains state in eight 32-bit variables. It processes input as 512-bit blocks, but internally splits these blocks into 32-bit chunks. We therefore designed the SHA-256 gadget to consume input in 32-bit chunks.","breadcrumbs":"Design » Gadgets » SHA-256 » Gadget interface","id":"128","title":"Gadget interface"},"129":{"body":"The SHA-256 gadget requires a chip with the following instructions: # extern crate halo2_proofs;\n# use halo2_proofs::plonk::Error;\n# use std::fmt;\n#\n# trait Chip: Sized {}\n# trait Layouter {}\nconst BLOCK_SIZE: usize = 16;\nconst DIGEST_SIZE: usize = 8; pub trait Sha256Instructions: Chip { /// Variable representing the SHA-256 internal state. type State: Clone + fmt::Debug; /// Variable representing a 32-bit word of the input block to the SHA-256 compression /// function. type BlockWord: Copy + fmt::Debug; /// Places the SHA-256 IV in the circuit, returning the initial state variable. fn initialization_vector(layouter: &mut impl Layouter) -> Result; /// Starting from the given initial state, processes a block of input and returns the /// final state. fn compress( layouter: &mut impl Layouter, initial_state: &Self::State, input: [Self::BlockWord; BLOCK_SIZE], ) -> Result; /// Converts the given state into a message digest. fn digest( layouter: &mut impl Layouter, state: &Self::State, ) -> Result<[Self::BlockWord; DIGEST_SIZE], Error>;\n} TODO: Add instruction for computing padding. This set of instructions was chosen to strike a balance between the reusability of the instructions, and the scope for chips to internally optimise them. In particular, we considered splitting the compression function into its constituent parts (Ch, Maj etc), and providing a compression function gadget that implemented the round logic. However, this would prevent chips from using relative references between the various parts of a compression round. Having an instruction that implements all compression rounds is also similar to the Intel SHA extensions, which provide an instruction that performs multiple compression rounds.","breadcrumbs":"Design » Gadgets » SHA-256 » Chip instructions","id":"129","title":"Chip instructions"},"13":{"body":"halo2_proofs::dev::MockProver is a tool for debugging circuits, as well as cheaply verifying their correctness in unit tests. The private and public inputs to the circuit are constructed as would normally be done to create a proof, but MockProver::run instead creates an object that will test every constraint in the circuit directly. It returns granular error messages that indicate which specific constraint (if any) is not satisfied.","breadcrumbs":"User Documentation » Developer tools » Mock prover","id":"13","title":"Mock prover"},"130":{"body":"This chip implementation is based around a single 16-bit lookup table. It requires a minimum of 216 circuit rows, and is therefore suitable for use in larger circuits. We target a maximum constraint degree of 9. That will allow us to handle constraining carries and \"small pieces\" to a range of up to {0..7} in one row.","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » 16-bit table chip for SHA-256","id":"130","title":"16-bit table chip for SHA-256"},"131":{"body":"There are 64 compression rounds. Each round takes 32-bit values A,B,C,D,E,F,G,H as input, and performs the following operations: Ch(E,F,G)Maj(A,B,C)Σ0​(A)Σ1​(E)H′Enew​Anew​​========​(E∧F)⊕(¬E∧G)(A∧B)⊕(A∧C)⊕(B∧C)count(A,B,C)≥2(A⋙2)⊕(A⋙13)⊕(A⋙22)(E⋙6)⊕(E⋙11)⊕(E⋙25)H+Ch(E,F,G)+Σ1​(E)+Kt​+Wt​reduce6​(H′+D)reduce7​(H′+Maj(A,B,C)+Σ0​(A))​ where reducei​ must handle a carry 0≤carry = MyCircuit { a: None, lookup_table, }; // Create the area you want to draw on. // Use SVGBackend if you want to render to .svg instead. use plotters::prelude::*; let root = BitMapBackend::new(\"layout.png\", (1024, 768)).into_drawing_area(); root.fill(&WHITE).unwrap(); let root = root .titled(\"Example Circuit Layout\", (\"sans-serif\", 60)) .unwrap(); halo2_proofs::dev::CircuitLayout::default() // You can optionally render only a section of the circuit. .view_width(0..2) .view_height(0..16) // You can hide labels, which can be useful with smaller areas. .show_labels(false) // Render the circuit onto your area! // The first argument is the size parameter for the circuit. .render(5, &circuit, &root) .unwrap();\n} Columns are laid out from left to right as instance, advice and fixed. The order of columns is otherwise without meaning. Instance columns have a white background. Advice columns have a red background. Fixed columns have a blue background. Regions are shown as labelled green boxes (overlaying the background colour). A region may appear as multiple boxes if some of its columns happen to not be adjacent. Cells that have been assigned to by the circuit will be shaded in grey. If any cells are assigned to more than once (which is usually a mistake), they will be shaded darker than the surrounding cells.","breadcrumbs":"User Documentation » Developer tools » Circuit layout","id":"15","title":"Circuit layout"},"150":{"body":"v1 v1 of the σ1​ gate takes in a word that's split into (10,7,2,13)-bit chunks (already constrained by message scheduling). We refer to these chunks respectively as (a(10),b(7),c(2),d(13)). b(7) is further split into (2,2,3)-bit chunks b(7)lo,b(7)mid,b(7)hi. We witness the spread versions of the small chunks. We already have spread(a(10)) and spread(d(13)) from the message scheduling. (X⋙17)⊕(X⋙19)⊕(X≫10) is equivalent to (X⋘15)⊕(X⋘13)⊕(X≫10). s_low_sigma_1 a0​ a1​ a2​ a3​ a4​ a5​ a6​ 0 {0,1,2,3,4,5} R0even​ spread(R0even​) b(7)lo spread(b(7)lo) b(7)mid spread(b(7)mid) 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(a(10)) spread(d(13)) b(7) 0 {0,1,2,3,4,5} R1even​ spread(R1even​) c(2) spread(c(2)) b(7)hi spread(b(7)hi) 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) Constraints: s_low_sigma_1 (σ1​ v1 constraint): LHS−RHS=0 LHS​=​spread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​)​ RHS=​429b(7)hi430c(2)​++​49d(13)427b(7)mid427b(7)hi​+++​47c(2)425b(7)lo425b(7)mid​+++​44b(7)hi415a(10)423b(7)lo​+++​42b(7)mid42d(13)413a(10)​+++​b(7)loc(2)d(13)​++​ check that b was properly split into subsections for 7-bit pieces. Wb(7)lo+22Wb(7)mid+24Wb(7)hi−W=0 2-bit range check and 2-bit spread check on b(7)lo,b(7)mid,c(2) 3-bit range check and 3-bit spread check on b(7)hi v2 v2 of the σ1​ gate takes in a word that's split into (3,4,3,7,1,1,13)-bit chunks (already constrained by message scheduling). We refer to these chunks respectively as (a(3),b(4),c(3),d(7),e(1),f(1),g(13)). We already have spread(d(7)),spread(g(13)) from the message scheduling. The 1-bit e(1),f(1) remain unchanged by the spread operation and can be used directly. We further split b(4) into two 2-bit chunks b(4)lo,b(4)hi. We witness the spread versions of the small chunks. (X⋙17)⊕(X⋙19)⊕(X≫10) is equivalent to (X⋘15)⊕(X⋘13)⊕(X≫10). s_low_sigma_1_v2 a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ 0 {0,1,2,3,4,5} R0even​ spread(R0even​) b(4)lo spread(b(4)lo) b(4)hi spread(b(4)hi) 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(7)) spread(g(13)) b(4) e(1) 0 {0,1,2,3,4,5} R1even​ spread(R1even​) a(3) spread(a(3)) c(3) spread(c(3)) f(1) 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) Constraints: s_low_sigma_1_v2 (σ1​ v2 constraint): LHS−RHS=0 LHS​=​spread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​)​ RHS=​425d(7)431f(1)​++​422c(3)430e(1)​++​420b(4)hi423d(7)​++​49g(13)418b(4)lo420c(3)​+++​48f(1)415a418b(4)hi​+++​47e(1)42g(13)416b(4)lo​+++​d(7)41f(1)413a​+++​e(1)g(13)​+​ check that b was properly split into subsections for 4-bit pieces. Wb(4)lo+22Wb(4)hi−W=0 2-bit range check and 2-bit spread check on b(4)lo,b(4)hi 3-bit range check and 3-bit spread check on a(3),c(3)","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » σ_1 gate","id":"150","title":"σ_1 gate"},"151":{"body":"Small range constraints Let constrainn​(x)=∏i=0n​(x−i). Constraining this expression to equal zero enforces that x is in [0..n]. 2-bit range check (a−3)(a−2)(a−1)(a)=0 sr2 a0​ 1 a 2-bit spread l1​(a)+4∗l2​(a)+5∗l3​(a)−a′=0 ss2 a0​ a1​ 1 a a' with interpolation polynomials: l0​(a)=(−3)(−2)(−1)(a−3)(a−2)(a−1)​ (spread(00)=0000) l1​(a)=(−2)(−1)(1)(a−3)(a−2)(a)​ (spread(01)=0001) l2​(a)=(−1)(1)(2)(a−3)(a−1)(a)​ (spread(10)=0100) l3​(a)=(1)(2)(3)(a−2)(a−1)(a)​ (spread(11)=0101) 3-bit range check (a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)(a)=0 sr3 a0​ 1 a 3-bit spread l1​(a)+4∗l2​(a)+5∗l3​(a)+16∗l4​(a)+17∗l5​(a)+20∗l6​(a)+21∗l7​(a)−a′=0 ss3 a0​ a1​ 1 a a' with interpolation polynomials: l0​(a)=(−7)(−6)(−5)(−4)(−3)(−2)(−1)(a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)​ (spread(000)=000000) l1​(a)=(−6)(−5)(−4)(−3)(−2)(−1)(1)(a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a)​ (spread(001)=000001) l2​(a)=(−5)(−4)(−3)(−2)(−1)(1)(2)(a−7)(a−6)(a−5)(a−4)(a−3)(a−1)(a)​ (spread(010)=000100) l3​(a)=(−4)(−3)(−2)(−1)(1)(2)(3)(a−7)(a−6)(a−5)(a−4)(a−2)(a−1)(a)​ (spread(011)=000101) l4​(a)=(−3)(−2)(−1)(1)(2)(3)(4)(a−7)(a−6)(a−5)(a−3)(a−2)(a−1)(a)​ (spread(100)=010000) l5​(a)=(−2)(−1)(1)(2)(3)(4)(5)(a−7)(a−6)(a−4)(a−3)(a−2)(a−1)(a)​ (spread(101)=010001) l6​(a)=(−1)(1)(2)(3)(4)(5)(6)(a−7)(a−5)(a−4)(a−3)(a−2)(a−1)(a)​ (spread(110)=010100) l7​(a)=(1)(2)(3)(4)(5)(6)(7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)(a)​ (spread(111)=010101) reduce_6 gate Addition (mod232) of 6 elements Input: E {eilo​,eihi​}i=05​ carry Check: E=e0​+e1​+e2​+e3​+e4​+e5​(mod32) Assume inputs are constrained to 16 bits. Addition gate (sa): a0​+a1​+a2​+a3​+a4​+a5​+a6​−a7​=0 Carry gate (sc): 216a6​ω−1+a6​+[(a6​−5)(a6​−4)(a6​−3)(a6​−2)(a6​−1)(a6​)]=0 sa sc a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ 1 0 e0lo​ e1lo​ e2lo​ e3lo​ e4lo​ e5lo​ −carry∗216 Elo 1 1 e0hi​ e1hi​ e2hi​ e3hi​ e4hi​ e5hi​ carry Ehi Assume inputs are constrained to 16 bits. Addition gate (sa): a0​ω−1+a1​ω−1+a2​ω−1+a0​+a1​+a2​+a3​ω−1−a3​=0 Carry gate (sc): 216a3​ω+a3​ω−1=0 sa sc a0​ a1​ a2​ a3​ 0 0 e0lo​ e1lo​ e2lo​ −carry∗216 1 1 e3lo​ e4lo​ e5lo​ Elo 0 0 e0hi​ e1hi​ e2hi​ carry 1 0 e3hi​ e4hi​ e5hi​ Ehi reduce_7 gate Addition (mod232) of 7 elements Input: E {eilo​,eihi​}i=06​ carry Check: E=e0​+e1​+e2​+e3​+e4​+e5​+e6​(mod32) Assume inputs are constrained to 16 bits. Addition gate (sa): a0​+a1​+a2​+a3​+a4​+a5​+a6​+a7​−a8​=0 Carry gate (sc): 216a7​ω−1+a7​+[(a7​−6)(a7​−5)(a7​−4)(a7​−3)(a7​−2)(a7​−1)(a7​)]=0 sa sc a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ 1 0 e0lo​ e1lo​ e2lo​ e3lo​ e4lo​ e5lo​ e6lo​ −carry∗216 Elo 1 1 e0hi​ e1hi​ e2hi​ e3hi​ e4hi​ e5hi​ e6hi​ carry Ehi","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » Helper gates","id":"151","title":"Helper gates"},"152":{"body":"For each block M∈{0,1}512 of the padded message, 64 words of 32 bits each are constructed as follows: the first 16 are obtained by splitting M into 32-bit blocks M=W0​∣∣W1​∣∣⋯∣∣W14​∣∣W15​; the remaining 48 words are constructed using the formula: Wi​=σ1​(Wi−2​)⊞Wi−7​⊞σ0​(Wi−15​)⊞Wi−16​, for 16≤i<64. sw sd0 sd1 sd2 sd3 ss0 ss0_v2 ss1 ss1_v2 a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W0lo​ spread(W0lo​) W0lo​ W0hi​ W0​ σ0​(W1​)lo σ1​(W14​)lo W9lo​ 1 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W0hi​ spread(W0hi​) W16​ σ0​(W1​)hi σ1​(W14​)hi W9hi​ carry16​ 0 1 1 0 0 0 0 0 0 {0,1,2,3,4} W1d(14)​ spread(W1d(14)​) W1lo​ W1hi​ W1​ σ0​(W2​)lo σ1​(W15​)lo W10lo​ 1 0 0 0 0 0 0 0 0 {0,1,2} W1c(11)​ spread(W1c(11)​) W1a(3)​ W1b(4)​ W17​ σ0​(W2​)hi σ1​(W15​)hi W10hi​ carry17​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W1b(4)lo​ spread(W1b(4)lo​) W1b(4)hi​ spread(W1b(4)hi​) 0 0 0 0 0 1 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R0odd​) spread(R1odd​) spread(W1c(11)​) spread(W1d(14)​) W1b(4)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R1even​) 0 0 W1a(3)​ spread(W1a(3)​) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1odd​) σ0​v1R0​ σ0​v1R1​ σ0​v1R0even​ σ0​v1R0odd​ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 0 0 0 0 0 0 0 0 {0,1,2,3} W14g(13)​ spread(W14g(13)​) W14a(3)​ W14c(3)​ 0 1 0 1 0 0 0 0 0 0 W14d(7)​ spread(W14d(7)​) W14lo​ W14hi​ W14​ σ0​(W15​)lo σ1​(W28​)lo W23lo​ 1 0 0 0 0 0 0 0 0 0 W14b(4)​ spread(W14b(4)​) W14e(1)​ W14f(1)​ W30​ σ0​(W15​)hi σ1​(W28​)hi W23hi​ carry30​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W14b(4)lo​ spread(W14b(4)lo​) W14b(4)hi​ spread(W14b(4)hi​) 0 0 0 0 0 0 1 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(W14d(7)​) spread(W14g(13)​) W1b(14)​ W14e(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) W14a(3)​ spread(W14a(3)​) W14c(3)​ spread(W14c(3)​) W14f(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) σ0​v2R0​ σ0​v2R1​ σ0​v2R0even​ σ0​v2R0odd​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W14b(4)lo​ spread(W14b(4)lo​) W14b(4)hi​ spread(W14b(4)hi​) 0 0 0 0 0 0 0 0 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d) spread(g) W14e(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) W14a(3)​ spread(W14a(3)​) W14c(3)​ spread(W14c(3)​) W14f(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) σ1​v2R0​ σ1​v2R1​ σ1​v2R0even​ σ1​v2R0odd​ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 1 0 0 1 0 0 0 0 {0,1,2,3} W49d(13)​ spread(W49d(13)​) W49lo​ W49hi​ W49​ 0 0 0 0 0 0 0 0 0 {0,1} W49a(10)​ spread(W49a(10)​) W49c(2)​ W49b(7)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W49b(7)lo​ spread(W49b(7)lo​) W49b(7)mid​ spread(W49b(7)mid​) 0 0 0 0 0 0 0 0 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(a) spread(d) W1b(49)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) W49c(2)​ spread(W49c(2)​) W49b(7)hi​ spread(W49b(7)hi​) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) σ1​v1R0​ σ1​v1R1​ σ1​v1R0even​ σ1​v1R0odd​ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W62lo​ spread(W62lo​) W62lo​ W62hi​ W62​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W62hi​ spread(W62hi​) 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W63lo​ spread(W63lo​) W63lo​ W63hi​ W63​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W63hi​ spread(W63hi​) Constraints: sw: construct word using reduce4​ sd0: decomposition gate for W0​,W62​,W63​ Wlo+216Whi−W=0 sd1: decomposition gate for W1..13​ (split into (3,4,11,14)-bit pieces) Wa(3)+23Wb(4)lo+25Wb(4)hi+27Wc(11)+218Wd(14)−W=0 sd2: decomposition gate for W14..48​ (split into (3,4,3,7,1,1,13)-bit pieces) Wa(3)+23Wb(4)lo+25Wb(4)hi+27Wc(11)+210Wd(14)+217We(1)+218Wf(1)+219Wg(13)−W=0 sd3: decomposition gate for W49..61​ (split into (10,7,2,13)-bit pieces) Wa(10)+210Wb(7)lo+212Wb(7)mid+215Wb(7)hi+217Wc(2)+219Wd(13)−W=0","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » Message scheduling region","id":"152","title":"Message scheduling region"},"153":{"body":"+----------------------------------------------------------+\n| |\n| decompose E, |\n| Σ_1(E) |\n| |\n| +---------------------------------------+\n| | |\n| | reduce_5() to get H' |\n| | |\n+----------------------------------------------------------+\n| decompose F, decompose G |\n| |\n| Ch(E,F,G) |\n| |\n+----------------------------------------------------------+\n| |\n| decompose A, |\n| Σ_0(A) |\n| |\n| |\n| +---------------------------------------+\n| | |\n| | reduce_7() to get A_new, |\n| | using H' |\n| | |\n+------------------+---------------------------------------+\n| decompose B, decompose C |\n| |\n| Maj(A,B,C) |\n| |\n| +---------------------------------------+\n| | reduce_6() to get E_new, |\n| | using H' |\n+------------------+---------------------------------------+ Initial round: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0​d(7) spread(E0​d(7)) E0​b(5)lo spread(E0​b(5)lo) E0​b(5)hi spread(E0​b(5)hi) E0lo​ spread(E0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} E0​c(14) spread(E0​c(14)) E0​a(6)lo spread(E0​a(6)lo) E0​a(6)hi spread(E0​a(6)hi) E0hi​ spread(E0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(E0​b(5)lo) spread(E0​b(5)hi) 0 0 0 0 1 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(E0​d(7)) spread(E0​c(14)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(E0​a(6)lo) spread(E0​a(6)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0​d(7) spread(F0​d(7)) F0​b(5)lo spread(F0​b(5)lo) F0​b(5)hi spread(F0​b(5)hi) F0lo​ spread(F0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} F0​c(14) spread(F0​c(14)) F0​a(6)lo spread(F0​a(6)lo) F0​a(6)hi spread(F0​a(6)hi) F0hi​ spread(F0hi​) 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} G0​d(7) spread(G0​d(7)) G0​b(5)lo spread(G0​b(5)lo) G0​b(5)hi spread(G0​b(5)hi) G0lo​ spread(G0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} G0​c(14) spread(G0​c(14)) G0​a(6)lo spread(G0​a(6)lo) G0​a(6)hi spread(G0​a(6)hi) G0hi​ spread(G0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P0even​ spread(P0even​) spread(Elo) spread(Ehi) Q0odd​ K0lo​ H0lo​ W0lo​ 0 0 0 0 0 0 0 1 0 0 1 {0,1,2,3,4,5} P0odd​ spread(P0odd​) spread(P1odd​) Σ1​(E0​)lo Σ1​(E0​)hi K0hi​ H0hi​ W0hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1even​ spread(P1even​) spread(Flo) spread(Fhi) Q1odd​ P1odd​ Hprime0lo​ Hprime0hi​ Hprime0​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1odd​ spread(P1odd​) D0lo​ E1lo​ 0 0 0 0 0 0 0 0 0 1 0 {0,1,2,3,4,5} Q0even​ spread(Q0even​) spread(Eneglo​) spread(Eneghi​) spread(Elo) D0hi​ E1hi​ E1​carry 0 0 0 0 0 0 1 0 0 0 0 {0,1,2,3,4,5} Q0odd​ spread(Q0odd​) spread(Q1odd​) spread(Ehi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1even​ spread(Q1even​) spread(Glo) spread(Ghi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1odd​ spread(Q1odd​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} A0​b(11) spread(A0​b(11)) A0​c(9)lo spread(A0​c(9)lo) A0​c(9)mid spread(A0​c(9)mid) A0lo​ spread(A0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} A0​d(10) spread(A0​d(10)) A0​a(2) spread(A0​a(2)) A0​c(9)hi spread(A0​c(9)hi) A0hi​ spread(A0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(c(9)lo) spread(c(9)mid) 0 0 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(10)) spread(b(11)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(a(2)) spread(c(9)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} B0​b(11) spread(B0​b(11)) B0​c(9)lo spread(B0​c(9)lo) B0​c(9)mid spread(B0​c(9)mid) B0lo​ spread(B0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} B0​d(10) spread(B0​d(10)) B0​a(2) spread(B0​a(2)) B0​c(9)hi spread(B0​c(9)hi) B0hi​ spread(B0hi​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} C0​b(11) spread(C0​b(11)) C0​c(9)lo spread(C0​c(9)lo) C0​c(9)mid spread(C0​c(9)mid) C0lo​ spread(C0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} C0​d(10) spread(C0​d(10)) C0​a(2) spread(C0​a(2)) C0​c(9)hi spread(C0​c(9)hi) C0hi​ spread(C0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M0even​ spread(M0even​) M1odd​ spread(A0lo​) spread(A0hi​) Hprime0lo​ Hprime0hi​ 0 0 0 0 0 1 0 0 1 0 0 {0,1,2,3,4,5} M0odd​ spread(M0odd​) spread(M1odd​) spread(B0lo​) spread(B0hi​) Σ0​(A0​)lo A1lo​ A1​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1even​ spread(M1even​) spread(C0lo​) spread(C0hi​) Σ0​(A0​)hi A1hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1odd​ spread(M1odd​) Steady-state: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0​d(7) spread(E0​d(7)) E0​b(5)lo spread(E0​b(5)lo) E0​b(5)hi spread(E0​b(5)hi) E0lo​ spread(E0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} E0​c(14) spread(E0​c(14)) E0​a(6)lo spread(E0​a(6)lo) E0​a(6)hi spread(E0​a(6)hi) E0hi​ spread(E0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(E0​b(5)lo) spread(E0​b(5)hi) 0 0 0 0 1 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(E0​d(7)) spread(E0​c(14)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(E0​a(6)lo) spread(E0​a(6)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P0even​ spread(P0even​) spread(Elo) spread(Ehi) Q0odd​ K0lo​ H0lo​ W0lo​ 0 0 0 0 0 0 0 1 0 0 1 {0,1,2,3,4,5} P0odd​ spread(P0odd​) spread(P1odd​) Σ1​(E0​)lo Σ1​(E0​)hi K0hi​ H0hi​ W0hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1even​ spread(P1even​) spread(Flo) spread(Fhi) Q1odd​ P1odd​ Hprime0lo​ Hprime0hi​ Hprime0​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1odd​ spread(P1odd​) D0lo​ E1lo​ 0 0 0 0 0 0 0 0 0 1 0 {0,1,2,3,4,5} Q0even​ spread(Q0even​) spread(Eneglo​) spread(Eneghi​) spread(Elo) D0hi​ E1hi​ E1​carry 0 0 0 0 0 0 1 0 0 0 0 {0,1,2,3,4,5} Q0odd​ spread(Q0odd​) spread(Q1odd​) spread(Ehi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1even​ spread(Q1even​) spread(Glo) spread(Ghi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1odd​ spread(Q1odd​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} A0​b(11) spread(A0​b(11)) A0​c(9)lo spread(A0​c(9)lo) A0​c(9)mid spread(A0​c(9)mid) A0lo​ spread(A0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} A0​d(10) spread(A0​d(10)) A0​a(2) spread(A0​a(2)) A0​c(9)hi spread(A0​c(9)hi) A0hi​ spread(A0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(c(9)lo) spread(c(9)mid) 0 0 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(10)) spread(b(11)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(a(2)) spread(c(9)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M0even​ spread(M0even​) M1odd​ spread(A0lo​) spread(A0hi​) Hprime0lo​ Hprime0hi​ 0 0 0 0 0 1 0 0 1 0 0 {0,1,2,3,4,5} M0odd​ spread(M0odd​) spread(M1odd​) spread(B0lo​) spread(B0hi​) Σ0​(A0​)lo A1lo​ A1​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1even​ spread(M1even​) spread(C0lo​) spread(C0hi​) Σ0​(A0​)hi A1hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1odd​ spread(M1odd​) Final digest: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 A63lo​ A63hi​ A63​ B63lo​ B63hi​ B63​ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C63lo​ C63hi​ C63​ C63lo​ C63hi​ C63​ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 E63lo​ E63hi​ E63​ G63lo​ G63hi​ G63​ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 F63lo​ F63hi​ F63​ H63lo​ H63hi​ H63​","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » Compression region","id":"153","title":"Compression region"},"154":{"body":"This section covers the background material required to understand the Halo 2 proving system. It is targeted at an ELI15 (Explain It Like I'm 15) level; if you think anything could do with additional explanation, let us know !","breadcrumbs":"Background Material » Background Material","id":"154","title":"Background Material"},"155":{"body":"A fundamental component of many cryptographic protocols is the algebraic structure known as a field . Fields are sets of objects (usually numbers) with two associated binary operators + and × such that various field axioms hold. The real numbers R are an example of a field with uncountably many elements. Halo makes use of finite fields which have a finite number of elements. Finite fields are fully classified as follows: if F is a finite field, it contains ∣F∣=pk elements for some integer k≥1 and some prime p; any two finite fields with the same number of elements are isomorphic. In particular, all of the arithmetic in a prime field Fp​ is isomorphic to addition and multiplication of integers modulo p, i.e. in Zp​. This is why we often refer to p as the modulus . We'll write a field as Fq​ where q=pk. The prime p is called its characteristic . In the cases where k>1 the field Fq​ is a k-degree extension of the field Fp​. (By analogy, the complex numbers C=R(i) are an extension of the real numbers.) However, in Halo we do not use extension fields. Whenever we write Fp​ we are referring to what we call a prime field which has a prime p number of elements, i.e. k=1. Important notes: There are two special elements in any field: 0, the additive identity, and 1, the multiplicative identity. The least significant bit of a field element, when represented as an integer in binary format, can be interpreted as its \"sign\" to help distinguish it from its additive inverse (negation). This is because for some nonzero element a which has a least significant bit 0 we have that −a=p−a has a least significant bit 1, and vice versa. We could also use whether or not an element is larger than (p−1)/2 to give it a \"sign.\" Finite fields will be useful later for constructing polynomials and elliptic curves . Elliptic curves are examples of groups, which we discuss next.","breadcrumbs":"Background Material » Fields » Fields","id":"155","title":"Fields"},"156":{"body":"Groups are simpler and more limited than fields; they have only one binary operator ⋅ and fewer axioms. They also have an identity, which we'll denote as 1. Any non-zero element a in a group has an inverse b=a−1, which is the unique element b such that a⋅b=1. For example, the set of nonzero elements of Fp​ forms a group, where the group operation is given by multiplication on the field. (aside) Additive vs multiplicative notation If ⋅ is written as × or omitted (i.e. a⋅b written as ab), the identity as 1, and inversion as a−1, as we did above, then we say that the group is \"written multiplicatively\". If ⋅ is written as +, the identity as 0 or O, and inversion as −a, then we say it is \"written additively\". It's conventional to use additive notation for elliptic curve groups, and multiplicative notation when the elements come from a finite field. When additive notation is used, we also write [k]A=k timesA+A+⋯+A​​ for nonnegative k and call this \"scalar multiplication\"; we also often use uppercase letters for variables denoting group elements. When multiplicative notation is used, we also write ak=k timesa×a×⋯×a​​ and call this \"exponentiation\". In either case we call the scalar k such that [k]g=a or gk=a the \"discrete logarithm\" of a to base g. We can extend scalars to negative integers by inversion, i.e. [−k]A+[k]A=O or a−k×ak=1. The order of an element a of a finite group is defined as the smallest positive integer k such that ak=1 (in multiplicative notation) or [k]a=O (in additive notation). The order of the group is the number of elements. Groups always have a generating set , which is a set of elements such that we can produce any element of the group as (in multiplicative terminology) a product of powers of those elements. So if the generating set is g1..k​, we can produce any element of the group as i=1∏k​giai​​. There can be many different generating sets for a given group. A group is called cyclic if it has a (not necessarily unique) generating set with only a single element — call it g. In that case we can say that g generates the group, and that the order of g is the order of the group. Any finite cyclic group G of order n is isomorphic to the integers modulo n (denoted Z/nZ), such that: the operation ⋅ in G corresponds to addition modulo n; the identity in G corresponds to 0; some generator g∈G corresponds to 1. Given a generator g, the isomorphism is always easy to compute in the Z/nZ→G direction; it is just a↦ga (or in additive notation, a↦[a]g). It may be difficult in general to compute in the G→Z/nZ direction; we'll discuss this further when we come to elliptic curves . If the order n of a finite group is prime, then the group is cyclic, and every non-identity element is a generator.","breadcrumbs":"Background Material » Fields » Groups","id":"156","title":"Groups"},"157":{"body":"We use the notation Fp×​ for the multiplicative group (i.e. the group operation is multiplication in Fp​) over the set Fp​−{0}. A quick way of obtaining the inverse in Fp×​ is a−1=ap−2. The reason for this stems from Fermat's little theorem , which states that ap=a(modp) for any integer a. If a is nonzero, we can divide by a twice to get ap−2=a−1. Let's assume that α is a generator of Fp×​, so it has order p−1 (equal to the number of elements in Fp×​). Therefore, for any element in a∈Fp×​ there is a unique integer i∈{0..p−2} such that a=αi. Notice that a×b where a,b∈Fp×​ can really be interpreted as αi×αj where a=αi and b=αj. Indeed, it holds that αi×αj=αi+j for all 0≤i,j = MyCircuit { a: None, lookup_table, }; // Generate the DOT graph string. let dot_string = halo2_proofs::dev::circuit_dot_graph(&circuit); // Now you can either handle it in Rust, or just // print it out to use with command-line tools. print!(\"{}\", dot_string);\n}","breadcrumbs":"User Documentation » Developer tools » Circuit structure","id":"16","title":"Circuit structure"},"160":{"body":"In a field Fp​ exactly half of all nonzero elements are squares; the remainder are non-squares or \"quadratic non-residues\". In order to see why, consider an α that generates the 2-order multiplicative subgroup of Fp×​ (this exists because p−1 is divisible by 2 since p is a prime greater than 2) and β that generates the t-order multiplicative subgroup of Fp×​ where p−1=2t. Then every element a∈Fp×​ can be written uniquely as αi⋅βj with i∈Z2​ and j∈Zt​. Half of all elements will have i=0 and the other half will have i=1. Let's consider the simple case where p≡3(mod4) and so t is odd (if t is even, then p−1 would be divisible by 4, which contradicts p being 3(mod4)). If a∈Fp×​ is a square, then there must exist b=αi⋅βj such that b2=a. But this means that a=(αi⋅βj)2=α2i⋅β2j=β2j. In other words, all squares in this particular field do not generate the 2-order multiplicative subgroup, and so since half of the elements generate the 2-order subgroup then at most half of the elements are square. In fact exactly half of the elements are square (since squaring each nonsquare element gives a unique square). This means we can assume all squares can be written as βm for some m, and therefore finding the square root is a matter of exponentiating by 2−1(modt). In the event that p≡1(mod4) then things get more complicated because 2−1(modt) does not exist. Let's write p−1 as 2k⋅t with t odd. The case k=0 is impossible, and the case k=1 is what we already described, so consider k≥2. α generates a 2k-order multiplicative subgroup and β generates the odd t-order multiplicative subgroup. Then every element a∈Fp×​ can be written as αi⋅βj for i∈Z2k​ and j∈Zt​. If the element is a square, then there exists some b=a​ which can be written b=αi′⋅βj′ for i′∈Z2k​ and j′∈Zt​. This means that a=b2=α2i′⋅β2j′, therefore we have i≡2i′(mod2k), and j≡2j′(modt). i would have to be even in this case because otherwise it would be impossible to have i≡2i′(mod2k) for any i′. In the case that a is not a square, then i is odd, and so half of all elements are squares. In order to compute the square root, we can first raise the element a=αi⋅βj to the power t to \"kill\" the t-order component, giving at=αit(mod2)k⋅βjt(modt)=αit(mod2)k and then raise this result to the power t−1(mod2k) to undo the effect of the original exponentiation on the 2k-order component: (αitmod2k)t−1(mod2k)=αi (since t is relatively prime to 2k). This leaves bare the αi value which we can trivially handle. We can similarly kill the 2k-order component to obtain βj⋅2−1(modt), and put the values together to obtain the square root. It turns out that in the cases k=2,3 there are simpler algorithms that merge several of these exponentiations together for efficiency. For other values of k, the only known way is to manually extract i by squaring until you obtain the identity for every single bit of i. This is the essence of the Tonelli-Shanks square root algorithm and describes the general strategy. (There is another square root algorithm that uses quadratic extension fields, but it doesn't pay off in efficiency until the prime becomes quite large.)","breadcrumbs":"Background Material » Fields » Square roots","id":"160","title":"Square roots"},"161":{"body":"In the previous sections we wrote p−1=2k⋅t with t odd, and stated that an element α∈Fp×​ generated the 2k-order subgroup. For convenience, let's denote n:=2k. The elements {1,α,…,αn−1} are known as the nth roots of unity . The primitive root of unity , ω, is an nth root of unity such that ωi=1 except when i≡0(modn). Important notes: If α is an nth root of unity, α satisfies αn−1=0. If α=1, then 1+α+α2+⋯+αn−1=0. Equivalently, the roots of unity are solutions to the equation Xn−1=(X−1)(X−α)(X−α2)⋯(X−αn−1). ω2n​+i=−ωi​ (\"Negation lemma\") . Proof: ωn=1​⟹ωn−1=0⟹(ωn/2+1)(ωn/2−1)=0.​ Since the order of ω is n, ωn/2=1. Therefore, ωn/2=−1. (ω2n​+i)2=(ωi)2​ (\"Halving lemma\") . Proof: (ω2n​+i)2=ωn+2i=ωn⋅ω2i=ω2i=(ωi)2. In other words, if we square each element in the nth roots of unity, we would get back only half the elements, {(ωni​)2}={ωn/2​} (i.e. the 2n​th roots of unity). There is a two-to-one mapping between the elements and their squares.","breadcrumbs":"Background Material » Fields » Roots of unity","id":"161","title":"Roots of unity"},"162":{"body":"Friedman, R. (n.d.) \"Cyclic Groups and Elementary Number Theory II\" (p. 5).","breadcrumbs":"Background Material » Fields » References","id":"162","title":"References"},"163":{"body":"Let A(X) be a polynomial over Fp​ with formal indeterminate X. As an example, A(X)=a0​+a1​X+a2​X2+a3​X3 defines a degree-3 polynomial. a0​ is referred to as the constant term. Polynomials of degree n−1 have n coefficients. We will often want to compute the result of replacing the formal indeterminate X with some concrete value x, which we denote by A(x). In mathematics this is commonly referred to as \"evaluating A(X) at a point x\". The word \"point\" here stems from the geometrical usage of polynomials in the form y=A(x), where (x,y) is the coordinate of a point in two-dimensional space. However, the polynomials we deal with are almost always constrained to equal zero, and x will be an element of some field . This should not be confused with points on an elliptic curve , which we also make use of, but never in the context of polynomial evaluation. Important notes: Multiplication of polynomials produces a product polynomial that is the sum of the degrees of its factors. Polynomial division subtracts from the degree. deg(A(X)B(X))=deg(A(X))+deg(B(X)), deg(A(X)/B(X))=deg(A(X))−deg(B(X)). Given a polynomial A(X) of degree n−1, if we obtain n evaluations of the polynomial at distinct points then these evaluations perfectly define the polynomial. In other words, given these evaluations we can obtain a unique polynomial A(X) of degree n−1 via polynomial interpolation. [a0​,a1​,⋯,an−1​] is the coefficient representation of the polynomial A(X). Equivalently, we could use its evaluation representation [(x0​,A(x0​)),(x1​,A(x1​)),⋯,(xn−1​,A(xn−1​))] at n distinct points. Either representation uniquely specifies the same polynomial. (aside) Horner's rule Horner's rule allows for efficient evaluation of a polynomial of degree n−1, using only n−1 multiplications and n−1 additions. It is the following identity: a0​​+a1​X+a2​X2+⋯+an−1​Xn−1=a0​+X(a1​+X(a2​+⋯+X(an−2​+Xan−1​))),​","breadcrumbs":"Background Material » Polynomials » Polynomials","id":"163","title":"Polynomials"},"164":{"body":"The FFT is an efficient way of converting between the coefficient and evaluation representations of a polynomial. It evaluates the polynomial at the nth roots of unity {ω0,ω1,⋯,ωn−1}, where ω is a primitive nth root of unity. By exploiting symmetries in the roots of unity, each round of the FFT reduces the evaluation into a problem only half the size. Most commonly we use polynomials of length some power of two, n=2k, and apply the halving reduction recursively.","breadcrumbs":"Background Material » Polynomials » Fast Fourier Transform (FFT)","id":"164","title":"Fast Fourier Transform (FFT)"},"165":{"body":"In the coefficient representation, it takes O(n2) operations to multiply two polynomials A(X)⋅B(X)=C(X): A(X)B(X)C(X)​=a0​+a1​X+a2​X2+⋯+an−1​Xn−1,=b0​+b1​X+b2​X2+⋯+bn−1​Xn−1,=a0​⋅(b0​+b1​X+b2​X2+⋯+bn−1​Xn−1)+a1​X⋅(b0​+b1​X+b2​X2+⋯+bn−1​Xn−1)+⋯+an−1​Xn−1⋅(b0​+b1​X+b2​X2+⋯+bn−1​Xn−1),​ where each of the n terms in the first polynomial has to be multiplied by the n terms of the second polynomial. In the evaluation representation, however, polynomial multiplication only requires O(n) operations: ABC​:{(x0​,A(x0​)),(x1​,A(x1​)),⋯,(xn−1​,A(xn−1​))},:{(x0​,B(x0​)),(x1​,B(x1​)),⋯,(xn−1​,B(xn−1​))},:{(x0​,A(x0​)B(x0​)),(x1​,A(x1​)B(x1​)),⋯,(xn−1​,A(xn−1​)B(xn−1​))},​ where each evaluation is multiplied pointwise. This suggests the following strategy for fast polynomial multiplication: Evaluate polynomials at all n points; Perform fast pointwise multiplication in the evaluation representation (O(n)); Convert back to the coefficient representation. The challenge now is how to evaluate and interpolate the polynomials efficiently. Naively, evaluating a polynomial at n points would require O(n2) operations (we use the O(n) Horner's rule at each point): ⎣⎡​A(1)A(ω)A(ω2)⋮A(ωn−1)​⎦⎤​=⎣⎡​111⋮1​1ωω2⋮ωn−1​1ω2ω2⋅2⋮ω2(n−1)​………⋯​1ωn−1ω2⋅(n−1)⋮ω(n−1)2​⎦⎤​⋅⎣⎡​a0​a1​a2​⋮an−1​​⎦⎤​. For convenience, we will denote the matrices above as: A^=Vω​⋅A. (A^ is known as the Discrete Fourier Transform of A; Vω​ is also called the Vandermonde matrix .)","breadcrumbs":"Background Material » Polynomials » Motivation: Fast polynomial multiplication","id":"165","title":"Motivation: Fast polynomial multiplication"},"166":{"body":"Our strategy is to divide a DFT of size n into two interleaved DFTs of size n/2. Given the polynomial A(X)=a0​+a1​X+a2​X2+⋯+an−1​Xn−1, we split it up into even and odd terms: Aeven​Aodd​​=a0​+a2​X+⋯+an−2​X2n​−1,=a1​+a3​X+⋯+an−1​X2n​−1.​ To recover the original polynomial, we do A(X)=Aeven​(X2)+XAodd​(X2). Trying this out on points ωni​ and ωn2n​+i​, i∈[0..2n​−1], we start to notice some symmetries: A(ωni​)A(ωn2n​+i​)​=Aeven​((ωni​)2)+ωni​Aodd​((ωni​)2),=Aeven​((ωn2n​+i​)2)+ωn2n​+i​Aodd​((ωn2n​+i​)2)=Aeven​((−ωni​)2)−ωni​Aodd​((−ωni​)2)←(negation lemma)=Aeven​((ωni​)2)−ωni​Aodd​((ωni​)2).​ Notice that we are only evaluating Aeven​(X) and Aodd​(X) over half the domain {(ωn0​)2,(ωn​)2,⋯,(ωn2n​−1​)2}={ωn/2i​},i=[0..2n​−1] (halving lemma). This gives us all the terms we need to reconstruct A(X) over the full domain {ω0,ω,⋯,ωn−1}: which means we have transformed a length-n DFT into two length-2n​ DFTs. We choose n=2k to be a power of two (by zero-padding if needed), and apply this divide-and-conquer strategy recursively. By the Master Theorem [1] , this gives us an evaluation algorithm with O(nlog2​n) operations, also known as the Fast Fourier Transform (FFT).","breadcrumbs":"Background Material » Polynomials » The (radix-2) Cooley-Tukey algorithm","id":"166","title":"The (radix-2) Cooley-Tukey algorithm"},"167":{"body":"So we've evaluated our polynomials and multiplied them pointwise. What remains is to convert the product from the evaluation representation back to coefficient representation. To do this, we simply call the FFT on the evaluation representation. However, this time we also: replace ωi by ω−i in the Vandermonde matrix, and multiply our final result by a factor of 1/n. In other words: A=n1​Vω−1​⋅A^. (To understand why the inverse FFT has a similar form to the FFT, refer to Slide 13-1 of [2] . The below image was also taken from [2] .)","breadcrumbs":"Background Material » Polynomials » Inverse FFT","id":"167","title":"Inverse FFT"},"168":{"body":"The Schwartz-Zippel lemma informally states that \"different polynomials are different at most points.\" Formally, it can be written as follows: Let p(x1​,x2​,⋯,xn​) be a nonzero polynomial of n variables with degree d. Let S be a finite set of numbers with at least d elements in it. If we choose random α1​,α1​,⋯,αn​ from S, Pr[p(α1​,α2​,⋯,αn​)=0]≤∣S∣d​. In the familiar univariate case p(X), this reduces to saying that a nonzero polynomial of degree d has at most d roots. The Schwartz-Zippel lemma is used in polynomial equality testing. Given two multi-variate polynomials p1​(x1​,⋯,xn​) and p2​(x1​,⋯,xn​) of degrees d1​,d2​ respectively, we can test if p1​(α1​,⋯,αn​)−p2​(α1​,⋯,αn​)=0 for random α1​,⋯,αn​←S, where the size of S is at least ∣S∣≥(d1​+d2​). If the two polynomials are identical, this will always be true, whereas if the two polynomials are different then the equality holds with probability at most ∣S∣max(d1​,d2​)​.","breadcrumbs":"Background Material » Polynomials » The Schwartz-Zippel lemma","id":"168","title":"The Schwartz-Zippel lemma"},"169":{"body":"Consider the order-n multiplicative subgroup H with primitive root of unity ω. For all ωi∈H,i∈[n−1], we have (ωi)n=(ωn)i=(ω0)i=1. In other words, every element of H fulfils the equation ZH​(X)​=Xn−1=(X−ω0)(X−ω1)(X−ω2)⋯(X−ωn−1),​ meaning every element is a root of ZH​(X). We call ZH​(X) the vanishing polynomial over H because it evaluates to zero on all elements of H. This comes in particularly handy when checking polynomial constraints. For instance, to check that A(X)+B(X)=C(X) over H, we simply have to check that A(X)+B(X)−C(X) is some multiple of ZH​(X). In other words, if dividing our constraint by the vanishing polynomial still yields some polynomial ZH​(X)A(X)+B(X)−C(X)​=H(X), we are satisfied that A(X)+B(X)−C(X)=0 over H.","breadcrumbs":"Background Material » Polynomials » Vanishing polynomial","id":"169","title":"Vanishing polynomial"},"17":{"body":"The cost-model binary takes high-level parameters for a circuit design, and estimates the verification cost, as well as resulting proof size. Usage: cargo run --example cost-model -- [OPTIONS] k Positional arguments: k 2^K bound on the number of rows. Optional arguments: -h, --help Print this message. -a, --advice R[,R..] An advice column with the given rotations. May be repeated. -i, --instance R[,R..] An instance column with the given rotations. May be repeated. -f, --fixed R[,R..] A fixed column with the given rotations. May be repeated. -g, --gate-degree D Maximum degree of the custom gates. -l, --lookup N,I,T A lookup over N columns with max input degree I and max table degree T. May be repeated. -p, --permutation N A permutation over N columns. May be repeated. For example, to estimate the cost of a circuit with three advice columns and one fixed column (with various rotations), and a maximum gate degree of 4: > cargo run --example cost-model -- -a 0,1 -a 0 -a-0,-1,1 -f 0 -g 4 11 Finished dev [unoptimized + debuginfo] target(s) in 0.03s Running `target/debug/examples/cost-model -a 0,1 -a 0 -a 0,-1,1 -f 0 -g 4 11`\nCircuit { k: 11, max_deg: 4, advice_columns: 3, lookups: 0, permutations: [], column_queries: 7, point_sets: 3, estimator: Estimator,\n}\nProof size: 1440 bytes\nVerification: at least 81.689ms","breadcrumbs":"User Documentation » Developer tools » Cost estimator","id":"17","title":"Cost estimator"},"170":{"body":"TODO: explain what a basis is in general (briefly). Polynomials are commonly written in the monomial basis (e.g. X,X2,...Xn). However, when working over a multiplicative subgroup of order n, we find a more natural expression in the Lagrange basis. Consider the order-n multiplicative subgroup H with primitive root of unity ω. The Lagrange basis corresponding to this subgroup is a set of functions {Li​}i=0n−1​, where Li​(ωj)={10​if i=j,otherwise.​ We can write this more compactly as Li​(ωj)=δij​, where δ is the Kronecker delta function. Now, we can write our polynomial as a linear combination of Lagrange basis functions, A(X)=i=0∑n−1​ai​Li​(X),X∈H, which is equivalent to saying that p(X) evaluates to a0​ at ω0, to a1​ at ω1, to a2​ at ω2,⋯, and so on. When working over a multiplicative subgroup, the Lagrange basis function has a convenient sparse representation of the form Li​(X)=X−ωici​⋅(Xn−1)​, where ci​ is the barycentric weight. (To understand how this form was derived, refer to [3] .) For i=0, we have c=1/n⟹L0​(X)=n1​X−1(Xn−1)​. Suppose we are given a set of evaluation points {x0​,x1​,⋯,xn−1​}. Since we cannot assume that the xi​'s form a multiplicative subgroup, we consider also the Lagrange polynomials Li​'s in the general case. Then we can construct: Li​(X)=j=i∏​xi​−xj​X−xj​​,i∈[0..n−1]. Here, every X=xj​=xi​ will produce a zero numerator term (xj​−xj​), causing the whole product to evaluate to zero. On the other hand, X=xi​ will evaluate to xi​−xj​xi​−xj​​ at every term, resulting in an overall product of one. This gives the desired Kronecker delta behaviour Li​(xj​)=δij​ on the set {x0​,x1​,⋯,xn−1​}.","breadcrumbs":"Background Material » Polynomials » Lagrange basis functions","id":"170","title":"Lagrange basis functions"},"171":{"body":"Given a polynomial in its evaluation representation A:{(x0​,A(x0​)),(x1​,A(x1​)),⋯,(xn−1​,A(xn−1​))}, we can reconstruct its coefficient form in the Lagrange basis: A(X)=i=0∑n−1​A(xi​)Li​(X), where X∈{x0​,x1​,⋯,x1−n​}.","breadcrumbs":"Background Material » Polynomials » Lagrange interpolation","id":"171","title":"Lagrange interpolation"},"172":{"body":"Dasgupta, S., Papadimitriou, C. H., & Vazirani, U. V. (2008). \"Algorithms\" (ch. 2). New York: McGraw-Hill Higher Education. Golin, M. (2016). \"The Fast Fourier Transform and Polynomial Multiplication\" [lecture notes], COMP 3711H Design and Analysis of Algorithms, Hong Kong University of Science and Technology. Berrut, J. and Trefethen, L. (2004). \"Barycentric Lagrange Interpolation.\"","breadcrumbs":"Background Material » Polynomials » References","id":"172","title":"References"},"173":{"body":"In the section Inverses and groups we introduced the concept of groups . A group has an identity and a group operation. In this section we will write groups additively, i.e. the identity is O and the group operation is +. Some groups can be used as cryptographic groups . At the risk of oversimplifying, this means that the problem of finding a discrete logarithm of a group element P to a given base G, i.e. finding x such that P=[x]G, is hard in general.","breadcrumbs":"Background Material » Cryptographic groups » Cryptographic groups","id":"173","title":"Cryptographic groups"},"174":{"body":"The Pedersen commitment [ P99 ] is a way to commit to a secret message in a verifiable way. It uses two random public generators G,H∈G, where G is a cryptographic group of order p. A random secret r is chosen in Zq​, and the message to commit to m is from any subset of Zq​. The commitment is c=Commit(m,r)=[m]G+[r]H. To open the commitment, the committer reveals m and r, thus allowing anyone to verify that c is indeed a commitment to m. Notice that the Pedersen commitment scheme is homomorphic: Commit(m,r)+Commit(m′,r′)​=[m]G+[r]H+[m′]G+[r′]H=[m+m′]G+[r+r′]H=Commit(m+m′,r+r′).​ Assuming the discrete log assumption holds, Pedersen commitments are also perfectly hiding and computationally binding: hiding : the adversary chooses messages m0​,m1​. The committer commits to one of these messages c=Commit(mb​;r),b∈{0,1}. Given c, the probability of the adversary guessing the correct b is no more than 21​. binding : the adversary cannot pick two different messages m0​=m1​, and randomness r0​,r1​, such that Commit(m0​,r0​)=Commit(m1​,r1​).","breadcrumbs":"Background Material » Cryptographic groups » Pedersen commitment","id":"174","title":"Pedersen commitment"},"175":{"body":"We can use a variant of the Pedersen commitment scheme to commit to multiple messages at once, m=(m1​,⋯,mn​). This time, we'll have to sample a corresponding number of random public generators G=(G0​,⋯,Gn−1​), along with a single random generator H as before (for use in hiding). Then, our commitment scheme is: Commit(m;r)​=Commit((m0​,⋯,mn−1​);r)=[r]H+[m0​]G0​+⋯+[mn−1​]Gn−1​=[r]H+i=0∑n−1​[mi​]Gi​.​ TODO: is this positionally binding?","breadcrumbs":"Background Material » Cryptographic groups » Vector Pedersen commitment","id":"175","title":"Vector Pedersen commitment"},"176":{"body":"An example of a protocol that uses cryptographic groups is Diffie--Hellman key agreement [ DH1976 ]. The Diffie--Hellman protocol is a method for two users, Alice and Bob, to generate a shared private key. It proceeds as follows: Alice and Bob publicly agree on two prime numbers, p and G, where p is large and G is a primitive root (modp). (Note that g is a generator of the group Fp×​.) Alice chooses a large random number a as her private key. She computes her public key A=[a]G(modp), and sends A to Bob. Similarly, Bob chooses a large random number b as his private key. He computes his public key B=[b]G(modp), and sends B to Alice. Now both Alice and Bob compute their shared key K=[ab]G(modp), which Alice computes as K=[a]B(modp)=[a]([b]G)(modp), and Bob computes as K=[b]A(modp)=[b]([a]G)(modp). A potential eavesdropper would need to derive K=[ab]g(modp) knowing only g,p,A=[a]G, and B=[b]G: in other words, they would need to either get the discrete logarithm a from A=[a]G or b from B=[b]G, which we assume to be computationally infeasible in Fp×​. More generally, protocols that use similar ideas to Diffie--Hellman are used throughout cryptography. One way of instantiating a cryptographic group is as an elliptic curve . Before we go into detail on elliptic curves, we'll describe some algorithms that can be used for any group.","breadcrumbs":"Background Material » Cryptographic groups » Diffie--Hellman","id":"176","title":"Diffie--Hellman"},"177":{"body":"","breadcrumbs":"Background Material » Cryptographic groups » Multiscalar multiplication","id":"177","title":"Multiscalar multiplication"},"178":{"body":"Reference: https://jbootle.github.io/Misc/pippenger.pdf","breadcrumbs":"Background Material » Cryptographic groups » TODO: Pippenger's algorithm","id":"178","title":"TODO: Pippenger's algorithm"},"179":{"body":"Elliptic curves constructed over finite fields are another important cryptographic tool. We use elliptic curves because they provide a cryptographic group , i.e. a group in which the discrete logarithm problem (discussed below) is hard. There are several ways to define the curve equation, but for our purposes, let Fp​ be a large (255-bit) field, and then let the set of solutions (x,y) to y2=x3+b for some constant b define the Fp​-rational points on an elliptic curve E(Fp​). These (x,y) coordinates are called \"affine coordinates\". Each of the Fp​-rational points, together with a \"point at infinity\" O that serves as the group identity, can be interpreted as an element of a group. By convention, elliptic curve groups are written additively. \"Three points on a line sum to zero, which is the point at infinity.\" The group addition law is simple: to add two points together, find the line that intersects both points and obtain the third point, and then negate its y-coordinate. The case that a point is being added to itself, called point doubling, requires special handling: we find the line tangent to the point, and then find the single other point that intersects this line and then negate. Otherwise, in the event that a point is being \"added\" to its negation, the result is the point at infinity. The ability to add and double points naturally gives us a way to scale them by integers, called scalars . The number of points on the curve is the group order. If this number is a prime q, then the scalars can be considered as elements of a scalar field , Fq​. Elliptic curves, when properly designed, have an important security property. Given two random elements G,H∈E(Fp​) finding a such that [a]G=H, otherwise known as the discrete log of H with respect to G, is considered computationally infeasible with classical computers. This is called the elliptic curve discrete log assumption. If an elliptic curve group G has prime order q (like the ones used in Halo 2), then it is a finite cyclic group. Recall from the section on groups that this implies it is isomorphic to Z/qZ, or equivalently, to the scalar field Fq​. Each possible generator G fixes the isomorphism; then an element on the scalar side is precisely the discrete log of the corresponding group element with respect to G. In the case of a cryptographically secure elliptic curve, the isomorphism is hard to compute in the G→Fq​ direction because the elliptic curve discrete log problem is hard. It is sometimes helpful to make use of this isomorphism by thinking of group-based cryptographic protocols and algorithms in terms of the scalars instead of in terms of the group elements. This can make proofs and notation simpler. For instance, it has become common in papers on proof systems to use the notation [x] to denote a group element with discrete log x, where the generator is implicit. We also used this idea in the \" distinct-x theorem \", in order to prove correctness of optimizations for elliptic curve scalar multiplication in Sapling, and an endomorphism-based optimization in Appendix C of the original Halo paper .","breadcrumbs":"Background Material » Elliptic curves » Elliptic curves","id":"179","title":"Elliptic curves"},"18":{"body":"Let's start with a simple circuit, to introduce you to the common APIs and how they are used. The circuit will take a public input c, and will prove knowledge of two private inputs a and b such that a2⋅b2=c.","breadcrumbs":"User Documentation » A simple example » A simple example","id":"18","title":"A simple example"},"180":{"body":"","breadcrumbs":"Background Material » Elliptic curves » Curve arithmetic","id":"180","title":"Curve arithmetic"},"181":{"body":"The simplest situation is doubling a point (x0​,y0​). Continuing with our example y2=x3+b, this is done first by computing the derivative λ=dxdy​=2y3x2​. To obtain expressions for (x1​,y1​)=(x0​,y0​)+(x0​,y0​), we consider x1​−x0​−y1​−y0​​=λ​⟹−y1​=λ(x1​−x0​)+y0​⟹y1​=λ(x0​−x1​)−y0​​.​ To get the expression for x1​, we substitute y=λ(x0​−x)−y0​ into the elliptic curve equation: y2=x3+b​⟹(λ(x0​−x)−y0​)2=x3+b⟹x3−λ2x2+⋯=0←(rearranging terms)=(x−x0​)(x−x0​)(x−x1​)←(known roots x0​,x0​,x1​)=x3−(x0​+x0​+x1​)x2+⋯.​ Comparing coefficients for the x2 term gives us λ2=x0​+x0​+x1​⟹x1​=λ2−2x0​​.","breadcrumbs":"Background Material » Elliptic curves » Point doubling","id":"181","title":"Point doubling"},"182":{"body":"This unfortunately requires an expensive inversion of 2y. We can avoid this by arranging our equations to \"defer\" the computation of the inverse, since we often do not need the actual affine (x′,y′) coordinate of the resulting point immediately after an individual curve operation. Let's introduce a third coordinate Z and scale our curve equation by Z3 like so: Z3y2=Z3x3+Z3b Our original curve is just this curve at the restriction Z=1. If we allow the affine point (x,y) to be represented by X=xZ, Y=yZ and Z=0 then we have the homogenous projective curve Y2Z=X3+Z3b. Obtaining (x,y) from (X,Y,Z) is as simple as computing (X/Z,Y/Z) when Z=0. (When Z=0, we are dealing with the point at infinity O:=(0:1:0).) In this form, we now have a convenient way to defer the inversion required by doubling a point. The general strategy is to express x′,y′ as rational functions using x=X/Z and y=Y/Z, rearrange to make their denominators the same, and then take the resulting point (X,Y,Z) to have Z be the shared denominator and X=x′Z,Y=y′Z. Projective coordinates are often, but not always, more efficient than affine coordinates. There may be exceptions to this when either we have a different way to apply Montgomery's trick, or when we're in the circuit setting where multiplications and inversions are about equally as expensive (at least in terms of circuit size). The following shows an example of doubling a point (X,Y,Z)=(xZ,yZ,Z) without an inversion. Substituting with X,Y,Z gives us λ=2y3x2​=2(Y/Z)3(X/Z)2​=2YZ3X2​ and gives us x′y′​=λ2−2x=λ2−Z2X​=4Y2Z29X4​−Z2X​=4Y2Z29X4−8XY2Z​=8Y3Z318X4YZ−16XY3Z2​=λ(x−x′)−y=λ(ZX​−4Y2Z29X4−8XY2Z​)−ZY​=2YZ3X2​(ZX​−4Y2Z29X4−8XY2Z​)−ZY​=2YZ23X3​−8Y3Z327X6−24X3Y2Z​−ZY​=8Y3Z312X3Y2Z−8Y4Z2−27X6+24X3Y2Z​​ Notice how the denominators of x′ and y′ are the same. Thus, instead of computing (x′,y′) we can compute (X,Y,Z) with Z=8Y3Z3 and X,Y set to the corresponding numerators such that X/Z=x′ and Y/Z=y′. This completely avoids the need to perform an inversion when doubling, and something analogous to this can be done when adding two distinct points.","breadcrumbs":"Background Material » Elliptic curves » Projective coordinates","id":"182","title":"Projective coordinates"},"183":{"body":"We now add two points with distinct x-coordinates, P=(x0​,y0​) and Q=(x1​,y1​), where x0​=x1​, to obtain R=P+Q=(x2​,y2​). The line PQ​ has slope λ=fracy1​−y0​x1​−x0​⟹y−y0​=λ⋅(x−x0​). Using the expression for PQ​, we compute y-coordinate −y2​ of −R as: −y2​−y0​=λ⋅(x2​−x0​)⟹y2​=(x0​−x2​)−y0​​. Plugging the expression for PQ​ into the curve equation y2=x3+b yields y2=x3+b​⟹(λ⋅(x−x0​)+y0​)2=x3+b⟹x3−λ2x2+⋯=0←(rearranging terms)=(x−x0​)(x−x1​)(x−x2​)←(known roots x0​,x1​,x2​)=x3−(x0​+x1​+x2​)x2+⋯.​ Comparing coefficients for the x2 term gives us λ2=x0​+x1​+x2​⟹x2​=λ2−x0​−x1​​. Important notes: There exist efficient formulae [1] for point addition that do not have edge cases (so-called \"complete\" formulae) and that unify the addition and doubling cases together. The result of adding a point to its negation using those formulae produces Z=0, which represents the point at infinity. In addition, there are other models like the Jacobian representation where (x,y)=(xZ2,yZ3,Z) where the curve is rescaled by Z6 instead of Z3, and this representation has even more efficient arithmetic but no unified/complete formulae. We can easily compare two curve points (X1​,Y1​,Z1​) and (X2​,Y2​,Z2​) for equality in the homogenous projective coordinate space by \"homogenizing\" their Z-coordinates; the checks become X1​Z2​=X2​Z1​ and Y1​Z2​=Y2​Z1​.","breadcrumbs":"Background Material » Elliptic curves » Point addition","id":"183","title":"Point addition"},"184":{"body":"Imagine that Fp​ has a primitive cube root of unity, or in other words that 3∣p−1 and so an element ζp​ generates a 3-order multiplicative subgroup. Notice that a point (x,y) on our example elliptic curve y2=x3+b has two cousin points: (ζp​x,ζp2​x), because the computation x3 effectively kills the ζ component of the x-coordinate. Applying the map (x,y)↦(ζp​x,y) is an application of an endomorphism over the curve. The exact mechanics involved are complicated, but when the curve has a prime q number of points (and thus a prime \"order\") the effect of the endomorphism is to multiply the point by a scalar in Fq​ which is also a primitive cube root ζq​ in the scalar field.","breadcrumbs":"Background Material » Elliptic curves » Curve endomorphisms","id":"184","title":"Curve endomorphisms"},"185":{"body":"Given a point on the curve P=(x,y), we know that its negation −P=(x,−y) is also on the curve. To uniquely specify a point, we need only encode its x-coordinate along with the sign of its y-coordinate.","breadcrumbs":"Background Material » Elliptic curves » Curve point compression","id":"185","title":"Curve point compression"},"186":{"body":"As mentioned in the Fields section, we can interpret the least significant bit of a field element as its \"sign\", since its additive inverse will always have the opposite LSB. So we record the LSB of the y-coordinate as sign. Pallas and Vesta are defined over the Fp​ and Fq​ fields, which elements can be expressed in 255 bits. This conveniently leaves one unused bit in a 32-byte representation. We pack the y-coordinate sign bit into the highest bit in the representation of the x-coordinate: <----------------------------------- x --------------------------------->\nEnc(P) = [_ _ _ _ _ _ _ _] [_ _ _ _ _ _ _ _] ... [_ _ _ _ _ _ _ _] [_ _ _ _ _ _ _ sign] ^ <-------------------------------------> ^ LSB 30 bytes MSB The \"point at infinity\" O that serves as the group identity, does not have an affine (x,y) representation. However, it turns out that there are no points on either the Pallas or Vesta curve with x=0 or y=0. We therefore use the \"fake\" affine coordinates (0,0) to encode O, which results in the all-zeroes 32-byte array.","breadcrumbs":"Background Material » Elliptic curves » Serialization","id":"186","title":"Serialization"},"187":{"body":"When deserializing a compressed curve point, we first read the most significant bit as ysign, the sign of the y-coordinate. Then, we set this bit to zero to recover the original x-coordinate. If x=0,y=0, we return the \"point at infinity\" O. Otherwise, we proceed to compute y=x3+b​. Here, we read the least significant bit of y as sign. If sign == ysign, we already have the correct sign and simply return the curve point (x,y). Otherwise, we negate y and return (x,−y).","breadcrumbs":"Background Material » Elliptic curves » Deserialization","id":"187","title":"Deserialization"},"188":{"body":"Let Ep​ be an elliptic curve over a finite field Fp​, where p is a prime. We denote this by Ep​/Fp​. and we denote the group of points of Ep​ over Fp​, with order q=#E(Fp​). For this curve, we call Fp​ the \"base field\" and Fq​ the \"scalar field\". We instantiate our proof system over the elliptic curve Ep​/Fp​. This allows us to prove statements about Fq​-arithmetic circuit satisfiability. (aside) If our curve Ep​ is over Fp​, why is the arithmetic circuit instead in Fq​? The proof system is basically working on encodings of the scalars in the circuit (or more precisely, commitments to polynomials whose coefficients are scalars). The scalars are in Fq​ when their encodings/commitments are elliptic curve points in Ep​/Fp​. However, most of the verifier's arithmetic computations are over the base field Fp​, and are thus efficiently expressed as an Fp​-arithmetic circuit. (aside) Why are the verifier's computations (mainly) over Fp​? The Halo 2 verifier actually has to perform group operations using information output by the circuit. Group operations like point doubling and addition use arithmetic in Fp​, because the coordinates of points are in Fp​. This motivates us to construct another curve with scalar field Fp​, which has an Fp​-arithmetic circuit that can efficiently verify proofs from the first curve. As a bonus, if this second curve had base field Eq​/Fq​, it would generate proofs that could be efficiently verified in the first curve's Fq​-arithmetic circuit. In other words, we instantiate a second proof system over Eq​/Fq​, forming a 2-cycle with the first:","breadcrumbs":"Background Material » Elliptic curves » Cycles of curves","id":"188","title":"Cycles of curves"},"189":{"body":"Reference: https://github.com/zcash/pasta","breadcrumbs":"Background Material » Elliptic curves » TODO: Pallas-Vesta curves","id":"189","title":"TODO: Pallas-Vesta curves"},"19":{"body":"Firstly, we need to define the instructions that our circuit will rely on. Instructions are the boundary between high-level gadgets and the low-level circuit operations. Instructions may be as coarse or as granular as desired, but in practice you want to strike a balance between an instruction being large enough to effectively optimize its implementation, and small enough that it is meaningfully reusable. For our circuit, we will use three instructions: Load a private number into the circuit. Multiply two numbers. Expose a number as a public input to the circuit. We also need a type for a variable representing a number. Instruction interfaces provide associated types for their inputs and outputs, to allow the implementations to represent these in a way that makes the most sense for their optimization goals. trait NumericInstructions: Chip { /// Variable representing a number. type Num; /// Loads a number into the circuit as a private input. fn load_private(&self, layouter: impl Layouter, a: Option) -> Result; /// Loads a number into the circuit as a fixed constant. fn load_constant(&self, layouter: impl Layouter, constant: F) -> Result; /// Returns `c = a * b`. fn mul( &self, layouter: impl Layouter, a: Self::Num, b: Self::Num, ) -> Result; /// Exposes a number as a public input to the circuit. fn expose_public( &self, layouter: impl Layouter, num: Self::Num, row: usize, ) -> Result<(), Error>;\n}","breadcrumbs":"User Documentation » A simple example » Define instructions","id":"19","title":"Define instructions"},"190":{"body":"Sometimes it is useful to be able to produce a random point on an elliptic curve Ep​/Fp​ corresponding to some input, in such a way that no-one will know its discrete logarithm (to any other base). This is described in detail in the Internet draft on Hashing to Elliptic Curves . Several algorithms can be used depending on efficiency and security requirements. The framework used in the Internet Draft makes use of several functions: hash_to_field: takes a byte sequence input and maps it to a element in the base field Fp​ map_to_curve: takes an Fp​ element and maps it to Ep​.","breadcrumbs":"Background Material » Elliptic curves » Hashing to curves","id":"190","title":"Hashing to curves"},"191":{"body":"Reference: https://eprint.iacr.org/2019/403.pdf","breadcrumbs":"Background Material » Elliptic curves » TODO: Simplified SWU","id":"191","title":"TODO: Simplified SWU"},"192":{"body":"Renes, J., Costello, C., & Batina, L. (2016, May). \"Complete addition formulas for prime order elliptic curves.\" In Annual International Conference on the Theory and Applications of Cryptographic Techniques (pp. 403-428). Springer, Berlin, Heidelberg.","breadcrumbs":"Background Material » Elliptic curves » References","id":"192","title":"References"},"193":{"body":"We want to commit to some polynomial p(X)∈Fp​[X], and be able to provably evaluate the committed polynomial at arbitrary points. The naive solution would be for the prover to simply send the polynomial's coefficients to the verifier: however, this requires O(n) communication. Our polynomial commitment scheme gets the job done using O(logn) communication.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Polynomial commitment using inner product argument","id":"193","title":"Polynomial commitment using inner product argument"},"194":{"body":"Given a parameter d=2k, we generate the common reference string σ=(G,G,H,Fp​) defining certain constants for this scheme: G is a group of prime order p; G∈Gd is a vector of d random group elements; H∈G is a random group element; and Fp​ is the finite field of order p.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Setup","id":"194","title":"Setup"},"195":{"body":"The Pedersen vector commitment Commit is defined as Commit(σ,p(X);r)=⟨a,G⟩+[r]H, for some polynomial p(X)∈Fp​[X] and some blinding factor r∈Fp​. Here, each element of the vector ai​∈Fp​ is the coefficient for the ith degree term of p(X), and p(X) is of maximal degree d−1.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Commit","id":"195","title":"Commit"},"196":{"body":"The modified inner product argument is an argument of knowledge for the relation {((P,x,v);(a,r)):P=⟨a,G⟩+[r]H,v=⟨a,b⟩}​, where b=(1,x,x2,⋯,xd−1) is composed of increasing powers of the evaluation point x. This allows a prover to demonstrate to a verifier that the polynomial contained “inside” the commitment P evaluates to v at x, and moreover, that the committed polynomial has maximum degree d−1. The inner product argument proceeds in k=log2​d rounds. For our purposes, it is sufficient to know about its final outputs, while merely providing intuition about the intermediate rounds. (Refer to Section 3 in the Halo paper for a full explanation.) Before beginning the argument, the verifier selects a random group element U and sends it to the prover. We initialize the argument at round k, with the vectors a(k):=a, G(k):=G and b(k):=b. In each round j=k,k−1,⋯,1: the prover computes two values Lj​ and Rj​ by taking some inner product of a(j) with G(j) and b(j). Note that are in some sense \"cross-terms\": the lower half of a is used with the higher half of G and b, and vice versa: Lj​Rj​​=⟨alo(j)​,Ghi(j)​⟩+[lj​]H+[⟨alo(j)​,bhi(j)​⟩]U=⟨ahi(j)​,Glo(j)​⟩+[lj​]H+[⟨ahi(j)​,blo(j)​⟩]U​ the verifier issues a random challenge uj​; the prover uses uj​ to compress the lower and higher halves of a(j), thus producing a new vector of half the original length a(j−1)=ahi(j)​⋅uj−1​+alo(j)​⋅uj​. The vectors G(j) and b(j) are similarly compressed to give G(j−1) and b(j−1). a(j−1), G(j−1) and b(j−1) are input to the next round j−1. Note that at the end of the last round j=1, we are left with a:=a(0), G:=G(0), b:=b(0), each of length 1. The intuition is that these final scalars, together with the challenges {uj​} and \"cross-terms\" {Lj​,Rj​} from each round, encode the compression in each round. Since the prover did not know the challenges U,{uj​} in advance, they would have been unable to manipulate the round compressions. Thus, checking a constraint on these final terms should enforce that the compression had been performed correctly, and that the original a satisfied the relation before undergoing compression. Note that G,b are simply rearrangements of the publicly known G,b, with the round challenges {uj​} mixed in: this means the verifier can compute G,b independently and verify that the prover had provided those same values.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Open (prover) and OpenVerify (verifier)","id":"196","title":"Open (prover) and OpenVerify (verifier)"},"197":{"body":"Alternative terms: Induction; Accumulation scheme; Proof-carrying data However, the computation of G requires a length-2k multiexponentiation ⟨G,s⟩, where s is composed of the round challenges u1​,⋯,uk​ arranged in a binary counting structure. This is the linear-time computation that we want to amortise across a batch of proof instances. Instead of computing G, notice that we can express G as a commitment to a polynomial G=Commit(σ,g(X,u1​,⋯,uk​)), where g(X,u1​,⋯,uk​):=∏i=1k​(ui​+ui−1​X2i−1) is a polynomial with degree 2k−1. Since G is a commitment, it can be checked in an inner product argument. The verifier circuit witnesses G and brings G,u1​,⋯,uk​ out as public inputs to the proof π. The next verifier instance checks π using the inner product argument; this includes checking that G=Commit(g(X,u1​,⋯,uk​)) evaluates at some random point to the expected value for the given challenges u1​,⋯,uk​. Recall from the previous section that this check only requires logd work. At the end of checking π and G, the circuit is left with a new G′, along with the u1′​,⋯,uk′​ challenges sampled for the check. To fully accept π as valid, we should perform a linear-time computation of G′=⟨G,s′⟩. Once again, we delay this computation by witnessing G′ and bringing G,u1​,⋯,uk​ out as public inputs to the proof π. This goes on from one proof instance to the next, until we are satisfied with the size of our batch of proofs. We finally perform a single linear-time computation, thus deciding the validity of the whole batch. We recall from the section Cycles of curves that we can instantiate this protocol over a two-cycle, where a proof produced by one curve is efficiently verified in the circuit of the other curve. However, some of these verifier checks can actually be efficiently performed in the native circuit; these are \"deferred\" to the next native circuit (see diagram below) instead of being immediately passed over to the other curve.","breadcrumbs":"Background Material » Recursion » Recursion","id":"197","title":"Recursion"},"2":{"body":"Requires Rust 1.51 or higher. Minimum supported Rust version can be changed in the future, but it will be done with a minor version bump.","breadcrumbs":"halo2 » Minimum Supported Rust Version","id":"2","title":"Minimum Supported Rust Version"},"20":{"body":"For our circuit, we will build a chip that provides the above numeric instructions for a finite field. /// The chip that will implement our instructions! Chips store their own\n/// config, as well as type markers if necessary.\nstruct FieldChip { config: FieldConfig, _marker: PhantomData,\n} Every chip needs to implement the Chip trait. This defines the properties of the chip that a Layouter may rely on when synthesizing a circuit, as well as enabling any initial state that the chip requires to be loaded into the circuit. impl Chip for FieldChip { type Config = FieldConfig; type Loaded = (); fn config(&self) -> &Self::Config { &self.config } fn loaded(&self) -> &Self::Loaded { &() }\n}","breadcrumbs":"User Documentation » A simple example » Define a chip implementation","id":"20","title":"Define a chip implementation"},"21":{"body":"The chip needs to be configured with the columns, permutations, and gates that will be required to implement all of the desired instructions. /// Chip state is stored in a config struct. This is generated by the chip\n/// during configuration, and then stored inside the chip.\n#[derive(Clone, Debug)]\nstruct FieldConfig { /// For this chip, we will use two advice columns to implement our instructions. /// These are also the columns through which we communicate with other parts of /// the circuit. advice: [Column; 2], /// This is the public input (instance) column. instance: Column, // We need a selector to enable the multiplication gate, so that we aren't placing // any constraints on cells where `NumericInstructions::mul` is not being used. // This is important when building larger circuits, where columns are used by // multiple sets of instructions. s_mul: Selector,\n} impl FieldChip { fn construct(config: >::Config) -> Self { Self { config, _marker: PhantomData, } } fn configure( meta: &mut ConstraintSystem, advice: [Column; 2], instance: Column, constant: Column, ) -> >::Config { meta.enable_equality(instance); meta.enable_constant(constant); for column in &advice { meta.enable_equality(*column); } let s_mul = meta.selector(); // Define our multiplication gate! meta.create_gate(\"mul\", |meta| { // To implement multiplication, we need three advice cells and a selector // cell. We arrange them like so: // // | a0 | a1 | s_mul | // |-----|-----|-------| // | lhs | rhs | s_mul | // | out | | | // // Gates may refer to any relative offsets we want, but each distinct // offset adds a cost to the proof. The most common offsets are 0 (the // current row), 1 (the next row), and -1 (the previous row), for which // `Rotation` has specific constructors. let lhs = meta.query_advice(advice[0], Rotation::cur()); let rhs = meta.query_advice(advice[1], Rotation::cur()); let out = meta.query_advice(advice[0], Rotation::next()); let s_mul = meta.query_selector(s_mul); // Finally, we return the polynomial expressions that constrain this gate. // For our multiplication gate, we only need a single polynomial constraint. // // The polynomial expressions returned from `create_gate` will be // constrained by the proving system to equal zero. Our expression // has the following properties: // - When s_mul = 0, any value is allowed in lhs, rhs, and out. // - When s_mul != 0, this constrains lhs * rhs = out. vec![s_mul * (lhs * rhs - out)] }); FieldConfig { advice, instance, s_mul, } }\n}","breadcrumbs":"User Documentation » A simple example » Configure the chip","id":"21","title":"Configure the chip"},"22":{"body":"/// A variable representing a number.\n#[derive(Clone)]\nstruct Number(AssignedCell); impl NumericInstructions for FieldChip { type Num = Number; fn load_private( &self, mut layouter: impl Layouter, value: Option, ) -> Result { let config = self.config(); layouter.assign_region( || \"load private\", |mut region| { region .assign_advice( || \"private input\", config.advice[0], 0, || value.ok_or(Error::Synthesis), ) .map(Number) }, ) } fn load_constant( &self, mut layouter: impl Layouter, constant: F, ) -> Result { let config = self.config(); layouter.assign_region( || \"load constant\", |mut region| { region .assign_advice_from_constant(|| \"constant value\", config.advice[0], 0, constant) .map(Number) }, ) } fn mul( &self, mut layouter: impl Layouter, a: Self::Num, b: Self::Num, ) -> Result { let config = self.config(); layouter.assign_region( || \"mul\", |mut region: Region<'_, F>| { // We only want to use a single multiplication gate in this region, // so we enable it at region offset 0; this means it will constrain // cells at offsets 0 and 1. config.s_mul.enable(&mut region, 0)?; // The inputs we've been given could be located anywhere in the circuit, // but we can only rely on relative offsets inside this region. So we // assign new cells inside the region and constrain them to have the // same values as the inputs. a.0.copy_advice(|| \"lhs\", &mut region, config.advice[0], 0)?; b.0.copy_advice(|| \"rhs\", &mut region, config.advice[1], 0)?; // Now we can assign the multiplication result, which is to be assigned // into the output position. let value = a.0.value().and_then(|a| b.0.value().map(|b| *a * *b)); // Finally, we do the assignment to the output, returning a // variable to be used in another part of the circuit. region .assign_advice( || \"lhs * rhs\", config.advice[0], 1, || value.ok_or(Error::Synthesis), ) .map(Number) }, ) } fn expose_public( &self, mut layouter: impl Layouter, num: Self::Num, row: usize, ) -> Result<(), Error> { let config = self.config(); layouter.constrain_instance(num.0.cell(), config.instance, row) }\n}","breadcrumbs":"User Documentation » A simple example » Implement chip traits","id":"22","title":"Implement chip traits"},"23":{"body":"Now that we have the instructions we need, and a chip that implements them, we can finally build our circuit! /// The full circuit implementation.\n///\n/// In this struct we store the private input variables. We use `Option` because\n/// they won't have any value during key generation. During proving, if any of these\n/// were `None` we would get an error.\n#[derive(Default)]\nstruct MyCircuit { constant: F, a: Option, b: Option,\n} impl Circuit for MyCircuit { // Since we are using a single chip for everything, we can just reuse its config. type Config = FieldConfig; type FloorPlanner = SimpleFloorPlanner; fn without_witnesses(&self) -> Self { Self::default() } fn configure(meta: &mut ConstraintSystem) -> Self::Config { // We create the two advice columns that FieldChip uses for I/O. let advice = [meta.advice_column(), meta.advice_column()]; // We also need an instance column to store public inputs. let instance = meta.instance_column(); // Create a fixed column to load constants. let constant = meta.fixed_column(); FieldChip::configure(meta, advice, instance, constant) } fn synthesize( &self, config: Self::Config, mut layouter: impl Layouter, ) -> Result<(), Error> { let field_chip = FieldChip::::construct(config); // Load our private values into the circuit. let a = field_chip.load_private(layouter.namespace(|| \"load a\"), self.a)?; let b = field_chip.load_private(layouter.namespace(|| \"load b\"), self.b)?; // Load the constant factor into the circuit. let constant = field_chip.load_constant(layouter.namespace(|| \"load constant\"), self.constant)?; // We only have access to plain multiplication. // We could implement our circuit as: // asq = a*a // bsq = b*b // absq = asq*bsq // c = constant*asq*bsq // // but it's more efficient to implement it as: // ab = a*b // absq = ab^2 // c = constant*absq let ab = field_chip.mul(layouter.namespace(|| \"a * b\"), a, b)?; let absq = field_chip.mul(layouter.namespace(|| \"ab * ab\"), ab.clone(), ab)?; let c = field_chip.mul(layouter.namespace(|| \"constant * absq\"), constant, absq)?; // Expose the result as a public input to the circuit. field_chip.expose_public(layouter.namespace(|| \"expose c\"), c, 0) }\n}","breadcrumbs":"User Documentation » A simple example » Build the circuit","id":"23","title":"Build the circuit"},"24":{"body":"halo2_proofs::dev::MockProver can be used to test that the circuit is working correctly. The private and public inputs to the circuit are constructed as we will do to create a proof, but by passing them to MockProver::run we get an object that can test every constraint in the circuit, and tell us exactly what is failing (if anything). // The number of rows in our circuit cannot exceed 2^k. Since our example // circuit is very small, we can pick a very small value here. let k = 4; // Prepare the private and public inputs to the circuit! let constant = Fp::from(7); let a = Fp::from(2); let b = Fp::from(3); let c = constant * a.square() * b.square(); // Instantiate the circuit with the private inputs. let circuit = MyCircuit { constant, a: Some(a), b: Some(b), }; // Arrange the public input. We expose the multiplication result in row 0 // of the instance column, so we position it there in our public inputs. let mut public_inputs = vec![c]; // Given the correct public input, our circuit will verify. let prover = MockProver::run(k, &circuit, vec![public_inputs.clone()]).unwrap(); assert_eq!(prover.verify(), Ok(())); // If we try some other public input, the proof will fail! public_inputs[0] += Fp::one(); let prover = MockProver::run(k, &circuit, vec![public_inputs]).unwrap(); assert!(prover.verify().is_err());","breadcrumbs":"User Documentation » A simple example » Testing the circuit","id":"24","title":"Testing the circuit"},"25":{"body":"You can find the source code for this example here .","breadcrumbs":"User Documentation » A simple example » Full example","id":"25","title":"Full example"},"26":{"body":"In normal programs, you can trade memory for CPU to improve performance, by pre-computing and storing lookup tables for some part of the computation. We can do the same thing in halo2 circuits! A lookup table can be thought of as enforcing a relation between variables, where the relation is expressed as a table. Assuming we have only one lookup argument in our constraint system, the total size of tables is constrained by the size of the circuit: each table entry costs one row, and it also costs one row to do each lookup. TODO","breadcrumbs":"User Documentation » Lookup tables » Lookup tables","id":"26","title":"Lookup tables"},"27":{"body":"","breadcrumbs":"User Documentation » Gadgets » Gadgets","id":"27","title":"Gadgets"},"28":{"body":"This section contains various ideas and snippets that you might find useful while writing halo2 circuits.","breadcrumbs":"User Documentation » Tips and tricks » Tips and tricks","id":"28","title":"Tips and tricks"},"29":{"body":"A common constraint used in R1CS circuits is the boolean constraint: b∗(1−b)=0. This constraint can only be satisfied by b=0 or b=1. In halo2 circuits, you can similarly constrain a cell to have one of a small set of values. For example, to constrain a to the range [0..5], you would create a gate of the form: a⋅(1−a)⋅(2−a)⋅(3−a)⋅(4−a)=0 while to constraint c to be either 7 or 13, you would use: (7−c)⋅(13−c)=0 The underlying principle here is that we create a polynomial constraint with roots at each value in the set of possible values we want to allow. In R1CS circuits, the maximum supported polynomial degree is 2 (due to all constraints being of the form a∗b=c). In halo2 circuits, you can use arbitrary-degree polynomials - with the proviso that higher-degree constraints are more expensive to use. Note that the roots don't have to be constants; for example (a−x)⋅(a−y)⋅(a−z)=0 will constrain a to be equal to one of {x,y,z} where the latter can be arbitrary polynomials, as long as the whole expression stays within the maximum degree bound.","breadcrumbs":"User Documentation » Tips and tricks » Small range constraints","id":"29","title":"Small range constraints"},"3":{"body":"halo2 currently uses rayon for parallel computation. The RAYON_NUM_THREADS environment variable can be used to set the number of threads.","breadcrumbs":"halo2 » Controlling parallelism","id":"3","title":"Controlling parallelism"},"30":{"body":"We can use Lagrange interpolation to create a polynomial constraint that maps f(X)=Y for small sets of X∈{xi​},Y∈{yi​}. For instance, say we want to map a 2-bit value to a \"spread\" version interleaved with zeros. We first precompute the evaluations at each point: 00→000001→000110→010011→0101​⟹⟹⟹⟹​0→01→12→43→5​ Then, we construct the Lagrange basis polynomial for each point using the identity: lj​(X)=0≤m B\n^ +\n| |\n+ v\nD <---+ C E +---> F ^ + | | + v H <---+ G After adding constraint B≡E the above algorithm produces the cycle: A +---> B +-------------+\n^ |\n| |\n+ v\nD <---+ C <---+ E F ^ + | | + v H <---+ G","breadcrumbs":"Design » Proving system » Permutation argument » Algorithm","id":"46","title":"Algorithm"},"47":{"body":"If we did not check whether left and right were already in the same cycle, then we could end up undoing an equality constraint. For example, if we have the following constraints: a≡b b≡c c≡d b≡d and we tried to implement adding an equality constraint just using step 5 of the above algorithm, then we would end up constructing the cycle (a b) (c d), rather than the correct (a b c d).","breadcrumbs":"Design » Proving system » Permutation argument » Broken alternatives","id":"47","title":"Broken alternatives"},"48":{"body":"We need to check a permutation of cells in m columns, represented in Lagrange basis by polynomials v0​,…,vm−1​. We will label each cell in those m columns with a unique element of F×. Suppose that we have a permutation on these labels, σ(column:i,row:j)=(column:i′,row:j′). in which the cycles correspond to equality-constraint sets. If we consider the set of pairs {(label,value)}, then the values within each cycle are equal if and only if permuting the label in each pair by σ yields the same set: An example for a cycle (A B C D). The set before permuting the labels is {(A, 7), (B, 7), (C, 7), (D, 7)}, and the set after is {(D, 7), (A, 7), (B, 7), (C, 7)} which is the same. If one of the 7s is replaced by 3, then the set after permuting labels is not the same. Since the labels are distinct, set equality is the same as multiset equality, which we can check using a product argument. Let ω be a 2k root of unity and let δ be a T root of unity, where T⋅2S+1=p with T odd and k≤S. We will use δi⋅ωj∈F× as the label for the cell in the jth row of the ith column of the permutation argument. We represent σ by a vector of m polynomials si​(X) such that si​(ωj)=δi′⋅ωj′. Notice that the identity permutation can be represented by the vector of m polynomials IDi​(ωj) such that IDi​(ωj)=δi⋅ωj. We will use a challenge β to compress each (label,value) pair to value+β⋅label. Just as in the product argument we used for lookups , we also use a challenge γ to randomize each term of the product. Now given our permutation represented by s0​,…,sm−1​ over columns represented by v0​,…,vm−1​, we want to ensure that: i=0∏m−1​j=0∏n−1​(vi​(ωj)+β⋅si​(ωj)+γvi​(ωj)+β⋅δi⋅ωj+γ​)=1 Here vi​(ωj)+β⋅δi⋅ωj represents the unpermuted (label,value) pair, and vi​(ωj)+β⋅si​(ωj) represents the permuted (σ(label),value) pair. Let ZP​ be such that ZP​(ω0)=ZP​(ωn)=1 and for 0≤j=r(λ) verifier challenges and let the ith challenge be sampled from Chi​. The advantage metric AdvIPSRS​(P,λ)=Pr[SRSPIP​(λ)] of a state restoration prover P is defined with respect to the following game. ​GameSRSIPP​(λ):​win←false;tr←ϵpp←IP.Setup(1λ)(x,stP​)←Pλ​(pp)RunPλOSRS​​(stP​)Return win​​​OracleOSRS​(τ=(a1​,c1​,...,ai−1​,ci−1​),ai​):​Ifτ∈trthenIfi≤rthenci​←Chi​;tr←tr∣∣(τ,ai​,ci​);Returnci​Else ifi=r+1thend←IP.V(pp,x,(τ,ai​));tr←(τ,ai​)Ifd=1then win←trueReturndReturn⊥​​ As shown in [GT20] (Theorem 1) state restoration soundness is tightly related to soundness after applying the Fiat-Shamir transformation. Knowledge Soundness We will show that our protocol satisfies a strengthened notion of knowledge soundness known as witness extended emulation . Informally, this notion states that for any successful prover algorithm there exists an efficient emulator that can extract a witness from it by rewinding it and supplying it with fresh randomness. However, we must slightly adjust our definition of witness extended emulation to account for the fact that our provers are state restoration provers and can rewind the verifier. Further, to avoid the need for rewinding the state restoration prover during witness extraction we study our protocol in the algebraic group model. Algebraic Group Model (AGM). An adversary Palg​ is said to be algebraic if whenever it outputs a group element X it also outputs a representation x∈Fn such that ⟨x,G⟩=X where G∈Gn is the vector of group elements that Palg​ has seen so far. Notationally, we write {X} to describe a group element X enhanced with this representation. We also write {X}iG​ to identify the component of the representation of X that corresponds with Gi​. In other words, X=i=0∑n−1​[{X}iG​]Gi​ The algebraic group model allows us to perform so-called \"online\" extraction for some protocols: the extractor can obtain the witness from the representations themselves for a single (accepting) transcript. State Restoration Witness Extended Emulation Let IP be an interactive argument for relation R with r=r(λ) challenges. We define for all non-uniform algebraic provers Palg​, extractors E, and computationally unbounded distinguishers D the advantage metric AdvIP,Rsr-wee​(Palg​,D,E,λ)=Pr[WEE-realIP,RP,D​(λ)]−Pr[WEE-idealIP,RE,P,D​(λ)] is defined with the respect to the following games. ​GameWEE-realIP,RPalg​,D​(λ):​tr←ϵpp←IP.Setup(1λ)(x,stP​)←Palg​(pp)RunPalg​Oreal​(stP​)b←D(tr)Returnb=1GameWEE-idealIP,RE,Palg​,D​(λ):​tr←ϵpp←IP.Setup(1λ)(x,stP​)←Palg​(pp)stE​←(1λ,pp,x)RunPalg​Oideal​(stP​)w←E(stE​,⊥)b←D(tr)Return(b=1)∧(Acc(tr)⟹(x,w)∈R)​​​OracleOreal​(τ=(a1​,c1​,...,ai−1​,ci−1​),ai​):​Ifτ∈trthenIfi≤rthenci​←Chi​;tr←tr∣∣(τ,ai​,ci​);Returnci​Else ifi=r+1thend←IP.V(pp,x,(τ,ai​));tr←(τ,ai​)Ifd=1then win←trueReturndReturn⊥OracleOideal​(τ,a):​Ifτ∈trthen(r,stE​)←E(stE​,[(τ,a)])tr←tr∣∣(τ,a,r)ReturnrReturn⊥​​ Zero Knowledge We say that an argument of knowledge is zero knowledge if the verifier also does not learn anything from their interaction besides that which can be learned from the existence of a valid w. More formally, Perfect Special Honest-Verifier Zero Knowledge. A public coin interactive argument (Setup,P,V) has perfect special honest-verifier zero knowledge (PSHVZK) if for all polynomial-time decidable relations R and for all (x,w)∈R and for all non-uniform polynomial-time adversaries A1​,A2​ there exists a probabilistic polynomial-time simulator S such that =​Pr⎣⎡​A1​(σ,x,tr)=1∣∣​​pp←Setup(1λ);(x,w,ρ)←A2​(pp);tr←⟨P(pp,x,w),V(pp,x,ρ)⟩​⎦⎤​Pr⎣⎡​A1​(σ,x,tr)=1∣∣​​pp←Setup(1λ);(x,w,ρ)←A2​(pp);tr←S(pp,x,ρ)​⎦⎤​​ where ρ is the internal randomness of the verifier. In this (common) definition of zero-knowledge the verifier is expected to act \"honestly\" and send challenges that correspond only with their internal randomness; they cannot adaptively respond to the prover based on the prover's messages. We use a strengthened form of this definition that forces the simulator to output a transcript with the same (adversarially provided) challenges that the verifier algorithm sends to the prover.","breadcrumbs":"Design » Protocol Description » Zero knowledge Arguments of Knowledge","id":"68","title":"Zero knowledge Arguments of Knowledge"},"69":{"body":"Let ω∈F be a n=2k primitive root of unity forming the domain D=(ω0,ω1,...,ωn−1) with t(X)=Xn−1 the vanishing polynomial over this domain. Let ng​,na​,ne​ be positive integers with na​,ne​p. That is, we must check that 0≤αq. Thus, given a scalar α, we witness the boolean decomposition of k=α+tq​. 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Typically, though, we will use them via gadgets . This indirection is useful because, for reasons of efficiency and limitations imposed by PLONKish circuits, the chip interfaces will often be dependent on low-level implementation details. The gadget interface can provide a more convenient and stable API that abstracts away from extraneous detail. For example, consider a hash function such as SHA-256. The interface of a chip supporting SHA-256 might be dependent on internals of the hash function design such as the separation between message schedule and compression function. The corresponding gadget interface can provide a more convenient and familiar update/finalize API, and can also handle parts of the hash function that do not need chip support, such as padding. This is similar to how accelerated instructions for cryptographic primitives on CPUs are typically accessed via software libraries, rather than directly. Gadgets can also provide modular and reusable abstractions for circuit programming at a higher level, similar to their use in libraries such as libsnark and bellman . As well as abstracting functions , they can also abstract types , such as elliptic curve points or integers of specific sizes.","breadcrumbs":"Concepts » Gadgets » Gadgets","id":"10","title":"Gadgets"},"100":{"body":"We use an optimized double-and-add algorithm, copied from \"Faster variable-base scalar multiplication in zk-SNARK circuits\" with some variable name changes: Acc := [2] T\nfor i from n-1 down to 0 { P := k_{i+1} ? T : −T Acc := (Acc + P) + Acc\n}\nreturn (k_0 = 0) ? (Acc - T) : Acc It remains to check that the x-coordinates of each pair of points to be added are distinct. When adding points in a prime-order group, we can rely on Theorem 3 from Appendix C of the Halo paper , which says that if we have two such points with nonzero indices wrt a given odd-prime order base, where the indices taken in the range −(q−1)/2..(q−1)/2 are distinct disregarding sign, then they have different x-coordinates. This is helpful, because it is easier to reason about the indices of points occurring in the scalar multiplication algorithm than it is to reason about their x-coordinates directly. So, the required check is equivalent to saying that the following \"indexed version\" of the above algorithm never asserts: acc := 2\nfor i from n-1 down to 0 { p = k_{i+1} ? 1 : −1 assert acc ≠ ± p assert (acc + p) ≠ acc // X acc := (acc + p) + acc assert 0 < acc ≤ (q-1)/2\n}\nif k_0 = 0 { assert acc ≠ 1 acc := acc - 1\n} The maximum value of acc is: <--- n 1s ---> 1011111...111111\n= 1100000...000000 - 1 = 2n+1+2n−1 The assertion labelled X obviously cannot fail because p=0. It is possible to see that acc is monotonically increasing except in the last conditional. It reaches its largest value when k is maximal, i.e. 2n+1+2n−1. So to entirely avoid exceptional cases, we would need 2n+1+2n−1<(q−1)/2. But we can use n larger by c if the last c iterations use complete addition . The first i for which the algorithm using only incomplete addition fails is going to be 252, since 2252+1+2252−1>(q−1)/2. We need n=254 to make the wraparound technique above work. sage: q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001\nsage: 2^253 + 2^252 - 1 < (q-1)//2\nFalse\nsage: 2^252 + 2^251 - 1 < (q-1)//2\nTrue So the last three iterations of the loop (i=2..0) need to use complete addition , as does the conditional subtraction at the end. Writing this out using ⸭ for incomplete addition (as we do in the spec), we have: Acc := [2] T\nfor i from 253 down to 3 { P := k_{i+1} ? T : −T Acc := (Acc ⸭ P) ⸭ Acc\n}\nfor i from 2 down to 0 { P := k_{i+1} ? T : −T Acc := (Acc + P) + Acc // complete addition\n}\nreturn (k_0 = 0) ? (Acc + (-T)) : Acc // complete addition","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Variable-base scalar multiplication","id":"100","title":"Variable-base scalar multiplication"},"101":{"body":"Define a running sum zj​=∑i=jn​(ki​⋅2i−j), where n=254 and: ​zn+1​=0,zn​=kn​,(most significant bit)z0​=k.​ Initialize A254​=[2]T.for i from 254 down to 4:bool_check(ki​)=0zi​=2zi+1​+ki​xP,i​=xT​yP,i​=(2ki​−1)⋅yT​(conditionally negate)λ1,i​⋅(xA,i​−xP,i​)=yA,i​−yP,i​λ1,i2​=xR,i​+xA,i​+xP,i​(λ1,i​+λ2,i​)⋅(xA,i​−xR,i​)=2yA,i​λ2,i2​=xA,i−1​+xR,i​+xA,i​λ2,i​⋅(xA,i​−xA,i−1​)=yA,i​+yA,i−1​,​ where xR,i​=(λ1,i2​−xA,i​−xT​). The helper bool_check(x)=x⋅(1−x). After substitution of xP,i​,yP,i​,xR,i​,yA,i​, and yA,i−1​, this becomes: Initialize A254​=[2]T.for i from 254 down to 4:// let ki​=zi​−2zi+1​// let yA,i​=2(λ1,i​+λ2,i​)⋅(xA,i​−(λ1,i2​−xA,i​−xT​))​bool_check(ki​)=0λ1,i​⋅(xA,i​−xT​)=yA,i​−(2ki​−1)⋅yT​λ2,i2​=xA,i−1​+λ1,i2​−xT​{λ2,i​⋅(xA,i​−xA,i−1​)=yA,i​+yA,i−1​,λ2,4​⋅(xA,4​−xA,3​)=yA,4​+yA,3witnessed​,​if i>4if i=4.​​ Here, yA,3witnessed​ is assigned to a cell. This is unlike previous yA,i​'s, which were implicitly derived from λ1,i​,λ2,i​,xA,i​,xT​, but never actually assigned. The bits k3…1​ are used in three further steps, using complete addition : for i from 3 down to 1:// let ki​=zi​−2zi+1​bool_check(ki​)=0(xA,i−1​,yA,i−1​)=((xA,i​,yA,i​)+(xT​,yT​))+(xA,i​,yA,i​)​ If the least significant bit k0​=1, we set B=O, otherwise we set B=−T. Then we return A+B using complete addition. Let B={(0,0),(xT​,−yT​),​ if k0​=1, otherwise.​ Output (xA,0​,yA,0​)+B. (Note that (0,0) represents O.)","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Constraint program for optimized double-and-add (incomplete addition)","id":"101","title":"Constraint program for optimized double-and-add (incomplete addition)"},"102":{"body":"We need six advice columns to witness (xT​,yT​,λ1​,λ2​,xA,i​,zi​). However, since (xT​,yT​) are the same, we can perform two incomplete additions in a single row, reusing the same (xT​,yT​). We split the scalar bits used in incomplete addition into hi and lo halves and process them in parallel. This means that we effectively have two for loops: the first, covering the hi half for i from 254 down to 130, with a special case at i=130; and the second, covering the lo half for the remaining i from 129 down to 4, with a special case at i=4. xT​xT​xT​⋮xT​​yT​yT​yT​⋮yT​​zhiz255​=0z254​z253​⋮z130​​xAhi​xA,254​=2[T]x​xA,253​⋮xA,130​xA,129​​λ1hi​yA,254​=2[T]y​λ1,254​λ1,253​⋮λ1,130​yA,129​​λ2hi​λ2,254​λ2,253​⋮λ2,130​​q1hi​100⋮0​q2hi​011⋮0​q3hi​000⋮1​zloz130​z129​z128​⋮z5​z4​​xAlo​xA,129​xA,128​⋮xA,5​xA,4​xA,3​​λ1lo​yA,129​λ1,129​λ1,128​⋮λ1,5​λ1,4​yA,3​​λ2lo​λ2,129​λ2,128​⋮λ2,5​λ2,4​​q1lo​100⋮00​q2lo​011⋮10​q3lo​000⋮01​​ For each hi and lo half, we have three sets of gates. Note that i is going from 255..=3; i is NOT indexing the rows. q1​=1 This gate is only used on the first row (before the for loop). We check that λ1​,λ2​ are initialized to values consistent with the initial yA​. Degree3​Constraintq1​⋅(yA,nwitnessed​−yA,n​)=0​​ where yA,n​yA,nwitnessed​​=2(λ1,n​+λ2,n​)⋅(xA,n​−(λ1,n2​−xA,n​−xT​))​, is witnessed.​ q2​=1 This gate is used on all rows corresponding to the for loop except the last. Degree223433​Constraintq2​⋅(xT,cur​−xT,next​)=0q2​⋅(yT,cur​−yT,next​)=0q2​⋅bool_check(ki​)=0, where ki​=zi​−2zi+1​q2​⋅(λ1,i​⋅(xA,i​−xT,i​)−yA,i​+(2ki​−1)⋅yT,i​)=0q2​⋅(λ2,i2​−xA,i−1​−λ1,i2​+xT,i​)=0q2​⋅(λ2,i​⋅(xA,i​−xA,i−1​)−yA,i​−yA,i−1​)=0​​ where yA,i​yA,i−1​​=2(λ1,i​+λ2,i​)⋅(xA,i​−(λ1,i2​−xA,i​−xT​))​,=2(λ1,i−1​+λ2,i−1​)⋅(xA,i−1​−(λ1,i−12​−xA,i−1​−xT​))​,​ q3​=1 This gate is used on the final iteration of the for loop, handling the special case where we check that the output yA​ has been witnessed correctly. Degree3433​Constraintq3​⋅bool_check(ki​)=0, where ki​=zi​−2zi+1​q3​⋅(λ1,i​⋅(xA,i​−xT,i​)−yA,i​+(2ki​−1)⋅yT,i​)=0q3​⋅(λ2,i2​−xA,i−1​−λ1,i2​+xT,i​)=0q3​⋅(λ2,i​⋅(xA,i​−xA,i−1​)−yA,i​−yA,i−1witnessed​)=0​​ where yA,i​yA,i−1witnessed​​=2(λ1,i​+λ2,i​)⋅(xA,i​−(λ1,i2​−xA,i​−xT​))​, is witnessed.​","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Circuit design","id":"102","title":"Circuit design"},"103":{"body":"zi​ cannot overflow for any i≥1, because it is a weighted sum of bits only up to 2n−1=2253, which is smaller than p (and also q). However, z0​=α+tq​ can overflow [0,p). Since overflow can only occur in the final step that constrains z0​=2⋅z1​+k0​, we have z0​=k(modp). It is then sufficient to also check that z0​=α+tq​(modp) (so that k=α+tq​(modp)) and that k∈[tq​,p+tq​). These conditions together imply that k=α+tq​ as an integer, and so 2254+k=α(modq) as required. Note: the bits k254..0​ do not represent a value reduced modulo q, but rather a representation of the unreduced α+tq​.","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Overflow check","id":"103","title":"Overflow check"},"104":{"body":"Since tp​+tq​<2130, we have [tq​,p+tq​)=[tq​,tq​+2130)∪[2130,2254)∪([2254,2254+2130)∩[p+tq​−2130,p+tq​)). We may assume that k=α+tq​(modp). Therefore, k∈[tq​,p+tq​)​⇔⇔⇔​(k∈[tq​,tq​+2130)∨k∈[2130,2254))∨(k∈[2254,2254+2130)∧k∈[p+tq​−2130,p+tq​))(k254​=0⟹(k∈[tq​,tq​+2130)∨k∈[2130,2254)))∧(k254​=1⟹(k∈[2254,2254+2130)∧k∈[p+tq​−2130,p+tq​))(k254​=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254​=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))Ⓐ​ Given k∈[2254,2254+2130), we prove equivalence of k∈[p+tq​−2130,p+tq​) and (α+2130)modp∈[0,2130) as follows: shift the range by 2130−p−tq​ to give k+2130−p−tq​∈[0,2130); observe that k+2130−p−tq​ is guaranteed to be in [2130−tp​−tq​,2131−tp​−tq​) and therefore cannot overflow or underflow modulo p; using the fact that k=α+tq​(modp), observe that (k+2130−p−tq​)modp=(α+tq​+2130−p−tq​)modp=(α+2130)modp. (We can see in a different way that this is correct by observing that it checks whether αmodp∈[p−2130,p), so the upper bound is aligned as we would expect.) Now, we can continue optimizing from Ⓐ: k∈[tq​,p+tq​)​⇔⇔​(k254​=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254​=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))(k254​=0⟹(α∈[0,2130)∨k253..130​ are not all 0))∧(k254​=1⟹(k253..130​ are all 0∧(α+2130)modp∈[0,2130)))​ Constraining k253..130​ to be all-0 or not-all-0 can be implemented almost \"for free\", as follows. Recall that zi​=∑h=in​(kh​⋅2h−i), so we have: z130​z130​z130​−k254​⋅2124​===​∑h=130254​(kh​⋅2h−130)k254​⋅2254−130+∑h=130253​(kh​⋅2h−130)∑h=130253​(kh​⋅2h−130)​ So k253..130​ are all 0 exactly when z130​=k254​⋅2124. Finally, we can merge the 130-bit decompositions for the k254​=0 and k254​=1 cases by checking that (α+k254​⋅2130)modp∈[0,2130).","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Optimized check for k∈[tq​,p+tq​)","id":"104","title":"Optimized check for k∈[tq​,p+tq​)"},"105":{"body":"Let s=α+k254​⋅2130. The constraints for the overflow check are: z0​k254​=1⟹(z130​k254​=0⟹(z130​​=α+tq​(modp)=2124∧smodp∈[0,2130))=0∨smodp∈[0,2130))​ Define inv0(x)={0,1/x,​if x=0otherwise.​ Witness η=inv0(z130​), and decompose smodp as s129..0​. Then the needed gates are: Degree22335​Constraintq_muloverflow⋅(s−(α+k254​⋅2130))=0q_muloverflow⋅(z0​−α−tq​)=0q_muloverflow⋅(k254​⋅(z130​−2124))=0q_muloverflow⋅(k254​⋅(s−i=0∑129​2i⋅si​)/2130)=0q_muloverflow⋅((1−k254​)⋅(1−z130​⋅η)⋅(s−i=0∑129​2i⋅si​)/2130)=0​​ where (s−i=0∑129​2i⋅si​)/2130 can be computed by another running sum. Note that the factor of 1/2130 has no effect on the constraint, since the RHS is zero. Running sum range check We make use of a 10-bit lookup range check in the circuit to subtract the low 130 bits of s. The range check subtracts the first 13⋅10 bits of s, and right-shifts the result to give (s−i=0∑129​2i⋅si​)/2130.","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Overflow check constraints","id":"105","title":"Overflow check constraints"},"106":{"body":"","breadcrumbs":"Design » Gadgets » Sinsemilla » Sinsemilla","id":"106","title":"Sinsemilla"},"107":{"body":"Sinsemilla is a collision-resistant hash function and commitment scheme designed to be efficient in algebraic circuit models that support lookups , such as PLONK or Halo 2. The security properties of Sinsemilla are similar to Pedersen hashes; it is not designed to be used where a random oracle, PRF, or preimage-resistant hash is required. The only claimed security property of the hash function is collision-resistance for fixed-length inputs. Sinsemilla is roughly 4 times less efficient than the algebraic hashes Rescue and Poseidon inside a circuit, but around 19 times more efficient than Rescue outside a circuit. Unlike either of these hashes, the collision resistance property of Sinsemilla can be proven based on cryptographic assumptions that have been well-established for at least 20 years. Sinsemilla can also be used as a computationally binding and perfectly hiding commitment scheme. The general approach is to split the message into k-bit pieces, and for each piece, select from a table of 2k bases in our cryptographic group. We combine the selected bases using a double-and-add algorithm. This ends up being provably as secure as a vector Pedersen hash, and makes advantageous use of the lookup facility supported by Halo 2.","breadcrumbs":"Design » Gadgets » Sinsemilla » Overview","id":"107","title":"Overview"},"108":{"body":"This section is an outline of how Sinsemilla works: for the normative specification, refer to §5.4.1.9 Sinsemilla Hash Function in the protocol spec. The incomplete point addition operator, ⸭, that we use below is also defined there. Let G be a cryptographic group of prime order q. We write G additively, with identity O, and using [m]P for scalar multiplication of P by m. Let k≥1 be an integer chosen based on efficiency considerations (the table size will be 2k). Let n be an integer, fixed for each instantiation, such that messages are kn bits, where 2n≤2q−1​. We use zero-padding to the next multiple of k bits if necessary. Setup: Choose Q and P[0..2k−1] as 2k+1 independent, verifiably random generators of G, using a suitable hash into G, such that none of Q or P[0..2k−1] are O. In Orchard, we define Q to be dependent on a domain separator D. The protocol specification uses Q(D) in place of Q and S(m) in place of P[m]. Hash(M): Split M into n groups of k bits. Interpret each group as a k-bit little-endian integer mi​. let Acc0​:=Q for i from 0 up to n−1: let Acci+1​:=(Acci​⸭P[mi+1​])⸭Acci​ return Accn​ Let ShortHash(M) be the x-coordinate of Hash(M). (This assumes that G is a prime-order elliptic curve in short Weierstrass form, as is the case for Pallas and Vesta.) It is slightly more efficient to express a double-and-add [2]A+R as (A+R)+A. We also use incomplete additions: it is shown in the Sinsemilla security argument that in the case where G is a prime-order short Weierstrass elliptic curve, an exceptional case for addition would lead to finding a discrete logarithm, which can be assumed to occur with negligible probability even for adversarial input.","breadcrumbs":"Design » Gadgets » Sinsemilla » Description","id":"108","title":"Description"},"109":{"body":"Choose another generator H independently of Q and P[0..2k−1]. The randomness r for a commitment is chosen uniformly on [0,q). Let Commitr​(M)=Hash(M)⸭[r]H. Let ShortCommitr​(M) be the x-coordinate of Commitr​(M). (This again assumes that G is a prime-order elliptic curve in short Weierstrass form.) Note that unlike a simple Pedersen commitment, this commitment scheme (Commit or ShortCommit) is not additively homomorphic.","breadcrumbs":"Design » Gadgets » Sinsemilla » Use as a commitment scheme","id":"109","title":"Use as a commitment scheme"},"11":{"body":"You're probably here because you want to write circuits? Excellent! This section will guide you through the process of creating circuits with halo2.","breadcrumbs":"User Documentation » User Documentation","id":"11","title":"User Documentation"},"110":{"body":"The aim of the design is to optimize the number of bits that can be processed for each step of the algorithm (which requires a doubling and addition in G) for a given table size. Using a single table of size 2k group elements, we can process k bits at a time.","breadcrumbs":"Design » Gadgets » Sinsemilla » Efficient implementation","id":"110","title":"Efficient implementation"},"111":{"body":"Let P={(j,xP[j]​,yP[j]​) for j∈{0..2k−1}}. Input: m1..=n​. (The message words are 1-indexed here, as in the protocol spec , but we start the loop from i=0 so that (xA,i​,yA,i​) corresponds to Acci​ in the protocol spec.) Output: (xA,n​,yA,n​). (xA,0​,yA,0​)=Q for i from 0 up to n−1: yP,i​=yA,i​−λ1,i​⋅(xA,i​−xP,i​) xR,i​=λ1,i2​−xA,i​−xP,i​ 2⋅yA,i​=(λ1,i​+λ2,i​)⋅(xA,i​−xR,i​) (mi+1​,xP,i​,yP,i​)∈P λ2,i2​=xA,i+1​+xR,i​+xA,i​ λ2,i​⋅(xA,i​−xA,i+1​)=yA,i​+yA,i+1​","breadcrumbs":"Design » Gadgets » Sinsemilla » Constraint program","id":"111","title":"Constraint program"},"112":{"body":"","breadcrumbs":"Design » Gadgets » Sinsemilla » PLONK / Halo 2 constraints","id":"112","title":"PLONK / Halo 2 constraints"},"113":{"body":"We have an n-bit message m=m1​+2km2​+...+2k⋅(n−1)mn​. (Note that the message words are 1-indexed as in the protocol spec .) Initialise the running sum z0​=α and define zi+1​:=2Kzi​−mi+1​​. We will end up with zn​=0. Rearranging gives us an expression for each word of the original message mi+1​=zi​−2k⋅zi+1​, which we can look up in the table. In other words, zn−i​=h=0∑i−1​2kh⋅mh+1​. For a little-endian decomposition as used here, the running sum is initialized to the scalar and ends at 0. For a big-endian decomposition as used in variable-base scalar multiplication , the running sum would start at 0 and end with recovering the original scalar. The running sum only applies to message words within a single field element, i.e. if n≥PrimeField::NUM_BITS then we will have several disjoint running sums. A longer message can be constructed by splitting the message words across several field elements, and then running several instances of the constraints below. An additional qS2​ selector is set to 0 for the last step of each element, except for the last element where it is set to 2. In order to support chaining multiple field elements without a gap, we will use a slightly more complicated expression for mi+1​ that effectively forces zn​ to zero for the last step of each element, as indicated by qS2​. This allows the cell that would have been zn​ to be used to reinitialize the running sum for the next element.","breadcrumbs":"Design » Gadgets » Sinsemilla » Message decomposition","id":"113","title":"Message decomposition"},"114":{"body":"The Sinsemilla circuit makes use of 210 pre-computed random generators. These are loaded into a lookup table: tableidx​012⋮210−1​tablex​xP[0]​xP[1]​xP[2]​⋮xP[210−1]​​tabley​yP[0]​yP[1]​yP[2]​⋮yP[210−1]​​​","breadcrumbs":"Design » Gadgets » Sinsemilla » Generator lookup table","id":"114","title":"Generator lookup table"},"115":{"body":"Note: qS3​ is synthesized from qS1​ and qS2​; it is shown here only for clarity. Step012⋮n−10′1′2′⋮n−1′n′​xA​xQ​xA,1​xA,2​⋮xA,n−1​xA,0′​xA,1′​xA,2′​⋮xA,n−1′​xA,n′​​xP​xP[m1​]​xP[m2​]​xP[m3​]​⋮xP[mn​]​xP[m1′​]​xP[m2′​]​xP[m3′​]​⋮xP[mn′​]​​bitsz0​z1​z2​⋮zn−1​z0′​z1′​z2′​⋮zn−1′​​λ1​λ1,0​λ1,1​λ1,2​⋮λ1,n−1​λ1,0′​λ1,1′​λ1,2′​⋮λ1,n−1′​yA,n​​λ2​λ2,0​λ2,1​λ2,2​⋮λ2,n−1​λ2,0′​λ2,1′​λ2,2′​⋮λ2,n−1′​​qS1​11111111110​qS2​11110111120​qS3​00000000020​qS4​10000000000​fixed_y_QyQ​0000000000​​ xQ​, z0​, z0′​, etc. would be copied in using equality constraints.","breadcrumbs":"Design » Gadgets » Sinsemilla » Layout","id":"115","title":"Layout"},"116":{"body":"For i∈[0,n), let​xR,i​YA,i​yP,i​mi+1​qS3​​=====​λ1,i2​−xA,i​−xP,i​(λ1,i​+λ2,i​)⋅(xA,i​−xR,i​)YA,i​/2−λ1,i​⋅(xA,i​−xP,i​)zi​−2k⋅(qS2,i​−qS3,i​)⋅zi+1​qS2​⋅(qS2​−1)​ The Halo 2 circuit API can automatically substitute yP,i​, xR,i​, yA,i​, and yA,i+1​, so we don't need to do that manually. xA,0​=xQ​ 2⋅yQ​=YA,0​ for i from 0 up to n−1: (mi+1​,xP,i​,yP,i​)∈P λ2,i2​=xA,i+1​+xR,i​+xA,i​ 4⋅λ2,i​⋅(xA,i​−xA,i+1​)=2⋅YA,i​+(2−qS3​)⋅YA,i+1​+2qS3​⋅yA,n​ Note that each term of the last constraint is multiplied by 4 relative to the constraint program given earlier. This is a small optimization that avoids divisions by 2. By gating the lookup expression on qS1​, we avoid the need to fill in unused cells with dummy values to pass the lookup argument. The optimized lookup value (using a default index of 0) is: (​qS1​⋅mi+1​,qS1​⋅xP,i​+(1−qS1​)⋅xP,0​,qS1​⋅yP,i​+(1−qS1​)⋅yP,0​)​ This increases the degree of the lookup argument to 6. Degree4635​ConstraintqS4​⋅(2⋅yQ​−YA,0​)=0qS1,i​⇒(mi+1​,xP,i​,yP,i​)∈PqS1,i​⋅(λ2,i2​−(xA,i+1​+xR,i​+xA,i​))qS1,i​⋅(4⋅λ2,i​⋅(xA,i​−xA,i+1​)−(2⋅YA,i​+(2−qS3,i​)⋅YA,i+1​+2⋅qS3,i​⋅yA,n​))=0​​","breadcrumbs":"Design » Gadgets » Sinsemilla » Optimized Sinsemilla gate","id":"116","title":"Optimized Sinsemilla gate"},"117":{"body":"","breadcrumbs":"Design » Gadgets » Sinsemilla » MerkleCRH » MerkleCRH","id":"117","title":"MerkleCRH"},"118":{"body":"SinsemillaHash is used in the MerkleCRHOrchard hash function . The input to SinsemillaHash is: l⋆∣∣left⋆∣∣right⋆, where: l⋆=I2LEBSP10​(l)=I2LEBSP10​(MerkleDepthOrchard−1−layer), left⋆=I2LEBSPℓMerkleOrchard​​(left), right⋆=I2LEBSPℓMerkleOrchard​​(right), with ℓMerkleOrchard​=255. left and right are allowed to be non-canonical 255-bit encodings. We break these inputs into the following MessagePieces: a (250 bits)b (20 bits)c (250 bits)​=a0​∣∣a1​=l⋆∣∣(bits 0..=239 of left )=b0​∣∣b1​∣∣b2​=(bits 240..=249 of left)∣∣(bits 250..=254 of left)∣∣(bits 0..=4 of right)=bits 5..=254 of right​ a,b,c are constrained by the SinsemillaHash to be 250 bits, 20 bits, and 250 bits respectively. In a custom gate, we check this message decomposition by enforcing the following constraints: a0​=l z1,a​, the index-1 running sum output of SinsemillaHash(a), is copied into the gate. z1,a​ has been constrained by the SinsemillaHash to be 240 bits. We recover the subpieces a0​,a1​ using a,z1,a​: z1,a​⟹a0​​=210a−a0​​=a1​=a−z1,a​⋅210.​ l+1 is loaded into a fixed column at each layer of the hash. It is used both as a gate selector, and to fix the value of l. We check that a0​=(l+1)−1. Note: The reason for using l+1 instead of l is that l=0 when layer=31 (hashing two leaves). We cannot have a zero-valued selector, since a constraint gated by a zero-valued selector is never checked. b1​+25⋅b2​=z1,b​ z1,b​, the index-1 running sum output of SinsemillaHash(b), is copied into the gate. z1,b​ has been constrained by the SinsemillaHash to be 10 bits. We witness the subpieces b1​,b2​ outside this gate, and constrain them each to be 5 bits. Inside the gate, we check that b1​+25⋅b2​=z1,b​. We also recover the subpiece b0​ using (b,z1,b​): z1,b​⟹b0​​=210b−b0..=10​​=b−(z1,b​⋅210).​ We have now derived or witnessed every subpiece, and range-constrained every subpiece: a0​ (10 bits), derived as a0​=a−210⋅z1,a​; a1​ (240 bits), equal to z1,a​; b0​ (10 bits), derived as b0​=b−210⋅z1,b​; b1​ (5 bits) is witnessed and constrained outside the gate; b2​ (5 bits) is witnessed and constrained outside the gate; b1​+25⋅b2​ is constrained to equal z1,b​, and we use them to reconstruct the original field element inputs: left=a1​+2240⋅b0​+2254⋅b1​ right=b2​+25⋅c","breadcrumbs":"Design » Gadgets » Sinsemilla » MerkleCRH » Message decomposition","id":"118","title":"Message decomposition"},"119":{"body":"The Orchard circuit spans 10 advice columns while the Sinsemilla chip only uses 5 advice columns. We distribute the path hashing evenly across two Sinsemilla chips to make better use of the available circuit area. Since the output from the previous layer hash is copied into the next layer hash, we maintain continuity even when moving from one chip to the other.","breadcrumbs":"Design » Gadgets » Sinsemilla » MerkleCRH » Circuit components","id":"119","title":"Circuit components"},"12":{"body":"The halo2 crate includes several utilities to help you design and implement your circuits.","breadcrumbs":"User Documentation » Developer tools » Developer tools","id":"12","title":"Developer tools"},"120":{"body":"Given a field element α, these gadgets decompose it into W K-bit windows α=k0​+2K⋅k1​+22K⋅k2​+⋯+2(W−1)K⋅kW−1​ where each ki​ a K-bit value. This is done using a running sum zi​,i∈[0..W). We initialize the running sum z0​=α, and compute subsequent terms zi+1​=2Kzi​−ki​​. This gives us: z0​z1​z2​↓zW​​=α=k0​+2K⋅k1​+22K⋅k2​+23K⋅k3​+⋯,=(z0​−k0​)/2K=k1​+2K⋅k2​+22K⋅k3​+⋯,=(z1​−k1​)/2K=k2​+2K⋅k3​+⋯,⋮ (in strict mode)=(zW−1​−kW−1​)/2K=0 (because zW−1​=kW−1​)​","breadcrumbs":"Design » Gadgets » Decomposition » Decomposition","id":"120","title":"Decomposition"},"121":{"body":"Strict mode constrains the running sum output zW​ to be zero, thus range-constraining the field element to be within W⋅K bits. In strict mode, we are also assured that zW−1​=kW−1​ gives us the last window in the decomposition.","breadcrumbs":"Design » Gadgets » Decomposition » Strict mode","id":"121","title":"Strict mode"},"122":{"body":"This gadget makes use of a K-bit lookup table to decompose a field element α into K-bit words. Each K-bit word ki​=zi​−2K⋅zi+1​ is range-constrained by a lookup in the K-bit table. The region layout for the lookup decomposition uses a single advice column z, and two selectors qlookup​ and qrunning​. zz0​z1​⋮zn−1​zn​​qlookup​11⋮10​qrunning​11⋮10​​","breadcrumbs":"Design » Gadgets » Decomposition » Lookup decomposition","id":"122","title":"Lookup decomposition"},"123":{"body":"Using two K-bit lookups, we can range-constrain a field element α to be n bits, where n≤K. To do this: Constrain 0≤α<2K to be within K bits using a K-bit lookup. Constrain 0≤α⋅2K−n<2K to be within K bits using a K-bit lookup. The short variant of the lookup decomposition introduces a qbitshift​ selector. The same advice column z has here been renamed to word for clarity: wordαα′2K−n​qlookup​110​qrunning​000​qbitshift​010​​ where α′=α⋅2K−n. Note that 2K−n is assigned to a fixed column at keygen, and copied in at proving time. This is used in the gate enabled by the qbitshift​ selector to check that α was shifted correctly: Degree2​Constraintqbitshift​⋅(α′−(α⋅2K−n))​​","breadcrumbs":"Design » Gadgets » Decomposition » Short range check","id":"123","title":"Short range check"},"124":{"body":"Since the lookup decomposition and its short variant both make use of the same lookup table, we combine their lookup input expressions into a single one: qlookup​⋅(qrunning​⋅(zi​−2K⋅zi+1​)+(1−qrunning​)⋅word) where zi​ and word are the same cell (but distinguished here for clarity of usage).","breadcrumbs":"Design » Gadgets » Decomposition » Combined lookup expression","id":"124","title":"Combined lookup expression"},"125":{"body":"For a short range (for instance, [0,range) where range≤8), we can range-constrain each word using a degree-range polynomial constraint instead of a lookup: range_check(word,range)=word⋅(1−word)⋯(range−1−word).","breadcrumbs":"Design » Gadgets » Decomposition » Short range decomposition","id":"125","title":"Short range decomposition"},"126":{"body":"","breadcrumbs":"Design » Gadgets » SHA-256 » SHA-256","id":"126","title":"SHA-256"},"127":{"body":"SHA-256 is specified in NIST FIPS PUB 180-4 . Unlike the specification, we use ⊞ for addition modulo 232, and + for field addition. ⊕ is used for XOR.","breadcrumbs":"Design » Gadgets » SHA-256 » Specification","id":"127","title":"Specification"},"128":{"body":"SHA-256 maintains state in eight 32-bit variables. It processes input as 512-bit blocks, but internally splits these blocks into 32-bit chunks. We therefore designed the SHA-256 gadget to consume input in 32-bit chunks.","breadcrumbs":"Design » Gadgets » SHA-256 » Gadget interface","id":"128","title":"Gadget interface"},"129":{"body":"The SHA-256 gadget requires a chip with the following instructions: # extern crate halo2_proofs;\n# use halo2_proofs::plonk::Error;\n# use std::fmt;\n#\n# trait Chip: Sized {}\n# trait Layouter {}\nconst BLOCK_SIZE: usize = 16;\nconst DIGEST_SIZE: usize = 8; pub trait Sha256Instructions: Chip { /// Variable representing the SHA-256 internal state. type State: Clone + fmt::Debug; /// Variable representing a 32-bit word of the input block to the SHA-256 compression /// function. type BlockWord: Copy + fmt::Debug; /// Places the SHA-256 IV in the circuit, returning the initial state variable. fn initialization_vector(layouter: &mut impl Layouter) -> Result; /// Starting from the given initial state, processes a block of input and returns the /// final state. fn compress( layouter: &mut impl Layouter, initial_state: &Self::State, input: [Self::BlockWord; BLOCK_SIZE], ) -> Result; /// Converts the given state into a message digest. fn digest( layouter: &mut impl Layouter, state: &Self::State, ) -> Result<[Self::BlockWord; DIGEST_SIZE], Error>;\n} TODO: Add instruction for computing padding. This set of instructions was chosen to strike a balance between the reusability of the instructions, and the scope for chips to internally optimise them. In particular, we considered splitting the compression function into its constituent parts (Ch, Maj etc), and providing a compression function gadget that implemented the round logic. However, this would prevent chips from using relative references between the various parts of a compression round. Having an instruction that implements all compression rounds is also similar to the Intel SHA extensions, which provide an instruction that performs multiple compression rounds.","breadcrumbs":"Design » Gadgets » SHA-256 » Chip instructions","id":"129","title":"Chip instructions"},"13":{"body":"halo2_proofs::dev::MockProver is a tool for debugging circuits, as well as cheaply verifying their correctness in unit tests. The private and public inputs to the circuit are constructed as would normally be done to create a proof, but MockProver::run instead creates an object that will test every constraint in the circuit directly. It returns granular error messages that indicate which specific constraint (if any) is not satisfied.","breadcrumbs":"User Documentation » Developer tools » Mock prover","id":"13","title":"Mock prover"},"130":{"body":"This chip implementation is based around a single 16-bit lookup table. It requires a minimum of 216 circuit rows, and is therefore suitable for use in larger circuits. We target a maximum constraint degree of 9. That will allow us to handle constraining carries and \"small pieces\" to a range of up to {0..7} in one row.","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » 16-bit table chip for SHA-256","id":"130","title":"16-bit table chip for SHA-256"},"131":{"body":"There are 64 compression rounds. Each round takes 32-bit values A,B,C,D,E,F,G,H as input, and performs the following operations: Ch(E,F,G)Maj(A,B,C)Σ0​(A)Σ1​(E)H′Enew​Anew​​========​(E∧F)⊕(¬E∧G)(A∧B)⊕(A∧C)⊕(B∧C)count(A,B,C)≥2(A⋙2)⊕(A⋙13)⊕(A⋙22)(E⋙6)⊕(E⋙11)⊕(E⋙25)H+Ch(E,F,G)+Σ1​(E)+Kt​+Wt​reduce6​(H′+D)reduce7​(H′+Maj(A,B,C)+Σ0​(A))​ where reducei​ must handle a carry 0≤carry = MyCircuit { a: None, lookup_table, }; // Create the area you want to draw on. // Use SVGBackend if you want to render to .svg instead. use plotters::prelude::*; let root = BitMapBackend::new(\"layout.png\", (1024, 768)).into_drawing_area(); root.fill(&WHITE).unwrap(); let root = root .titled(\"Example Circuit Layout\", (\"sans-serif\", 60)) .unwrap(); halo2_proofs::dev::CircuitLayout::default() // You can optionally render only a section of the circuit. .view_width(0..2) .view_height(0..16) // You can hide labels, which can be useful with smaller areas. .show_labels(false) // Render the circuit onto your area! // The first argument is the size parameter for the circuit. .render(5, &circuit, &root) .unwrap();\n} Columns are laid out from left to right as instance, advice and fixed. The order of columns is otherwise without meaning. Instance columns have a white background. Advice columns have a red background. Fixed columns have a blue background. Regions are shown as labelled green boxes (overlaying the background colour). A region may appear as multiple boxes if some of its columns happen to not be adjacent. Cells that have been assigned to by the circuit will be shaded in grey. If any cells are assigned to more than once (which is usually a mistake), they will be shaded darker than the surrounding cells.","breadcrumbs":"User Documentation » Developer tools » Circuit layout","id":"15","title":"Circuit layout"},"150":{"body":"v1 v1 of the σ1​ gate takes in a word that's split into (10,7,2,13)-bit chunks (already constrained by message scheduling). We refer to these chunks respectively as (a(10),b(7),c(2),d(13)). b(7) is further split into (2,2,3)-bit chunks b(7)lo,b(7)mid,b(7)hi. We witness the spread versions of the small chunks. We already have spread(a(10)) and spread(d(13)) from the message scheduling. (X⋙17)⊕(X⋙19)⊕(X≫10) is equivalent to (X⋘15)⊕(X⋘13)⊕(X≫10). s_low_sigma_1 a0​ a1​ a2​ a3​ a4​ a5​ a6​ 0 {0,1,2,3,4,5} R0even​ spread(R0even​) b(7)lo spread(b(7)lo) b(7)mid spread(b(7)mid) 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(a(10)) spread(d(13)) b(7) 0 {0,1,2,3,4,5} R1even​ spread(R1even​) c(2) spread(c(2)) b(7)hi spread(b(7)hi) 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) Constraints: s_low_sigma_1 (σ1​ v1 constraint): LHS−RHS=0 LHS​=​spread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​)​ RHS=​429b(7)hi430c(2)​++​49d(13)427b(7)mid427b(7)hi​+++​47c(2)425b(7)lo425b(7)mid​+++​44b(7)hi415a(10)423b(7)lo​+++​42b(7)mid42d(13)413a(10)​+++​b(7)loc(2)d(13)​++​ check that b was properly split into subsections for 7-bit pieces. Wb(7)lo+22Wb(7)mid+24Wb(7)hi−W=0 2-bit range check and 2-bit spread check on b(7)lo,b(7)mid,c(2) 3-bit range check and 3-bit spread check on b(7)hi v2 v2 of the σ1​ gate takes in a word that's split into (3,4,3,7,1,1,13)-bit chunks (already constrained by message scheduling). We refer to these chunks respectively as (a(3),b(4),c(3),d(7),e(1),f(1),g(13)). We already have spread(d(7)),spread(g(13)) from the message scheduling. The 1-bit e(1),f(1) remain unchanged by the spread operation and can be used directly. We further split b(4) into two 2-bit chunks b(4)lo,b(4)hi. We witness the spread versions of the small chunks. (X⋙17)⊕(X⋙19)⊕(X≫10) is equivalent to (X⋘15)⊕(X⋘13)⊕(X≫10). s_low_sigma_1_v2 a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ 0 {0,1,2,3,4,5} R0even​ spread(R0even​) b(4)lo spread(b(4)lo) b(4)hi spread(b(4)hi) 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(7)) spread(g(13)) b(4) e(1) 0 {0,1,2,3,4,5} R1even​ spread(R1even​) a(3) spread(a(3)) c(3) spread(c(3)) f(1) 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) Constraints: s_low_sigma_1_v2 (σ1​ v2 constraint): LHS−RHS=0 LHS​=​spread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​)​ RHS=​425d(7)431f(1)​++​422c(3)430e(1)​++​420b(4)hi423d(7)​++​49g(13)418b(4)lo420c(3)​+++​48f(1)415a418b(4)hi​+++​47e(1)42g(13)416b(4)lo​+++​d(7)41f(1)413a​+++​e(1)g(13)​+​ check that b was properly split into subsections for 4-bit pieces. Wb(4)lo+22Wb(4)hi−W=0 2-bit range check and 2-bit spread check on b(4)lo,b(4)hi 3-bit range check and 3-bit spread check on a(3),c(3)","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » σ_1 gate","id":"150","title":"σ_1 gate"},"151":{"body":"Small range constraints Let constrainn​(x)=∏i=0n​(x−i). Constraining this expression to equal zero enforces that x is in [0..n]. 2-bit range check (a−3)(a−2)(a−1)(a)=0 sr2 a0​ 1 a 2-bit spread l1​(a)+4∗l2​(a)+5∗l3​(a)−a′=0 ss2 a0​ a1​ 1 a a' with interpolation polynomials: l0​(a)=(−3)(−2)(−1)(a−3)(a−2)(a−1)​ (spread(00)=0000) l1​(a)=(−2)(−1)(1)(a−3)(a−2)(a)​ (spread(01)=0001) l2​(a)=(−1)(1)(2)(a−3)(a−1)(a)​ (spread(10)=0100) l3​(a)=(1)(2)(3)(a−2)(a−1)(a)​ (spread(11)=0101) 3-bit range check (a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)(a)=0 sr3 a0​ 1 a 3-bit spread l1​(a)+4∗l2​(a)+5∗l3​(a)+16∗l4​(a)+17∗l5​(a)+20∗l6​(a)+21∗l7​(a)−a′=0 ss3 a0​ a1​ 1 a a' with interpolation polynomials: l0​(a)=(−7)(−6)(−5)(−4)(−3)(−2)(−1)(a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)​ (spread(000)=000000) l1​(a)=(−6)(−5)(−4)(−3)(−2)(−1)(1)(a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a)​ (spread(001)=000001) l2​(a)=(−5)(−4)(−3)(−2)(−1)(1)(2)(a−7)(a−6)(a−5)(a−4)(a−3)(a−1)(a)​ (spread(010)=000100) l3​(a)=(−4)(−3)(−2)(−1)(1)(2)(3)(a−7)(a−6)(a−5)(a−4)(a−2)(a−1)(a)​ (spread(011)=000101) l4​(a)=(−3)(−2)(−1)(1)(2)(3)(4)(a−7)(a−6)(a−5)(a−3)(a−2)(a−1)(a)​ (spread(100)=010000) l5​(a)=(−2)(−1)(1)(2)(3)(4)(5)(a−7)(a−6)(a−4)(a−3)(a−2)(a−1)(a)​ (spread(101)=010001) l6​(a)=(−1)(1)(2)(3)(4)(5)(6)(a−7)(a−5)(a−4)(a−3)(a−2)(a−1)(a)​ (spread(110)=010100) l7​(a)=(1)(2)(3)(4)(5)(6)(7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)(a)​ (spread(111)=010101) reduce_6 gate Addition (mod232) of 6 elements Input: E {eilo​,eihi​}i=05​ carry Check: E=e0​+e1​+e2​+e3​+e4​+e5​(mod32) Assume inputs are constrained to 16 bits. Addition gate (sa): a0​+a1​+a2​+a3​+a4​+a5​+a6​−a7​=0 Carry gate (sc): 216a6​ω−1+a6​+[(a6​−5)(a6​−4)(a6​−3)(a6​−2)(a6​−1)(a6​)]=0 sa sc a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ 1 0 e0lo​ e1lo​ e2lo​ e3lo​ e4lo​ e5lo​ −carry∗216 Elo 1 1 e0hi​ e1hi​ e2hi​ e3hi​ e4hi​ e5hi​ carry Ehi Assume inputs are constrained to 16 bits. Addition gate (sa): a0​ω−1+a1​ω−1+a2​ω−1+a0​+a1​+a2​+a3​ω−1−a3​=0 Carry gate (sc): 216a3​ω+a3​ω−1=0 sa sc a0​ a1​ a2​ a3​ 0 0 e0lo​ e1lo​ e2lo​ −carry∗216 1 1 e3lo​ e4lo​ e5lo​ Elo 0 0 e0hi​ e1hi​ e2hi​ carry 1 0 e3hi​ e4hi​ e5hi​ Ehi reduce_7 gate Addition (mod232) of 7 elements Input: E {eilo​,eihi​}i=06​ carry Check: E=e0​+e1​+e2​+e3​+e4​+e5​+e6​(mod32) Assume inputs are constrained to 16 bits. Addition gate (sa): a0​+a1​+a2​+a3​+a4​+a5​+a6​+a7​−a8​=0 Carry gate (sc): 216a7​ω−1+a7​+[(a7​−6)(a7​−5)(a7​−4)(a7​−3)(a7​−2)(a7​−1)(a7​)]=0 sa sc a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ 1 0 e0lo​ e1lo​ e2lo​ e3lo​ e4lo​ e5lo​ e6lo​ −carry∗216 Elo 1 1 e0hi​ e1hi​ e2hi​ e3hi​ e4hi​ e5hi​ e6hi​ carry Ehi","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » Helper gates","id":"151","title":"Helper gates"},"152":{"body":"For each block M∈{0,1}512 of the padded message, 64 words of 32 bits each are constructed as follows: the first 16 are obtained by splitting M into 32-bit blocks M=W0​∣∣W1​∣∣⋯∣∣W14​∣∣W15​; the remaining 48 words are constructed using the formula: Wi​=σ1​(Wi−2​)⊞Wi−7​⊞σ0​(Wi−15​)⊞Wi−16​, for 16≤i<64. sw sd0 sd1 sd2 sd3 ss0 ss0_v2 ss1 ss1_v2 a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W0lo​ spread(W0lo​) W0lo​ W0hi​ W0​ σ0​(W1​)lo σ1​(W14​)lo W9lo​ 1 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W0hi​ spread(W0hi​) W16​ σ0​(W1​)hi σ1​(W14​)hi W9hi​ carry16​ 0 1 1 0 0 0 0 0 0 {0,1,2,3,4} W1d(14)​ spread(W1d(14)​) W1lo​ W1hi​ W1​ σ0​(W2​)lo σ1​(W15​)lo W10lo​ 1 0 0 0 0 0 0 0 0 {0,1,2} W1c(11)​ spread(W1c(11)​) W1a(3)​ W1b(4)​ W17​ σ0​(W2​)hi σ1​(W15​)hi W10hi​ carry17​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W1b(4)lo​ spread(W1b(4)lo​) W1b(4)hi​ spread(W1b(4)hi​) 0 0 0 0 0 1 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R0odd​) spread(R1odd​) spread(W1c(11)​) spread(W1d(14)​) W1b(4)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R1even​) 0 0 W1a(3)​ spread(W1a(3)​) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1odd​) σ0​v1R0​ σ0​v1R1​ σ0​v1R0even​ σ0​v1R0odd​ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 0 0 0 0 0 0 0 0 {0,1,2,3} W14g(13)​ spread(W14g(13)​) W14a(3)​ W14c(3)​ 0 1 0 1 0 0 0 0 0 0 W14d(7)​ spread(W14d(7)​) W14lo​ W14hi​ W14​ σ0​(W15​)lo σ1​(W28​)lo W23lo​ 1 0 0 0 0 0 0 0 0 0 W14b(4)​ spread(W14b(4)​) W14e(1)​ W14f(1)​ W30​ σ0​(W15​)hi σ1​(W28​)hi W23hi​ carry30​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W14b(4)lo​ spread(W14b(4)lo​) W14b(4)hi​ spread(W14b(4)hi​) 0 0 0 0 0 0 1 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(W14d(7)​) spread(W14g(13)​) W1b(14)​ W14e(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) W14a(3)​ spread(W14a(3)​) W14c(3)​ spread(W14c(3)​) W14f(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) σ0​v2R0​ σ0​v2R1​ σ0​v2R0even​ σ0​v2R0odd​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W14b(4)lo​ spread(W14b(4)lo​) W14b(4)hi​ spread(W14b(4)hi​) 0 0 0 0 0 0 0 0 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d) spread(g) W14e(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) W14a(3)​ spread(W14a(3)​) W14c(3)​ spread(W14c(3)​) W14f(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) σ1​v2R0​ σ1​v2R1​ σ1​v2R0even​ σ1​v2R0odd​ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 1 0 0 1 0 0 0 0 {0,1,2,3} W49d(13)​ spread(W49d(13)​) W49lo​ W49hi​ W49​ 0 0 0 0 0 0 0 0 0 {0,1} W49a(10)​ spread(W49a(10)​) W49c(2)​ W49b(7)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W49b(7)lo​ spread(W49b(7)lo​) W49b(7)mid​ spread(W49b(7)mid​) 0 0 0 0 0 0 0 0 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(a) spread(d) W1b(49)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) W49c(2)​ spread(W49c(2)​) W49b(7)hi​ spread(W49b(7)hi​) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) σ1​v1R0​ σ1​v1R1​ σ1​v1R0even​ σ1​v1R0odd​ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W62lo​ spread(W62lo​) W62lo​ W62hi​ W62​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W62hi​ spread(W62hi​) 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W63lo​ spread(W63lo​) W63lo​ W63hi​ W63​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W63hi​ spread(W63hi​) Constraints: sw: construct word using reduce4​ sd0: decomposition gate for W0​,W62​,W63​ Wlo+216Whi−W=0 sd1: decomposition gate for W1..13​ (split into (3,4,11,14)-bit pieces) Wa(3)+23Wb(4)lo+25Wb(4)hi+27Wc(11)+218Wd(14)−W=0 sd2: decomposition gate for W14..48​ (split into (3,4,3,7,1,1,13)-bit pieces) Wa(3)+23Wb(4)lo+25Wb(4)hi+27Wc(11)+210Wd(14)+217We(1)+218Wf(1)+219Wg(13)−W=0 sd3: decomposition gate for W49..61​ (split into (10,7,2,13)-bit pieces) Wa(10)+210Wb(7)lo+212Wb(7)mid+215Wb(7)hi+217Wc(2)+219Wd(13)−W=0","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » Message scheduling region","id":"152","title":"Message scheduling region"},"153":{"body":"+----------------------------------------------------------+\n| |\n| decompose E, |\n| Σ_1(E) |\n| |\n| +---------------------------------------+\n| | |\n| | reduce_5() to get H' |\n| | |\n+----------------------------------------------------------+\n| decompose F, decompose G |\n| |\n| Ch(E,F,G) |\n| |\n+----------------------------------------------------------+\n| |\n| decompose A, |\n| Σ_0(A) |\n| |\n| |\n| +---------------------------------------+\n| | |\n| | reduce_7() to get A_new, |\n| | using H' |\n| | |\n+------------------+---------------------------------------+\n| decompose B, decompose C |\n| |\n| Maj(A,B,C) |\n| |\n| +---------------------------------------+\n| | reduce_6() to get E_new, |\n| | using H' |\n+------------------+---------------------------------------+ Initial round: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0​d(7) spread(E0​d(7)) E0​b(5)lo spread(E0​b(5)lo) E0​b(5)hi spread(E0​b(5)hi) E0lo​ spread(E0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} E0​c(14) spread(E0​c(14)) E0​a(6)lo spread(E0​a(6)lo) E0​a(6)hi spread(E0​a(6)hi) E0hi​ spread(E0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(E0​b(5)lo) spread(E0​b(5)hi) 0 0 0 0 1 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(E0​d(7)) spread(E0​c(14)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(E0​a(6)lo) spread(E0​a(6)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0​d(7) spread(F0​d(7)) F0​b(5)lo spread(F0​b(5)lo) F0​b(5)hi spread(F0​b(5)hi) F0lo​ spread(F0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} F0​c(14) spread(F0​c(14)) F0​a(6)lo spread(F0​a(6)lo) F0​a(6)hi spread(F0​a(6)hi) F0hi​ spread(F0hi​) 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} G0​d(7) spread(G0​d(7)) G0​b(5)lo spread(G0​b(5)lo) G0​b(5)hi spread(G0​b(5)hi) G0lo​ spread(G0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} G0​c(14) spread(G0​c(14)) G0​a(6)lo spread(G0​a(6)lo) G0​a(6)hi spread(G0​a(6)hi) G0hi​ spread(G0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P0even​ spread(P0even​) spread(Elo) spread(Ehi) Q0odd​ K0lo​ H0lo​ W0lo​ 0 0 0 0 0 0 0 1 0 0 1 {0,1,2,3,4,5} P0odd​ spread(P0odd​) spread(P1odd​) Σ1​(E0​)lo Σ1​(E0​)hi K0hi​ H0hi​ W0hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1even​ spread(P1even​) spread(Flo) spread(Fhi) Q1odd​ P1odd​ Hprime0lo​ Hprime0hi​ Hprime0​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1odd​ spread(P1odd​) D0lo​ E1lo​ 0 0 0 0 0 0 0 0 0 1 0 {0,1,2,3,4,5} Q0even​ spread(Q0even​) spread(Eneglo​) spread(Eneghi​) spread(Elo) D0hi​ E1hi​ E1​carry 0 0 0 0 0 0 1 0 0 0 0 {0,1,2,3,4,5} Q0odd​ spread(Q0odd​) spread(Q1odd​) spread(Ehi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1even​ spread(Q1even​) spread(Glo) spread(Ghi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1odd​ spread(Q1odd​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} A0​b(11) spread(A0​b(11)) A0​c(9)lo spread(A0​c(9)lo) A0​c(9)mid spread(A0​c(9)mid) A0lo​ spread(A0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} A0​d(10) spread(A0​d(10)) A0​a(2) spread(A0​a(2)) A0​c(9)hi spread(A0​c(9)hi) A0hi​ spread(A0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(c(9)lo) spread(c(9)mid) 0 0 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(10)) spread(b(11)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(a(2)) spread(c(9)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} B0​b(11) spread(B0​b(11)) B0​c(9)lo spread(B0​c(9)lo) B0​c(9)mid spread(B0​c(9)mid) B0lo​ spread(B0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} B0​d(10) spread(B0​d(10)) B0​a(2) spread(B0​a(2)) B0​c(9)hi spread(B0​c(9)hi) B0hi​ spread(B0hi​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} C0​b(11) spread(C0​b(11)) C0​c(9)lo spread(C0​c(9)lo) C0​c(9)mid spread(C0​c(9)mid) C0lo​ spread(C0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} C0​d(10) spread(C0​d(10)) C0​a(2) spread(C0​a(2)) C0​c(9)hi spread(C0​c(9)hi) C0hi​ spread(C0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M0even​ spread(M0even​) M1odd​ spread(A0lo​) spread(A0hi​) Hprime0lo​ Hprime0hi​ 0 0 0 0 0 1 0 0 1 0 0 {0,1,2,3,4,5} M0odd​ spread(M0odd​) spread(M1odd​) spread(B0lo​) spread(B0hi​) Σ0​(A0​)lo A1lo​ A1​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1even​ spread(M1even​) spread(C0lo​) spread(C0hi​) Σ0​(A0​)hi A1hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1odd​ spread(M1odd​) Steady-state: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0​d(7) spread(E0​d(7)) E0​b(5)lo spread(E0​b(5)lo) E0​b(5)hi spread(E0​b(5)hi) E0lo​ spread(E0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} E0​c(14) spread(E0​c(14)) E0​a(6)lo spread(E0​a(6)lo) E0​a(6)hi spread(E0​a(6)hi) E0hi​ spread(E0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(E0​b(5)lo) spread(E0​b(5)hi) 0 0 0 0 1 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(E0​d(7)) spread(E0​c(14)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(E0​a(6)lo) spread(E0​a(6)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P0even​ spread(P0even​) spread(Elo) spread(Ehi) Q0odd​ K0lo​ H0lo​ W0lo​ 0 0 0 0 0 0 0 1 0 0 1 {0,1,2,3,4,5} P0odd​ spread(P0odd​) spread(P1odd​) Σ1​(E0​)lo Σ1​(E0​)hi K0hi​ H0hi​ W0hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1even​ spread(P1even​) spread(Flo) spread(Fhi) Q1odd​ P1odd​ Hprime0lo​ Hprime0hi​ Hprime0​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1odd​ spread(P1odd​) D0lo​ E1lo​ 0 0 0 0 0 0 0 0 0 1 0 {0,1,2,3,4,5} Q0even​ spread(Q0even​) spread(Eneglo​) spread(Eneghi​) spread(Elo) D0hi​ E1hi​ E1​carry 0 0 0 0 0 0 1 0 0 0 0 {0,1,2,3,4,5} Q0odd​ spread(Q0odd​) spread(Q1odd​) spread(Ehi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1even​ spread(Q1even​) spread(Glo) spread(Ghi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1odd​ spread(Q1odd​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} A0​b(11) spread(A0​b(11)) A0​c(9)lo spread(A0​c(9)lo) A0​c(9)mid spread(A0​c(9)mid) A0lo​ spread(A0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} A0​d(10) spread(A0​d(10)) A0​a(2) spread(A0​a(2)) A0​c(9)hi spread(A0​c(9)hi) A0hi​ spread(A0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(c(9)lo) spread(c(9)mid) 0 0 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(10)) spread(b(11)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(a(2)) spread(c(9)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M0even​ spread(M0even​) M1odd​ spread(A0lo​) spread(A0hi​) Hprime0lo​ Hprime0hi​ 0 0 0 0 0 1 0 0 1 0 0 {0,1,2,3,4,5} M0odd​ spread(M0odd​) spread(M1odd​) spread(B0lo​) spread(B0hi​) Σ0​(A0​)lo A1lo​ A1​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1even​ spread(M1even​) spread(C0lo​) spread(C0hi​) Σ0​(A0​)hi A1hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1odd​ spread(M1odd​) Final digest: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 A63lo​ A63hi​ A63​ B63lo​ B63hi​ B63​ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C63lo​ C63hi​ C63​ C63lo​ C63hi​ C63​ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 E63lo​ E63hi​ E63​ G63lo​ G63hi​ G63​ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 F63lo​ F63hi​ F63​ H63lo​ H63hi​ H63​","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » Compression region","id":"153","title":"Compression region"},"154":{"body":"This section covers the background material required to understand the Halo 2 proving system. It is targeted at an ELI15 (Explain It Like I'm 15) level; if you think anything could do with additional explanation, let us know !","breadcrumbs":"Background Material » Background Material","id":"154","title":"Background Material"},"155":{"body":"A fundamental component of many cryptographic protocols is the algebraic structure known as a field . Fields are sets of objects (usually numbers) with two associated binary operators + and × such that various field axioms hold. The real numbers R are an example of a field with uncountably many elements. Halo makes use of finite fields which have a finite number of elements. Finite fields are fully classified as follows: if F is a finite field, it contains ∣F∣=pk elements for some integer k≥1 and some prime p; any two finite fields with the same number of elements are isomorphic. In particular, all of the arithmetic in a prime field Fp​ is isomorphic to addition and multiplication of integers modulo p, i.e. in Zp​. This is why we often refer to p as the modulus . We'll write a field as Fq​ where q=pk. The prime p is called its characteristic . In the cases where k>1 the field Fq​ is a k-degree extension of the field Fp​. (By analogy, the complex numbers C=R(i) are an extension of the real numbers.) However, in Halo we do not use extension fields. Whenever we write Fp​ we are referring to what we call a prime field which has a prime p number of elements, i.e. k=1. Important notes: There are two special elements in any field: 0, the additive identity, and 1, the multiplicative identity. The least significant bit of a field element, when represented as an integer in binary format, can be interpreted as its \"sign\" to help distinguish it from its additive inverse (negation). This is because for some nonzero element a which has a least significant bit 0 we have that −a=p−a has a least significant bit 1, and vice versa. We could also use whether or not an element is larger than (p−1)/2 to give it a \"sign.\" Finite fields will be useful later for constructing polynomials and elliptic curves . Elliptic curves are examples of groups, which we discuss next.","breadcrumbs":"Background Material » Fields » Fields","id":"155","title":"Fields"},"156":{"body":"Groups are simpler and more limited than fields; they have only one binary operator ⋅ and fewer axioms. They also have an identity, which we'll denote as 1. Any non-zero element a in a group has an inverse b=a−1, which is the unique element b such that a⋅b=1. For example, the set of nonzero elements of Fp​ forms a group, where the group operation is given by multiplication on the field. (aside) Additive vs multiplicative notation If ⋅ is written as × or omitted (i.e. a⋅b written as ab), the identity as 1, and inversion as a−1, as we did above, then we say that the group is \"written multiplicatively\". If ⋅ is written as +, the identity as 0 or O, and inversion as −a, then we say it is \"written additively\". It's conventional to use additive notation for elliptic curve groups, and multiplicative notation when the elements come from a finite field. When additive notation is used, we also write [k]A=k timesA+A+⋯+A​​ for nonnegative k and call this \"scalar multiplication\"; we also often use uppercase letters for variables denoting group elements. When multiplicative notation is used, we also write ak=k timesa×a×⋯×a​​ and call this \"exponentiation\". In either case we call the scalar k such that [k]g=a or gk=a the \"discrete logarithm\" of a to base g. We can extend scalars to negative integers by inversion, i.e. [−k]A+[k]A=O or a−k×ak=1. The order of an element a of a finite group is defined as the smallest positive integer k such that ak=1 (in multiplicative notation) or [k]a=O (in additive notation). The order of the group is the number of elements. Groups always have a generating set , which is a set of elements such that we can produce any element of the group as (in multiplicative terminology) a product of powers of those elements. So if the generating set is g1..k​, we can produce any element of the group as i=1∏k​giai​​. There can be many different generating sets for a given group. A group is called cyclic if it has a (not necessarily unique) generating set with only a single element — call it g. In that case we can say that g generates the group, and that the order of g is the order of the group. Any finite cyclic group G of order n is isomorphic to the integers modulo n (denoted Z/nZ), such that: the operation ⋅ in G corresponds to addition modulo n; the identity in G corresponds to 0; some generator g∈G corresponds to 1. Given a generator g, the isomorphism is always easy to compute in the Z/nZ→G direction; it is just a↦ga (or in additive notation, a↦[a]g). It may be difficult in general to compute in the G→Z/nZ direction; we'll discuss this further when we come to elliptic curves . If the order n of a finite group is prime, then the group is cyclic, and every non-identity element is a generator.","breadcrumbs":"Background Material » Fields » Groups","id":"156","title":"Groups"},"157":{"body":"We use the notation Fp×​ for the multiplicative group (i.e. the group operation is multiplication in Fp​) over the set Fp​−{0}. A quick way of obtaining the inverse in Fp×​ is a−1=ap−2. The reason for this stems from Fermat's little theorem , which states that ap=a(modp) for any integer a. If a is nonzero, we can divide by a twice to get ap−2=a−1. Let's assume that α is a generator of Fp×​, so it has order p−1 (equal to the number of elements in Fp×​). Therefore, for any element in a∈Fp×​ there is a unique integer i∈{0..p−2} such that a=αi. Notice that a×b where a,b∈Fp×​ can really be interpreted as αi×αj where a=αi and b=αj. Indeed, it holds that αi×αj=αi+j for all 0≤i,j = MyCircuit { a: None, lookup_table, }; // Generate the DOT graph string. let dot_string = halo2_proofs::dev::circuit_dot_graph(&circuit); // Now you can either handle it in Rust, or just // print it out to use with command-line tools. print!(\"{}\", dot_string);\n}","breadcrumbs":"User Documentation » Developer tools » Circuit structure","id":"16","title":"Circuit structure"},"160":{"body":"In a field Fp​ exactly half of all nonzero elements are squares; the remainder are non-squares or \"quadratic non-residues\". In order to see why, consider an α that generates the 2-order multiplicative subgroup of Fp×​ (this exists because p−1 is divisible by 2 since p is a prime greater than 2) and β that generates the t-order multiplicative subgroup of Fp×​ where p−1=2t. Then every element a∈Fp×​ can be written uniquely as αi⋅βj with i∈Z2​ and j∈Zt​. Half of all elements will have i=0 and the other half will have i=1. Let's consider the simple case where p≡3(mod4) and so t is odd (if t is even, then p−1 would be divisible by 4, which contradicts p being 3(mod4)). If a∈Fp×​ is a square, then there must exist b=αi⋅βj such that b2=a. But this means that a=(αi⋅βj)2=α2i⋅β2j=β2j. In other words, all squares in this particular field do not generate the 2-order multiplicative subgroup, and so since half of the elements generate the 2-order subgroup then at most half of the elements are square. In fact exactly half of the elements are square (since squaring each nonsquare element gives a unique square). This means we can assume all squares can be written as βm for some m, and therefore finding the square root is a matter of exponentiating by 2−1(modt). In the event that p≡1(mod4) then things get more complicated because 2−1(modt) does not exist. Let's write p−1 as 2k⋅t with t odd. The case k=0 is impossible, and the case k=1 is what we already described, so consider k≥2. α generates a 2k-order multiplicative subgroup and β generates the odd t-order multiplicative subgroup. Then every element a∈Fp×​ can be written as αi⋅βj for i∈Z2k​ and j∈Zt​. If the element is a square, then there exists some b=a​ which can be written b=αi′⋅βj′ for i′∈Z2k​ and j′∈Zt​. This means that a=b2=α2i′⋅β2j′, therefore we have i≡2i′(mod2k), and j≡2j′(modt). i would have to be even in this case because otherwise it would be impossible to have i≡2i′(mod2k) for any i′. In the case that a is not a square, then i is odd, and so half of all elements are squares. In order to compute the square root, we can first raise the element a=αi⋅βj to the power t to \"kill\" the t-order component, giving at=αit(mod2)k⋅βjt(modt)=αit(mod2)k and then raise this result to the power t−1(mod2k) to undo the effect of the original exponentiation on the 2k-order component: (αitmod2k)t−1(mod2k)=αi (since t is relatively prime to 2k). This leaves bare the αi value which we can trivially handle. We can similarly kill the 2k-order component to obtain βj⋅2−1(modt), and put the values together to obtain the square root. It turns out that in the cases k=2,3 there are simpler algorithms that merge several of these exponentiations together for efficiency. For other values of k, the only known way is to manually extract i by squaring until you obtain the identity for every single bit of i. This is the essence of the Tonelli-Shanks square root algorithm and describes the general strategy. (There is another square root algorithm that uses quadratic extension fields, but it doesn't pay off in efficiency until the prime becomes quite large.)","breadcrumbs":"Background Material » Fields » Square roots","id":"160","title":"Square roots"},"161":{"body":"In the previous sections we wrote p−1=2k⋅t with t odd, and stated that an element α∈Fp×​ generated the 2k-order subgroup. For convenience, let's denote n:=2k. The elements {1,α,…,αn−1} are known as the nth roots of unity . The primitive root of unity , ω, is an nth root of unity such that ωi=1 except when i≡0(modn). Important notes: If α is an nth root of unity, α satisfies αn−1=0. If α=1, then 1+α+α2+⋯+αn−1=0. Equivalently, the roots of unity are solutions to the equation Xn−1=(X−1)(X−α)(X−α2)⋯(X−αn−1). ω2n​+i=−ωi​ (\"Negation lemma\") . Proof: ωn=1​⟹ωn−1=0⟹(ωn/2+1)(ωn/2−1)=0.​ Since the order of ω is n, ωn/2=1. Therefore, ωn/2=−1. (ω2n​+i)2=(ωi)2​ (\"Halving lemma\") . Proof: (ω2n​+i)2=ωn+2i=ωn⋅ω2i=ω2i=(ωi)2. In other words, if we square each element in the nth roots of unity, we would get back only half the elements, {(ωni​)2}={ωn/2​} (i.e. the 2n​th roots of unity). There is a two-to-one mapping between the elements and their squares.","breadcrumbs":"Background Material » Fields » Roots of unity","id":"161","title":"Roots of unity"},"162":{"body":"Friedman, R. (n.d.) \"Cyclic Groups and Elementary Number Theory II\" (p. 5).","breadcrumbs":"Background Material » Fields » References","id":"162","title":"References"},"163":{"body":"Let A(X) be a polynomial over Fp​ with formal indeterminate X. As an example, A(X)=a0​+a1​X+a2​X2+a3​X3 defines a degree-3 polynomial. a0​ is referred to as the constant term. Polynomials of degree n−1 have n coefficients. We will often want to compute the result of replacing the formal indeterminate X with some concrete value x, which we denote by A(x). In mathematics this is commonly referred to as \"evaluating A(X) at a point x\". The word \"point\" here stems from the geometrical usage of polynomials in the form y=A(x), where (x,y) is the coordinate of a point in two-dimensional space. However, the polynomials we deal with are almost always constrained to equal zero, and x will be an element of some field . This should not be confused with points on an elliptic curve , which we also make use of, but never in the context of polynomial evaluation. Important notes: Multiplication of polynomials produces a product polynomial that is the sum of the degrees of its factors. Polynomial division subtracts from the degree. deg(A(X)B(X))=deg(A(X))+deg(B(X)), deg(A(X)/B(X))=deg(A(X))−deg(B(X)). Given a polynomial A(X) of degree n−1, if we obtain n evaluations of the polynomial at distinct points then these evaluations perfectly define the polynomial. In other words, given these evaluations we can obtain a unique polynomial A(X) of degree n−1 via polynomial interpolation. [a0​,a1​,⋯,an−1​] is the coefficient representation of the polynomial A(X). Equivalently, we could use its evaluation representation [(x0​,A(x0​)),(x1​,A(x1​)),⋯,(xn−1​,A(xn−1​))] at n distinct points. Either representation uniquely specifies the same polynomial. (aside) Horner's rule Horner's rule allows for efficient evaluation of a polynomial of degree n−1, using only n−1 multiplications and n−1 additions. It is the following identity: a0​​+a1​X+a2​X2+⋯+an−1​Xn−1=a0​+X(a1​+X(a2​+⋯+X(an−2​+Xan−1​))),​","breadcrumbs":"Background Material » Polynomials » Polynomials","id":"163","title":"Polynomials"},"164":{"body":"The FFT is an efficient way of converting between the coefficient and evaluation representations of a polynomial. It evaluates the polynomial at the nth roots of unity {ω0,ω1,⋯,ωn−1}, where ω is a primitive nth root of unity. By exploiting symmetries in the roots of unity, each round of the FFT reduces the evaluation into a problem only half the size. Most commonly we use polynomials of length some power of two, n=2k, and apply the halving reduction recursively.","breadcrumbs":"Background Material » Polynomials » Fast Fourier Transform (FFT)","id":"164","title":"Fast Fourier Transform (FFT)"},"165":{"body":"In the coefficient representation, it takes O(n2) operations to multiply two polynomials A(X)⋅B(X)=C(X): A(X)B(X)C(X)​=a0​+a1​X+a2​X2+⋯+an−1​Xn−1,=b0​+b1​X+b2​X2+⋯+bn−1​Xn−1,=a0​⋅(b0​+b1​X+b2​X2+⋯+bn−1​Xn−1)+a1​X⋅(b0​+b1​X+b2​X2+⋯+bn−1​Xn−1)+⋯+an−1​Xn−1⋅(b0​+b1​X+b2​X2+⋯+bn−1​Xn−1),​ where each of the n terms in the first polynomial has to be multiplied by the n terms of the second polynomial. In the evaluation representation, however, polynomial multiplication only requires O(n) operations: ABC​:{(x0​,A(x0​)),(x1​,A(x1​)),⋯,(xn−1​,A(xn−1​))},:{(x0​,B(x0​)),(x1​,B(x1​)),⋯,(xn−1​,B(xn−1​))},:{(x0​,A(x0​)B(x0​)),(x1​,A(x1​)B(x1​)),⋯,(xn−1​,A(xn−1​)B(xn−1​))},​ where each evaluation is multiplied pointwise. This suggests the following strategy for fast polynomial multiplication: Evaluate polynomials at all n points; Perform fast pointwise multiplication in the evaluation representation (O(n)); Convert back to the coefficient representation. The challenge now is how to evaluate and interpolate the polynomials efficiently. Naively, evaluating a polynomial at n points would require O(n2) operations (we use the O(n) Horner's rule at each point): ⎣⎡​A(1)A(ω)A(ω2)⋮A(ωn−1)​⎦⎤​=⎣⎡​111⋮1​1ωω2⋮ωn−1​1ω2ω2⋅2⋮ω2(n−1)​………⋯​1ωn−1ω2⋅(n−1)⋮ω(n−1)2​⎦⎤​⋅⎣⎡​a0​a1​a2​⋮an−1​​⎦⎤​. For convenience, we will denote the matrices above as: A^=Vω​⋅A. (A^ is known as the Discrete Fourier Transform of A; Vω​ is also called the Vandermonde matrix .)","breadcrumbs":"Background Material » Polynomials » Motivation: Fast polynomial multiplication","id":"165","title":"Motivation: Fast polynomial multiplication"},"166":{"body":"Our strategy is to divide a DFT of size n into two interleaved DFTs of size n/2. Given the polynomial A(X)=a0​+a1​X+a2​X2+⋯+an−1​Xn−1, we split it up into even and odd terms: Aeven​Aodd​​=a0​+a2​X+⋯+an−2​X2n​−1,=a1​+a3​X+⋯+an−1​X2n​−1.​ To recover the original polynomial, we do A(X)=Aeven​(X2)+XAodd​(X2). Trying this out on points ωni​ and ωn2n​+i​, i∈[0..2n​−1], we start to notice some symmetries: A(ωni​)A(ωn2n​+i​)​=Aeven​((ωni​)2)+ωni​Aodd​((ωni​)2),=Aeven​((ωn2n​+i​)2)+ωn2n​+i​Aodd​((ωn2n​+i​)2)=Aeven​((−ωni​)2)−ωni​Aodd​((−ωni​)2)←(negation lemma)=Aeven​((ωni​)2)−ωni​Aodd​((ωni​)2).​ Notice that we are only evaluating Aeven​(X) and Aodd​(X) over half the domain {(ωn0​)2,(ωn​)2,⋯,(ωn2n​−1​)2}={ωn/2i​},i=[0..2n​−1] (halving lemma). This gives us all the terms we need to reconstruct A(X) over the full domain {ω0,ω,⋯,ωn−1}: which means we have transformed a length-n DFT into two length-2n​ DFTs. We choose n=2k to be a power of two (by zero-padding if needed), and apply this divide-and-conquer strategy recursively. By the Master Theorem [1] , this gives us an evaluation algorithm with O(nlog2​n) operations, also known as the Fast Fourier Transform (FFT).","breadcrumbs":"Background Material » Polynomials » The (radix-2) Cooley-Tukey algorithm","id":"166","title":"The (radix-2) Cooley-Tukey algorithm"},"167":{"body":"So we've evaluated our polynomials and multiplied them pointwise. What remains is to convert the product from the evaluation representation back to coefficient representation. To do this, we simply call the FFT on the evaluation representation. However, this time we also: replace ωi by ω−i in the Vandermonde matrix, and multiply our final result by a factor of 1/n. In other words: A=n1​Vω−1​⋅A^. (To understand why the inverse FFT has a similar form to the FFT, refer to Slide 13-1 of [2] . The below image was also taken from [2] .)","breadcrumbs":"Background Material » Polynomials » Inverse FFT","id":"167","title":"Inverse FFT"},"168":{"body":"The Schwartz-Zippel lemma informally states that \"different polynomials are different at most points.\" Formally, it can be written as follows: Let p(x1​,x2​,⋯,xn​) be a nonzero polynomial of n variables with degree d. Let S be a finite set of numbers with at least d elements in it. If we choose random α1​,α1​,⋯,αn​ from S, Pr[p(α1​,α2​,⋯,αn​)=0]≤∣S∣d​. In the familiar univariate case p(X), this reduces to saying that a nonzero polynomial of degree d has at most d roots. The Schwartz-Zippel lemma is used in polynomial equality testing. Given two multi-variate polynomials p1​(x1​,⋯,xn​) and p2​(x1​,⋯,xn​) of degrees d1​,d2​ respectively, we can test if p1​(α1​,⋯,αn​)−p2​(α1​,⋯,αn​)=0 for random α1​,⋯,αn​←S, where the size of S is at least ∣S∣≥(d1​+d2​). If the two polynomials are identical, this will always be true, whereas if the two polynomials are different then the equality holds with probability at most ∣S∣max(d1​,d2​)​.","breadcrumbs":"Background Material » Polynomials » The Schwartz-Zippel lemma","id":"168","title":"The Schwartz-Zippel lemma"},"169":{"body":"Consider the order-n multiplicative subgroup H with primitive root of unity ω. For all ωi∈H,i∈[n−1], we have (ωi)n=(ωn)i=(ω0)i=1. In other words, every element of H fulfils the equation ZH​(X)​=Xn−1=(X−ω0)(X−ω1)(X−ω2)⋯(X−ωn−1),​ meaning every element is a root of ZH​(X). We call ZH​(X) the vanishing polynomial over H because it evaluates to zero on all elements of H. This comes in particularly handy when checking polynomial constraints. For instance, to check that A(X)+B(X)=C(X) over H, we simply have to check that A(X)+B(X)−C(X) is some multiple of ZH​(X). In other words, if dividing our constraint by the vanishing polynomial still yields some polynomial ZH​(X)A(X)+B(X)−C(X)​=H(X), we are satisfied that A(X)+B(X)−C(X)=0 over H.","breadcrumbs":"Background Material » Polynomials » Vanishing polynomial","id":"169","title":"Vanishing polynomial"},"17":{"body":"The cost-model binary takes high-level parameters for a circuit design, and estimates the verification cost, as well as resulting proof size. Usage: cargo run --example cost-model -- [OPTIONS] k Positional arguments: k 2^K bound on the number of rows. Optional arguments: -h, --help Print this message. -a, --advice R[,R..] An advice column with the given rotations. May be repeated. -i, --instance R[,R..] An instance column with the given rotations. May be repeated. -f, --fixed R[,R..] A fixed column with the given rotations. May be repeated. -g, --gate-degree D Maximum degree of the custom gates. -l, --lookup N,I,T A lookup over N columns with max input degree I and max table degree T. May be repeated. -p, --permutation N A permutation over N columns. May be repeated. For example, to estimate the cost of a circuit with three advice columns and one fixed column (with various rotations), and a maximum gate degree of 4: > cargo run --example cost-model -- -a 0,1 -a 0 -a-0,-1,1 -f 0 -g 4 11 Finished dev [unoptimized + debuginfo] target(s) in 0.03s Running `target/debug/examples/cost-model -a 0,1 -a 0 -a 0,-1,1 -f 0 -g 4 11`\nCircuit { k: 11, max_deg: 4, advice_columns: 3, lookups: 0, permutations: [], column_queries: 7, point_sets: 3, estimator: Estimator,\n}\nProof size: 1440 bytes\nVerification: at least 81.689ms","breadcrumbs":"User Documentation » Developer tools » Cost estimator","id":"17","title":"Cost estimator"},"170":{"body":"TODO: explain what a basis is in general (briefly). Polynomials are commonly written in the monomial basis (e.g. X,X2,...Xn). However, when working over a multiplicative subgroup of order n, we find a more natural expression in the Lagrange basis. Consider the order-n multiplicative subgroup H with primitive root of unity ω. The Lagrange basis corresponding to this subgroup is a set of functions {Li​}i=0n−1​, where Li​(ωj)={10​if i=j,otherwise.​ We can write this more compactly as Li​(ωj)=δij​, where δ is the Kronecker delta function. Now, we can write our polynomial as a linear combination of Lagrange basis functions, A(X)=i=0∑n−1​ai​Li​(X),X∈H, which is equivalent to saying that p(X) evaluates to a0​ at ω0, to a1​ at ω1, to a2​ at ω2,⋯, and so on. When working over a multiplicative subgroup, the Lagrange basis function has a convenient sparse representation of the form Li​(X)=X−ωici​⋅(Xn−1)​, where ci​ is the barycentric weight. (To understand how this form was derived, refer to [3] .) For i=0, we have c=1/n⟹L0​(X)=n1​X−1(Xn−1)​. Suppose we are given a set of evaluation points {x0​,x1​,⋯,xn−1​}. Since we cannot assume that the xi​'s form a multiplicative subgroup, we consider also the Lagrange polynomials Li​'s in the general case. Then we can construct: Li​(X)=j=i∏​xi​−xj​X−xj​​,i∈[0..n−1]. Here, every X=xj​=xi​ will produce a zero numerator term (xj​−xj​), causing the whole product to evaluate to zero. On the other hand, X=xi​ will evaluate to xi​−xj​xi​−xj​​ at every term, resulting in an overall product of one. This gives the desired Kronecker delta behaviour Li​(xj​)=δij​ on the set {x0​,x1​,⋯,xn−1​}.","breadcrumbs":"Background Material » Polynomials » Lagrange basis functions","id":"170","title":"Lagrange basis functions"},"171":{"body":"Given a polynomial in its evaluation representation A:{(x0​,A(x0​)),(x1​,A(x1​)),⋯,(xn−1​,A(xn−1​))}, we can reconstruct its coefficient form in the Lagrange basis: A(X)=i=0∑n−1​A(xi​)Li​(X), where X∈{x0​,x1​,⋯,x1−n​}.","breadcrumbs":"Background Material » Polynomials » Lagrange interpolation","id":"171","title":"Lagrange interpolation"},"172":{"body":"Dasgupta, S., Papadimitriou, C. H., & Vazirani, U. V. (2008). \"Algorithms\" (ch. 2). New York: McGraw-Hill Higher Education. Golin, M. (2016). \"The Fast Fourier Transform and Polynomial Multiplication\" [lecture notes], COMP 3711H Design and Analysis of Algorithms, Hong Kong University of Science and Technology. Berrut, J. and Trefethen, L. (2004). \"Barycentric Lagrange Interpolation.\"","breadcrumbs":"Background Material » Polynomials » References","id":"172","title":"References"},"173":{"body":"In the section Inverses and groups we introduced the concept of groups . A group has an identity and a group operation. In this section we will write groups additively, i.e. the identity is O and the group operation is +. Some groups can be used as cryptographic groups . At the risk of oversimplifying, this means that the problem of finding a discrete logarithm of a group element P to a given base G, i.e. finding x such that P=[x]G, is hard in general.","breadcrumbs":"Background Material » Cryptographic groups » Cryptographic groups","id":"173","title":"Cryptographic groups"},"174":{"body":"The Pedersen commitment [ P99 ] is a way to commit to a secret message in a verifiable way. It uses two random public generators G,H∈G, where G is a cryptographic group of order p. A random secret r is chosen in Zq​, and the message to commit to m is from any subset of Zq​. The commitment is c=Commit(m,r)=[m]G+[r]H. To open the commitment, the committer reveals m and r, thus allowing anyone to verify that c is indeed a commitment to m. Notice that the Pedersen commitment scheme is homomorphic: Commit(m,r)+Commit(m′,r′)​=[m]G+[r]H+[m′]G+[r′]H=[m+m′]G+[r+r′]H=Commit(m+m′,r+r′).​ Assuming the discrete log assumption holds, Pedersen commitments are also perfectly hiding and computationally binding: hiding : the adversary chooses messages m0​,m1​. The committer commits to one of these messages c=Commit(mb​;r),b∈{0,1}. Given c, the probability of the adversary guessing the correct b is no more than 21​. binding : the adversary cannot pick two different messages m0​=m1​, and randomness r0​,r1​, such that Commit(m0​,r0​)=Commit(m1​,r1​).","breadcrumbs":"Background Material » Cryptographic groups » Pedersen commitment","id":"174","title":"Pedersen commitment"},"175":{"body":"We can use a variant of the Pedersen commitment scheme to commit to multiple messages at once, m=(m1​,⋯,mn​). This time, we'll have to sample a corresponding number of random public generators G=(G0​,⋯,Gn−1​), along with a single random generator H as before (for use in hiding). Then, our commitment scheme is: Commit(m;r)​=Commit((m0​,⋯,mn−1​);r)=[r]H+[m0​]G0​+⋯+[mn−1​]Gn−1​=[r]H+i=0∑n−1​[mi​]Gi​.​ TODO: is this positionally binding?","breadcrumbs":"Background Material » Cryptographic groups » Vector Pedersen commitment","id":"175","title":"Vector Pedersen commitment"},"176":{"body":"An example of a protocol that uses cryptographic groups is Diffie--Hellman key agreement [ DH1976 ]. The Diffie--Hellman protocol is a method for two users, Alice and Bob, to generate a shared private key. It proceeds as follows: Alice and Bob publicly agree on two prime numbers, p and G, where p is large and G is a primitive root (modp). (Note that g is a generator of the group Fp×​.) Alice chooses a large random number a as her private key. She computes her public key A=[a]G(modp), and sends A to Bob. Similarly, Bob chooses a large random number b as his private key. He computes his public key B=[b]G(modp), and sends B to Alice. Now both Alice and Bob compute their shared key K=[ab]G(modp), which Alice computes as K=[a]B(modp)=[a]([b]G)(modp), and Bob computes as K=[b]A(modp)=[b]([a]G)(modp). A potential eavesdropper would need to derive K=[ab]g(modp) knowing only g,p,A=[a]G, and B=[b]G: in other words, they would need to either get the discrete logarithm a from A=[a]G or b from B=[b]G, which we assume to be computationally infeasible in Fp×​. More generally, protocols that use similar ideas to Diffie--Hellman are used throughout cryptography. One way of instantiating a cryptographic group is as an elliptic curve . Before we go into detail on elliptic curves, we'll describe some algorithms that can be used for any group.","breadcrumbs":"Background Material » Cryptographic groups » Diffie--Hellman","id":"176","title":"Diffie--Hellman"},"177":{"body":"","breadcrumbs":"Background Material » Cryptographic groups » Multiscalar multiplication","id":"177","title":"Multiscalar multiplication"},"178":{"body":"Reference: https://jbootle.github.io/Misc/pippenger.pdf","breadcrumbs":"Background Material » Cryptographic groups » TODO: Pippenger's algorithm","id":"178","title":"TODO: Pippenger's algorithm"},"179":{"body":"Elliptic curves constructed over finite fields are another important cryptographic tool. We use elliptic curves because they provide a cryptographic group , i.e. a group in which the discrete logarithm problem (discussed below) is hard. There are several ways to define the curve equation, but for our purposes, let Fp​ be a large (255-bit) field, and then let the set of solutions (x,y) to y2=x3+b for some constant b define the Fp​-rational points on an elliptic curve E(Fp​). These (x,y) coordinates are called \"affine coordinates\". Each of the Fp​-rational points, together with a \"point at infinity\" O that serves as the group identity, can be interpreted as an element of a group. By convention, elliptic curve groups are written additively. \"Three points on a line sum to zero, which is the point at infinity.\" The group addition law is simple: to add two points together, find the line that intersects both points and obtain the third point, and then negate its y-coordinate. The case that a point is being added to itself, called point doubling, requires special handling: we find the line tangent to the point, and then find the single other point that intersects this line and then negate. Otherwise, in the event that a point is being \"added\" to its negation, the result is the point at infinity. The ability to add and double points naturally gives us a way to scale them by integers, called scalars . The number of points on the curve is the group order. If this number is a prime q, then the scalars can be considered as elements of a scalar field , Fq​. Elliptic curves, when properly designed, have an important security property. Given two random elements G,H∈E(Fp​) finding a such that [a]G=H, otherwise known as the discrete log of H with respect to G, is considered computationally infeasible with classical computers. This is called the elliptic curve discrete log assumption. If an elliptic curve group G has prime order q (like the ones used in Halo 2), then it is a finite cyclic group. Recall from the section on groups that this implies it is isomorphic to Z/qZ, or equivalently, to the scalar field Fq​. Each possible generator G fixes the isomorphism; then an element on the scalar side is precisely the discrete log of the corresponding group element with respect to G. In the case of a cryptographically secure elliptic curve, the isomorphism is hard to compute in the G→Fq​ direction because the elliptic curve discrete log problem is hard. It is sometimes helpful to make use of this isomorphism by thinking of group-based cryptographic protocols and algorithms in terms of the scalars instead of in terms of the group elements. This can make proofs and notation simpler. For instance, it has become common in papers on proof systems to use the notation [x] to denote a group element with discrete log x, where the generator is implicit. We also used this idea in the \" distinct-x theorem \", in order to prove correctness of optimizations for elliptic curve scalar multiplication in Sapling, and an endomorphism-based optimization in Appendix C of the original Halo paper .","breadcrumbs":"Background Material » Elliptic curves » Elliptic curves","id":"179","title":"Elliptic curves"},"18":{"body":"Let's start with a simple circuit, to introduce you to the common APIs and how they are used. The circuit will take a public input c, and will prove knowledge of two private inputs a and b such that a2⋅b2=c.","breadcrumbs":"User Documentation » A simple example » A simple example","id":"18","title":"A simple example"},"180":{"body":"","breadcrumbs":"Background Material » Elliptic curves » Curve arithmetic","id":"180","title":"Curve arithmetic"},"181":{"body":"The simplest situation is doubling a point (x0​,y0​). Continuing with our example y2=x3+b, this is done first by computing the derivative λ=dxdy​=2y3x2​. To obtain expressions for (x1​,y1​)=(x0​,y0​)+(x0​,y0​), we consider x1​−x0​−y1​−y0​​=λ​⟹−y1​=λ(x1​−x0​)+y0​⟹y1​=λ(x0​−x1​)−y0​​.​ To get the expression for x1​, we substitute y=λ(x0​−x)−y0​ into the elliptic curve equation: y2=x3+b​⟹(λ(x0​−x)−y0​)2=x3+b⟹x3−λ2x2+⋯=0←(rearranging terms)=(x−x0​)(x−x0​)(x−x1​)←(known roots x0​,x0​,x1​)=x3−(x0​+x0​+x1​)x2+⋯.​ Comparing coefficients for the x2 term gives us λ2=x0​+x0​+x1​⟹x1​=λ2−2x0​​.","breadcrumbs":"Background Material » Elliptic curves » Point doubling","id":"181","title":"Point doubling"},"182":{"body":"This unfortunately requires an expensive inversion of 2y. We can avoid this by arranging our equations to \"defer\" the computation of the inverse, since we often do not need the actual affine (x′,y′) coordinate of the resulting point immediately after an individual curve operation. Let's introduce a third coordinate Z and scale our curve equation by Z3 like so: Z3y2=Z3x3+Z3b Our original curve is just this curve at the restriction Z=1. If we allow the affine point (x,y) to be represented by X=xZ, Y=yZ and Z=0 then we have the homogenous projective curve Y2Z=X3+Z3b. Obtaining (x,y) from (X,Y,Z) is as simple as computing (X/Z,Y/Z) when Z=0. (When Z=0, we are dealing with the point at infinity O:=(0:1:0).) In this form, we now have a convenient way to defer the inversion required by doubling a point. The general strategy is to express x′,y′ as rational functions using x=X/Z and y=Y/Z, rearrange to make their denominators the same, and then take the resulting point (X,Y,Z) to have Z be the shared denominator and X=x′Z,Y=y′Z. Projective coordinates are often, but not always, more efficient than affine coordinates. There may be exceptions to this when either we have a different way to apply Montgomery's trick, or when we're in the circuit setting where multiplications and inversions are about equally as expensive (at least in terms of circuit size). The following shows an example of doubling a point (X,Y,Z)=(xZ,yZ,Z) without an inversion. Substituting with X,Y,Z gives us λ=2y3x2​=2(Y/Z)3(X/Z)2​=2YZ3X2​ and gives us x′y′​=λ2−2x=λ2−Z2X​=4Y2Z29X4​−Z2X​=4Y2Z29X4−8XY2Z​=8Y3Z318X4YZ−16XY3Z2​=λ(x−x′)−y=λ(ZX​−4Y2Z29X4−8XY2Z​)−ZY​=2YZ3X2​(ZX​−4Y2Z29X4−8XY2Z​)−ZY​=2YZ23X3​−8Y3Z327X6−24X3Y2Z​−ZY​=8Y3Z312X3Y2Z−8Y4Z2−27X6+24X3Y2Z​​ Notice how the denominators of x′ and y′ are the same. Thus, instead of computing (x′,y′) we can compute (X,Y,Z) with Z=8Y3Z3 and X,Y set to the corresponding numerators such that X/Z=x′ and Y/Z=y′. This completely avoids the need to perform an inversion when doubling, and something analogous to this can be done when adding two distinct points.","breadcrumbs":"Background Material » Elliptic curves » Projective coordinates","id":"182","title":"Projective coordinates"},"183":{"body":"We now add two points with distinct x-coordinates, P=(x0​,y0​) and Q=(x1​,y1​), where x0​=x1​, to obtain R=P+Q=(x2​,y2​). The line PQ​ has slope λ=fracy1​−y0​x1​−x0​⟹y−y0​=λ⋅(x−x0​). Using the expression for PQ​, we compute y-coordinate −y2​ of −R as: −y2​−y0​=λ⋅(x2​−x0​)⟹y2​=(x0​−x2​)−y0​​. Plugging the expression for PQ​ into the curve equation y2=x3+b yields y2=x3+b​⟹(λ⋅(x−x0​)+y0​)2=x3+b⟹x3−λ2x2+⋯=0←(rearranging terms)=(x−x0​)(x−x1​)(x−x2​)←(known roots x0​,x1​,x2​)=x3−(x0​+x1​+x2​)x2+⋯.​ Comparing coefficients for the x2 term gives us λ2=x0​+x1​+x2​⟹x2​=λ2−x0​−x1​​. Important notes: There exist efficient formulae [1] for point addition that do not have edge cases (so-called \"complete\" formulae) and that unify the addition and doubling cases together. The result of adding a point to its negation using those formulae produces Z=0, which represents the point at infinity. In addition, there are other models like the Jacobian representation where (x,y)=(xZ2,yZ3,Z) where the curve is rescaled by Z6 instead of Z3, and this representation has even more efficient arithmetic but no unified/complete formulae. We can easily compare two curve points (X1​,Y1​,Z1​) and (X2​,Y2​,Z2​) for equality in the homogenous projective coordinate space by \"homogenizing\" their Z-coordinates; the checks become X1​Z2​=X2​Z1​ and Y1​Z2​=Y2​Z1​.","breadcrumbs":"Background Material » Elliptic curves » Point addition","id":"183","title":"Point addition"},"184":{"body":"Imagine that Fp​ has a primitive cube root of unity, or in other words that 3∣p−1 and so an element ζp​ generates a 3-order multiplicative subgroup. Notice that a point (x,y) on our example elliptic curve y2=x3+b has two cousin points: (ζp​x,ζp2​x), because the computation x3 effectively kills the ζ component of the x-coordinate. Applying the map (x,y)↦(ζp​x,y) is an application of an endomorphism over the curve. The exact mechanics involved are complicated, but when the curve has a prime q number of points (and thus a prime \"order\") the effect of the endomorphism is to multiply the point by a scalar in Fq​ which is also a primitive cube root ζq​ in the scalar field.","breadcrumbs":"Background Material » Elliptic curves » Curve endomorphisms","id":"184","title":"Curve endomorphisms"},"185":{"body":"Given a point on the curve P=(x,y), we know that its negation −P=(x,−y) is also on the curve. To uniquely specify a point, we need only encode its x-coordinate along with the sign of its y-coordinate.","breadcrumbs":"Background Material » Elliptic curves » Curve point compression","id":"185","title":"Curve point compression"},"186":{"body":"As mentioned in the Fields section, we can interpret the least significant bit of a field element as its \"sign\", since its additive inverse will always have the opposite LSB. So we record the LSB of the y-coordinate as sign. Pallas and Vesta are defined over the Fp​ and Fq​ fields, which elements can be expressed in 255 bits. This conveniently leaves one unused bit in a 32-byte representation. We pack the y-coordinate sign bit into the highest bit in the representation of the x-coordinate: <----------------------------------- x --------------------------------->\nEnc(P) = [_ _ _ _ _ _ _ _] [_ _ _ _ _ _ _ _] ... [_ _ _ _ _ _ _ _] [_ _ _ _ _ _ _ sign] ^ <-------------------------------------> ^ LSB 30 bytes MSB The \"point at infinity\" O that serves as the group identity, does not have an affine (x,y) representation. However, it turns out that there are no points on either the Pallas or Vesta curve with x=0 or y=0. We therefore use the \"fake\" affine coordinates (0,0) to encode O, which results in the all-zeroes 32-byte array.","breadcrumbs":"Background Material » Elliptic curves » Serialization","id":"186","title":"Serialization"},"187":{"body":"When deserializing a compressed curve point, we first read the most significant bit as ysign, the sign of the y-coordinate. Then, we set this bit to zero to recover the original x-coordinate. If x=0,y=0, we return the \"point at infinity\" O. Otherwise, we proceed to compute y=x3+b​. Here, we read the least significant bit of y as sign. If sign == ysign, we already have the correct sign and simply return the curve point (x,y). Otherwise, we negate y and return (x,−y).","breadcrumbs":"Background Material » Elliptic curves » Deserialization","id":"187","title":"Deserialization"},"188":{"body":"Let Ep​ be an elliptic curve over a finite field Fp​, where p is a prime. We denote this by Ep​/Fp​. and we denote the group of points of Ep​ over Fp​, with order q=#E(Fp​). For this curve, we call Fp​ the \"base field\" and Fq​ the \"scalar field\". We instantiate our proof system over the elliptic curve Ep​/Fp​. This allows us to prove statements about Fq​-arithmetic circuit satisfiability. (aside) If our curve Ep​ is over Fp​, why is the arithmetic circuit instead in Fq​? The proof system is basically working on encodings of the scalars in the circuit (or more precisely, commitments to polynomials whose coefficients are scalars). The scalars are in Fq​ when their encodings/commitments are elliptic curve points in Ep​/Fp​. However, most of the verifier's arithmetic computations are over the base field Fp​, and are thus efficiently expressed as an Fp​-arithmetic circuit. (aside) Why are the verifier's computations (mainly) over Fp​? The Halo 2 verifier actually has to perform group operations using information output by the circuit. Group operations like point doubling and addition use arithmetic in Fp​, because the coordinates of points are in Fp​. This motivates us to construct another curve with scalar field Fp​, which has an Fp​-arithmetic circuit that can efficiently verify proofs from the first curve. As a bonus, if this second curve had base field Eq​/Fq​, it would generate proofs that could be efficiently verified in the first curve's Fq​-arithmetic circuit. In other words, we instantiate a second proof system over Eq​/Fq​, forming a 2-cycle with the first:","breadcrumbs":"Background Material » Elliptic curves » Cycles of curves","id":"188","title":"Cycles of curves"},"189":{"body":"Reference: https://github.com/zcash/pasta","breadcrumbs":"Background Material » Elliptic curves » TODO: Pallas-Vesta curves","id":"189","title":"TODO: Pallas-Vesta curves"},"19":{"body":"Firstly, we need to define the instructions that our circuit will rely on. Instructions are the boundary between high-level gadgets and the low-level circuit operations. Instructions may be as coarse or as granular as desired, but in practice you want to strike a balance between an instruction being large enough to effectively optimize its implementation, and small enough that it is meaningfully reusable. For our circuit, we will use three instructions: Load a private number into the circuit. Multiply two numbers. Expose a number as a public input to the circuit. We also need a type for a variable representing a number. Instruction interfaces provide associated types for their inputs and outputs, to allow the implementations to represent these in a way that makes the most sense for their optimization goals. trait NumericInstructions: Chip { /// Variable representing a number. type Num; /// Loads a number into the circuit as a private input. fn load_private(&self, layouter: impl Layouter, a: Option) -> Result; /// Loads a number into the circuit as a fixed constant. fn load_constant(&self, layouter: impl Layouter, constant: F) -> Result; /// Returns `c = a * b`. fn mul( &self, layouter: impl Layouter, a: Self::Num, b: Self::Num, ) -> Result; /// Exposes a number as a public input to the circuit. fn expose_public( &self, layouter: impl Layouter, num: Self::Num, row: usize, ) -> Result<(), Error>;\n}","breadcrumbs":"User Documentation » A simple example » Define instructions","id":"19","title":"Define instructions"},"190":{"body":"Sometimes it is useful to be able to produce a random point on an elliptic curve Ep​/Fp​ corresponding to some input, in such a way that no-one will know its discrete logarithm (to any other base). This is described in detail in the Internet draft on Hashing to Elliptic Curves . Several algorithms can be used depending on efficiency and security requirements. The framework used in the Internet Draft makes use of several functions: hash_to_field: takes a byte sequence input and maps it to a element in the base field Fp​ map_to_curve: takes an Fp​ element and maps it to Ep​.","breadcrumbs":"Background Material » Elliptic curves » Hashing to curves","id":"190","title":"Hashing to curves"},"191":{"body":"Reference: https://eprint.iacr.org/2019/403.pdf","breadcrumbs":"Background Material » Elliptic curves » TODO: Simplified SWU","id":"191","title":"TODO: Simplified SWU"},"192":{"body":"Renes, J., Costello, C., & Batina, L. (2016, May). \"Complete addition formulas for prime order elliptic curves.\" In Annual International Conference on the Theory and Applications of Cryptographic Techniques (pp. 403-428). Springer, Berlin, Heidelberg.","breadcrumbs":"Background Material » Elliptic curves » References","id":"192","title":"References"},"193":{"body":"We want to commit to some polynomial p(X)∈Fp​[X], and be able to provably evaluate the committed polynomial at arbitrary points. The naive solution would be for the prover to simply send the polynomial's coefficients to the verifier: however, this requires O(n) communication. Our polynomial commitment scheme gets the job done using O(logn) communication.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Polynomial commitment using inner product argument","id":"193","title":"Polynomial commitment using inner product argument"},"194":{"body":"Given a parameter d=2k, we generate the common reference string σ=(G,G,H,Fp​) defining certain constants for this scheme: G is a group of prime order p; G∈Gd is a vector of d random group elements; H∈G is a random group element; and Fp​ is the finite field of order p.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Setup","id":"194","title":"Setup"},"195":{"body":"The Pedersen vector commitment Commit is defined as Commit(σ,p(X);r)=⟨a,G⟩+[r]H, for some polynomial p(X)∈Fp​[X] and some blinding factor r∈Fp​. Here, each element of the vector ai​∈Fp​ is the coefficient for the ith degree term of p(X), and p(X) is of maximal degree d−1.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Commit","id":"195","title":"Commit"},"196":{"body":"The modified inner product argument is an argument of knowledge for the relation {((P,x,v);(a,r)):P=⟨a,G⟩+[r]H,v=⟨a,b⟩}​, where b=(1,x,x2,⋯,xd−1) is composed of increasing powers of the evaluation point x. This allows a prover to demonstrate to a verifier that the polynomial contained “inside” the commitment P evaluates to v at x, and moreover, that the committed polynomial has maximum degree d−1. The inner product argument proceeds in k=log2​d rounds. For our purposes, it is sufficient to know about its final outputs, while merely providing intuition about the intermediate rounds. (Refer to Section 3 in the Halo paper for a full explanation.) Before beginning the argument, the verifier selects a random group element U and sends it to the prover. We initialize the argument at round k, with the vectors a(k):=a, G(k):=G and b(k):=b. In each round j=k,k−1,⋯,1: the prover computes two values Lj​ and Rj​ by taking some inner product of a(j) with G(j) and b(j). Note that are in some sense \"cross-terms\": the lower half of a is used with the higher half of G and b, and vice versa: Lj​Rj​​=⟨alo(j)​,Ghi(j)​⟩+[lj​]H+[⟨alo(j)​,bhi(j)​⟩]U=⟨ahi(j)​,Glo(j)​⟩+[lj​]H+[⟨ahi(j)​,blo(j)​⟩]U​ the verifier issues a random challenge uj​; the prover uses uj​ to compress the lower and higher halves of a(j), thus producing a new vector of half the original length a(j−1)=ahi(j)​⋅uj−1​+alo(j)​⋅uj​. The vectors G(j) and b(j) are similarly compressed to give G(j−1) and b(j−1). a(j−1), G(j−1) and b(j−1) are input to the next round j−1. Note that at the end of the last round j=1, we are left with a:=a(0), G:=G(0), b:=b(0), each of length 1. The intuition is that these final scalars, together with the challenges {uj​} and \"cross-terms\" {Lj​,Rj​} from each round, encode the compression in each round. Since the prover did not know the challenges U,{uj​} in advance, they would have been unable to manipulate the round compressions. Thus, checking a constraint on these final terms should enforce that the compression had been performed correctly, and that the original a satisfied the relation before undergoing compression. Note that G,b are simply rearrangements of the publicly known G,b, with the round challenges {uj​} mixed in: this means the verifier can compute G,b independently and verify that the prover had provided those same values.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Open (prover) and OpenVerify (verifier)","id":"196","title":"Open (prover) and OpenVerify (verifier)"},"197":{"body":"Alternative terms: Induction; Accumulation scheme; Proof-carrying data However, the computation of G requires a length-2k multiexponentiation ⟨G,s⟩, where s is composed of the round challenges u1​,⋯,uk​ arranged in a binary counting structure. This is the linear-time computation that we want to amortise across a batch of proof instances. Instead of computing G, notice that we can express G as a commitment to a polynomial G=Commit(σ,g(X,u1​,⋯,uk​)), where g(X,u1​,⋯,uk​):=∏i=1k​(ui​+ui−1​X2i−1) is a polynomial with degree 2k−1. Since G is a commitment, it can be checked in an inner product argument. The verifier circuit witnesses G and brings G,u1​,⋯,uk​ out as public inputs to the proof π. The next verifier instance checks π using the inner product argument; this includes checking that G=Commit(g(X,u1​,⋯,uk​)) evaluates at some random point to the expected value for the given challenges u1​,⋯,uk​. Recall from the previous section that this check only requires logd work. At the end of checking π and G, the circuit is left with a new G′, along with the u1′​,⋯,uk′​ challenges sampled for the check. To fully accept π as valid, we should perform a linear-time computation of G′=⟨G,s′⟩. Once again, we delay this computation by witnessing G′ and bringing G,u1​,⋯,uk​ out as public inputs to the proof π. This goes on from one proof instance to the next, until we are satisfied with the size of our batch of proofs. We finally perform a single linear-time computation, thus deciding the validity of the whole batch. We recall from the section Cycles of curves that we can instantiate this protocol over a two-cycle, where a proof produced by one curve is efficiently verified in the circuit of the other curve. However, some of these verifier checks can actually be efficiently performed in the native circuit; these are \"deferred\" to the next native circuit (see diagram below) instead of being immediately passed over to the other curve.","breadcrumbs":"Background Material » Recursion » Recursion","id":"197","title":"Recursion"},"2":{"body":"Requires Rust 1.56.1 or higher. Minimum supported Rust version can be changed in the future, but it will be done with a minor version bump.","breadcrumbs":"halo2 » Minimum Supported Rust Version","id":"2","title":"Minimum Supported Rust Version"},"20":{"body":"For our circuit, we will build a chip that provides the above numeric instructions for a finite field. /// The chip that will implement our instructions! Chips store their own\n/// config, as well as type markers if necessary.\nstruct FieldChip { config: FieldConfig, _marker: PhantomData,\n} Every chip needs to implement the Chip trait. This defines the properties of the chip that a Layouter may rely on when synthesizing a circuit, as well as enabling any initial state that the chip requires to be loaded into the circuit. impl Chip for FieldChip { type Config = FieldConfig; type Loaded = (); fn config(&self) -> &Self::Config { &self.config } fn loaded(&self) -> &Self::Loaded { &() }\n}","breadcrumbs":"User Documentation » A simple example » Define a chip implementation","id":"20","title":"Define a chip implementation"},"21":{"body":"The chip needs to be configured with the columns, permutations, and gates that will be required to implement all of the desired instructions. /// Chip state is stored in a config struct. This is generated by the chip\n/// during configuration, and then stored inside the chip.\n#[derive(Clone, Debug)]\nstruct FieldConfig { /// For this chip, we will use two advice columns to implement our instructions. /// These are also the columns through which we communicate with other parts of /// the circuit. advice: [Column; 2], /// This is the public input (instance) column. instance: Column, // We need a selector to enable the multiplication gate, so that we aren't placing // any constraints on cells where `NumericInstructions::mul` is not being used. // This is important when building larger circuits, where columns are used by // multiple sets of instructions. s_mul: Selector,\n} impl FieldChip { fn construct(config: >::Config) -> Self { Self { config, _marker: PhantomData, } } fn configure( meta: &mut ConstraintSystem, advice: [Column; 2], instance: Column, constant: Column, ) -> >::Config { meta.enable_equality(instance); meta.enable_constant(constant); for column in &advice { meta.enable_equality(*column); } let s_mul = meta.selector(); // Define our multiplication gate! meta.create_gate(\"mul\", |meta| { // To implement multiplication, we need three advice cells and a selector // cell. We arrange them like so: // // | a0 | a1 | s_mul | // |-----|-----|-------| // | lhs | rhs | s_mul | // | out | | | // // Gates may refer to any relative offsets we want, but each distinct // offset adds a cost to the proof. The most common offsets are 0 (the // current row), 1 (the next row), and -1 (the previous row), for which // `Rotation` has specific constructors. let lhs = meta.query_advice(advice[0], Rotation::cur()); let rhs = meta.query_advice(advice[1], Rotation::cur()); let out = meta.query_advice(advice[0], Rotation::next()); let s_mul = meta.query_selector(s_mul); // Finally, we return the polynomial expressions that constrain this gate. // For our multiplication gate, we only need a single polynomial constraint. // // The polynomial expressions returned from `create_gate` will be // constrained by the proving system to equal zero. Our expression // has the following properties: // - When s_mul = 0, any value is allowed in lhs, rhs, and out. // - When s_mul != 0, this constrains lhs * rhs = out. vec![s_mul * (lhs * rhs - out)] }); FieldConfig { advice, instance, s_mul, } }\n}","breadcrumbs":"User Documentation » A simple example » Configure the chip","id":"21","title":"Configure the chip"},"22":{"body":"/// A variable representing a number.\n#[derive(Clone)]\nstruct Number(AssignedCell); impl NumericInstructions for FieldChip { type Num = Number; fn load_private( &self, mut layouter: impl Layouter, value: Option, ) -> Result { let config = self.config(); layouter.assign_region( || \"load private\", |mut region| { region .assign_advice( || \"private input\", config.advice[0], 0, || value.ok_or(Error::Synthesis), ) .map(Number) }, ) } fn load_constant( &self, mut layouter: impl Layouter, constant: F, ) -> Result { let config = self.config(); layouter.assign_region( || \"load constant\", |mut region| { region .assign_advice_from_constant(|| \"constant value\", config.advice[0], 0, constant) .map(Number) }, ) } fn mul( &self, mut layouter: impl Layouter, a: Self::Num, b: Self::Num, ) -> Result { let config = self.config(); layouter.assign_region( || \"mul\", |mut region: Region<'_, F>| { // We only want to use a single multiplication gate in this region, // so we enable it at region offset 0; this means it will constrain // cells at offsets 0 and 1. config.s_mul.enable(&mut region, 0)?; // The inputs we've been given could be located anywhere in the circuit, // but we can only rely on relative offsets inside this region. So we // assign new cells inside the region and constrain them to have the // same values as the inputs. a.0.copy_advice(|| \"lhs\", &mut region, config.advice[0], 0)?; b.0.copy_advice(|| \"rhs\", &mut region, config.advice[1], 0)?; // Now we can assign the multiplication result, which is to be assigned // into the output position. let value = a.0.value().and_then(|a| b.0.value().map(|b| *a * *b)); // Finally, we do the assignment to the output, returning a // variable to be used in another part of the circuit. region .assign_advice( || \"lhs * rhs\", config.advice[0], 1, || value.ok_or(Error::Synthesis), ) .map(Number) }, ) } fn expose_public( &self, mut layouter: impl Layouter, num: Self::Num, row: usize, ) -> Result<(), Error> { let config = self.config(); layouter.constrain_instance(num.0.cell(), config.instance, row) }\n}","breadcrumbs":"User Documentation » A simple example » Implement chip traits","id":"22","title":"Implement chip traits"},"23":{"body":"Now that we have the instructions we need, and a chip that implements them, we can finally build our circuit! /// The full circuit implementation.\n///\n/// In this struct we store the private input variables. We use `Option` because\n/// they won't have any value during key generation. During proving, if any of these\n/// were `None` we would get an error.\n#[derive(Default)]\nstruct MyCircuit { constant: F, a: Option, b: Option,\n} impl Circuit for MyCircuit { // Since we are using a single chip for everything, we can just reuse its config. type Config = FieldConfig; type FloorPlanner = SimpleFloorPlanner; fn without_witnesses(&self) -> Self { Self::default() } fn configure(meta: &mut ConstraintSystem) -> Self::Config { // We create the two advice columns that FieldChip uses for I/O. let advice = [meta.advice_column(), meta.advice_column()]; // We also need an instance column to store public inputs. let instance = meta.instance_column(); // Create a fixed column to load constants. let constant = meta.fixed_column(); FieldChip::configure(meta, advice, instance, constant) } fn synthesize( &self, config: Self::Config, mut layouter: impl Layouter, ) -> Result<(), Error> { let field_chip = FieldChip::::construct(config); // Load our private values into the circuit. let a = field_chip.load_private(layouter.namespace(|| \"load a\"), self.a)?; let b = field_chip.load_private(layouter.namespace(|| \"load b\"), self.b)?; // Load the constant factor into the circuit. let constant = field_chip.load_constant(layouter.namespace(|| \"load constant\"), self.constant)?; // We only have access to plain multiplication. // We could implement our circuit as: // asq = a*a // bsq = b*b // absq = asq*bsq // c = constant*asq*bsq // // but it's more efficient to implement it as: // ab = a*b // absq = ab^2 // c = constant*absq let ab = field_chip.mul(layouter.namespace(|| \"a * b\"), a, b)?; let absq = field_chip.mul(layouter.namespace(|| \"ab * ab\"), ab.clone(), ab)?; let c = field_chip.mul(layouter.namespace(|| \"constant * absq\"), constant, absq)?; // Expose the result as a public input to the circuit. field_chip.expose_public(layouter.namespace(|| \"expose c\"), c, 0) }\n}","breadcrumbs":"User Documentation » A simple example » Build the circuit","id":"23","title":"Build the circuit"},"24":{"body":"halo2_proofs::dev::MockProver can be used to test that the circuit is working correctly. The private and public inputs to the circuit are constructed as we will do to create a proof, but by passing them to MockProver::run we get an object that can test every constraint in the circuit, and tell us exactly what is failing (if anything). // The number of rows in our circuit cannot exceed 2^k. Since our example // circuit is very small, we can pick a very small value here. let k = 4; // Prepare the private and public inputs to the circuit! let constant = Fp::from(7); let a = Fp::from(2); let b = Fp::from(3); let c = constant * a.square() * b.square(); // Instantiate the circuit with the private inputs. let circuit = MyCircuit { constant, a: Some(a), b: Some(b), }; // Arrange the public input. We expose the multiplication result in row 0 // of the instance column, so we position it there in our public inputs. let mut public_inputs = vec![c]; // Given the correct public input, our circuit will verify. let prover = MockProver::run(k, &circuit, vec![public_inputs.clone()]).unwrap(); assert_eq!(prover.verify(), Ok(())); // If we try some other public input, the proof will fail! public_inputs[0] += Fp::one(); let prover = MockProver::run(k, &circuit, vec![public_inputs]).unwrap(); assert!(prover.verify().is_err());","breadcrumbs":"User Documentation » A simple example » Testing the circuit","id":"24","title":"Testing the circuit"},"25":{"body":"You can find the source code for this example here .","breadcrumbs":"User Documentation » A simple example » Full example","id":"25","title":"Full example"},"26":{"body":"In normal programs, you can trade memory for CPU to improve performance, by pre-computing and storing lookup tables for some part of the computation. We can do the same thing in halo2 circuits! A lookup table can be thought of as enforcing a relation between variables, where the relation is expressed as a table. Assuming we have only one lookup argument in our constraint system, the total size of tables is constrained by the size of the circuit: each table entry costs one row, and it also costs one row to do each lookup. TODO","breadcrumbs":"User Documentation » Lookup tables » Lookup tables","id":"26","title":"Lookup tables"},"27":{"body":"","breadcrumbs":"User Documentation » Gadgets » Gadgets","id":"27","title":"Gadgets"},"28":{"body":"This section contains various ideas and snippets that you might find useful while writing halo2 circuits.","breadcrumbs":"User Documentation » Tips and tricks » Tips and tricks","id":"28","title":"Tips and tricks"},"29":{"body":"A common constraint used in R1CS circuits is the boolean constraint: b∗(1−b)=0. This constraint can only be satisfied by b=0 or b=1. In halo2 circuits, you can similarly constrain a cell to have one of a small set of values. For example, to constrain a to the range [0..5], you would create a gate of the form: a⋅(1−a)⋅(2−a)⋅(3−a)⋅(4−a)=0 while to constraint c to be either 7 or 13, you would use: (7−c)⋅(13−c)=0 The underlying principle here is that we create a polynomial constraint with roots at each value in the set of possible values we want to allow. In R1CS circuits, the maximum supported polynomial degree is 2 (due to all constraints being of the form a∗b=c). In halo2 circuits, you can use arbitrary-degree polynomials - with the proviso that higher-degree constraints are more expensive to use. Note that the roots don't have to be constants; for example (a−x)⋅(a−y)⋅(a−z)=0 will constrain a to be equal to one of {x,y,z} where the latter can be arbitrary polynomials, as long as the whole expression stays within the maximum degree bound.","breadcrumbs":"User Documentation » Tips and tricks » Small range constraints","id":"29","title":"Small range constraints"},"3":{"body":"halo2 currently uses rayon for parallel computation. The RAYON_NUM_THREADS environment variable can be used to set the number of threads.","breadcrumbs":"halo2 » Controlling parallelism","id":"3","title":"Controlling parallelism"},"30":{"body":"We can use Lagrange interpolation to create a polynomial constraint that maps f(X)=Y for small sets of X∈{xi​},Y∈{yi​}. For instance, say we want to map a 2-bit value to a \"spread\" version interleaved with zeros. We first precompute the evaluations at each point: 00→000001→000110→010011→0101​⟹⟹⟹⟹​0→01→12→43→5​ Then, we construct the Lagrange basis polynomial for each point using the identity: lj​(X)=0≤m B\n^ +\n| |\n+ v\nD <---+ C E +---> F ^ + | | + v H <---+ G After adding constraint B≡E the above algorithm produces the cycle: A +---> B +-------------+\n^ |\n| |\n+ v\nD <---+ C <---+ E F ^ + | | + v H <---+ G","breadcrumbs":"Design » Proving system » Permutation argument » Algorithm","id":"46","title":"Algorithm"},"47":{"body":"If we did not check whether left and right were already in the same cycle, then we could end up undoing an equality constraint. For example, if we have the following constraints: a≡b b≡c c≡d b≡d and we tried to implement adding an equality constraint just using step 5 of the above algorithm, then we would end up constructing the cycle (a b) (c d), rather than the correct (a b c d).","breadcrumbs":"Design » Proving system » Permutation argument » Broken alternatives","id":"47","title":"Broken alternatives"},"48":{"body":"We need to check a permutation of cells in m columns, represented in Lagrange basis by polynomials v0​,…,vm−1​. We will label each cell in those m columns with a unique element of F×. Suppose that we have a permutation on these labels, σ(column:i,row:j)=(column:i′,row:j′). in which the cycles correspond to equality-constraint sets. If we consider the set of pairs {(label,value)}, then the values within each cycle are equal if and only if permuting the label in each pair by σ yields the same set: An example for a cycle (A B C D). The set before permuting the labels is {(A, 7), (B, 7), (C, 7), (D, 7)}, and the set after is {(D, 7), (A, 7), (B, 7), (C, 7)} which is the same. If one of the 7s is replaced by 3, then the set after permuting labels is not the same. Since the labels are distinct, set equality is the same as multiset equality, which we can check using a product argument. Let ω be a 2k root of unity and let δ be a T root of unity, where T⋅2S+1=p with T odd and k≤S. We will use δi⋅ωj∈F× as the label for the cell in the jth row of the ith column of the permutation argument. We represent σ by a vector of m polynomials si​(X) such that si​(ωj)=δi′⋅ωj′. Notice that the identity permutation can be represented by the vector of m polynomials IDi​(ωj) such that IDi​(ωj)=δi⋅ωj. We will use a challenge β to compress each (label,value) pair to value+β⋅label. Just as in the product argument we used for lookups , we also use a challenge γ to randomize each term of the product. Now given our permutation represented by s0​,…,sm−1​ over columns represented by v0​,…,vm−1​, we want to ensure that: i=0∏m−1​j=0∏n−1​(vi​(ωj)+β⋅si​(ωj)+γvi​(ωj)+β⋅δi⋅ωj+γ​)=1 Here vi​(ωj)+β⋅δi⋅ωj represents the unpermuted (label,value) pair, and vi​(ωj)+β⋅si​(ωj) represents the permuted (σ(label),value) pair. Let ZP​ be such that ZP​(ω0)=ZP​(ωn)=1 and for 0≤j=r(λ) verifier challenges and let the ith challenge be sampled from Chi​. The advantage metric AdvIPSRS​(P,λ)=Pr[SRSPIP​(λ)] of a state restoration prover P is defined with respect to the following game. ​GameSRSIPP​(λ):​win←false;tr←ϵpp←IP.Setup(1λ)(x,stP​)←Pλ​(pp)RunPλOSRS​​(stP​)Return win​​​OracleOSRS​(τ=(a1​,c1​,...,ai−1​,ci−1​),ai​):​Ifτ∈trthenIfi≤rthenci​←Chi​;tr←tr∣∣(τ,ai​,ci​);Returnci​Else ifi=r+1thend←IP.V(pp,x,(τ,ai​));tr←(τ,ai​)Ifd=1then win←trueReturndReturn⊥​​ As shown in [GT20] (Theorem 1) state restoration soundness is tightly related to soundness after applying the Fiat-Shamir transformation. Knowledge Soundness We will show that our protocol satisfies a strengthened notion of knowledge soundness known as witness extended emulation . Informally, this notion states that for any successful prover algorithm there exists an efficient emulator that can extract a witness from it by rewinding it and supplying it with fresh randomness. However, we must slightly adjust our definition of witness extended emulation to account for the fact that our provers are state restoration provers and can rewind the verifier. Further, to avoid the need for rewinding the state restoration prover during witness extraction we study our protocol in the algebraic group model. Algebraic Group Model (AGM). An adversary Palg​ is said to be algebraic if whenever it outputs a group element X it also outputs a representation x∈Fn such that ⟨x,G⟩=X where G∈Gn is the vector of group elements that Palg​ has seen so far. Notationally, we write {X} to describe a group element X enhanced with this representation. We also write {X}iG​ to identify the component of the representation of X that corresponds with Gi​. In other words, X=i=0∑n−1​[{X}iG​]Gi​ The algebraic group model allows us to perform so-called \"online\" extraction for some protocols: the extractor can obtain the witness from the representations themselves for a single (accepting) transcript. State Restoration Witness Extended Emulation Let IP be an interactive argument for relation R with r=r(λ) challenges. We define for all non-uniform algebraic provers Palg​, extractors E, and computationally unbounded distinguishers D the advantage metric AdvIP,Rsr-wee​(Palg​,D,E,λ)=Pr[WEE-realIP,RP,D​(λ)]−Pr[WEE-idealIP,RE,P,D​(λ)] is defined with the respect to the following games. ​GameWEE-realIP,RPalg​,D​(λ):​tr←ϵpp←IP.Setup(1λ)(x,stP​)←Palg​(pp)RunPalg​Oreal​(stP​)b←D(tr)Returnb=1GameWEE-idealIP,RE,Palg​,D​(λ):​tr←ϵpp←IP.Setup(1λ)(x,stP​)←Palg​(pp)stE​←(1λ,pp,x)RunPalg​Oideal​(stP​)w←E(stE​,⊥)b←D(tr)Return(b=1)∧(Acc(tr)⟹(x,w)∈R)​​​OracleOreal​(τ=(a1​,c1​,...,ai−1​,ci−1​),ai​):​Ifτ∈trthenIfi≤rthenci​←Chi​;tr←tr∣∣(τ,ai​,ci​);Returnci​Else ifi=r+1thend←IP.V(pp,x,(τ,ai​));tr←(τ,ai​)Ifd=1then win←trueReturndReturn⊥OracleOideal​(τ,a):​Ifτ∈trthen(r,stE​)←E(stE​,[(τ,a)])tr←tr∣∣(τ,a,r)ReturnrReturn⊥​​ Zero Knowledge We say that an argument of knowledge is zero knowledge if the verifier also does not learn anything from their interaction besides that which can be learned from the existence of a valid w. More formally, Perfect Special Honest-Verifier Zero Knowledge. A public coin interactive argument (Setup,P,V) has perfect special honest-verifier zero knowledge (PSHVZK) if for all polynomial-time decidable relations R and for all (x,w)∈R and for all non-uniform polynomial-time adversaries A1​,A2​ there exists a probabilistic polynomial-time simulator S such that =​Pr⎣⎡​A1​(σ,x,tr)=1∣∣​​pp←Setup(1λ);(x,w,ρ)←A2​(pp);tr←⟨P(pp,x,w),V(pp,x,ρ)⟩​⎦⎤​Pr⎣⎡​A1​(σ,x,tr)=1∣∣​​pp←Setup(1λ);(x,w,ρ)←A2​(pp);tr←S(pp,x,ρ)​⎦⎤​​ where ρ is the internal randomness of the verifier. In this (common) definition of zero-knowledge the verifier is expected to act \"honestly\" and send challenges that correspond only with their internal randomness; they cannot adaptively respond to the prover based on the prover's messages. We use a strengthened form of this definition that forces the simulator to output a transcript with the same (adversarially provided) challenges that the verifier algorithm sends to the prover.","breadcrumbs":"Design » Protocol Description » Zero knowledge Arguments of Knowledge","id":"68","title":"Zero knowledge Arguments of Knowledge"},"69":{"body":"Let ω∈F be a n=2k primitive root of unity forming the domain D=(ω0,ω1,...,ωn−1) with t(X)=Xn−1 the vanishing polynomial over this domain. Let ng​,na​,ne​ be positive integers with na​,ne​p. That is, we must check that 0≤αq. Thus, given a scalar α, we witness the boolean decomposition of k=α+tq​. 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