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//! Constants required for the ECC chip.
use arrayvec::ArrayVec;
use group::{
ff::{Field, PrimeField},
Curve,
};
use halo2_proofs::arithmetic::lagrange_interpolate;
use pasta_curves::{arithmetic::CurveAffine, pallas};
/// Window size for fixed-base scalar multiplication
pub const FIXED_BASE_WINDOW_SIZE: usize = 3;
/// $2^{`FIXED_BASE_WINDOW_SIZE`}$
pub const H: usize = 1 << FIXED_BASE_WINDOW_SIZE;
/// Number of windows for a full-width scalar
pub const NUM_WINDOWS: usize =
(pallas::Scalar::NUM_BITS as usize + FIXED_BASE_WINDOW_SIZE - 1) / FIXED_BASE_WINDOW_SIZE;
/// Number of windows for a short signed scalar
pub const NUM_WINDOWS_SHORT: usize =
(L_SCALAR_SHORT + FIXED_BASE_WINDOW_SIZE - 1) / FIXED_BASE_WINDOW_SIZE;
/// $\ell_\mathsf{value}$
/// Number of bits in an unsigned short scalar.
pub(crate) const L_SCALAR_SHORT: usize = 64;
/// The Pallas scalar field modulus is $q = 2^{254} + \mathsf{t_q}$.
/// <https://github.com/zcash/pasta>
pub(crate) const T_Q: u128 = 45560315531506369815346746415080538113;
/// The Pallas base field modulus is $p = 2^{254} + \mathsf{t_p}$.
/// <https://github.com/zcash/pasta>
pub(crate) const T_P: u128 = 45560315531419706090280762371685220353;
/// For each fixed base, we calculate its scalar multiples in three-bit windows.
/// Each window will have $2^3 = 8$ points. The tables are computed as described in
/// [the Halo 2 book](https://zcash.github.io/halo2/design/gadgets/ecc/fixed-base-scalar-mul.html#load-fixed-base).
fn compute_window_table<C: CurveAffine>(base: C, num_windows: usize) -> Vec<[C; H]> {
let mut window_table: Vec<[C; H]> = Vec::with_capacity(num_windows);
// Generate window table entries for all windows but the last.
// For these first `num_windows - 1` windows, we compute the multiple [(k+2)*(2^3)^w]B.
// Here, w ranges from [0..`num_windows - 1`)
for w in 0..(num_windows - 1) {
window_table.push(
(0..H)
.map(|k| {
// scalar = (k+2)*(8^w)
let scalar = C::Scalar::from(k as u64 + 2)
* C::Scalar::from(H as u64).pow(&[w as u64, 0, 0, 0]);
(base * scalar).to_affine()
})
.collect::<ArrayVec<C, H>>()
.into_inner()
.unwrap(),
);
}
// Generate window table entries for the last window, w = `num_windows - 1`.
// For the last window, we compute [k * (2^3)^w - sum]B, where sum is defined
// as sum = \sum_{j = 0}^{`num_windows - 2`} 2^{3j+1}
let sum = (0..(num_windows - 1)).fold(C::Scalar::ZERO, |acc, j| {
acc + C::Scalar::from(2).pow(&[FIXED_BASE_WINDOW_SIZE as u64 * j as u64 + 1, 0, 0, 0])
});
window_table.push(
(0..H)
.map(|k| {
// scalar = k * (2^3)^w - sum, where w = `num_windows - 1`
let scalar = C::Scalar::from(k as u64)
* C::Scalar::from(H as u64).pow(&[(num_windows - 1) as u64, 0, 0, 0])
- sum;
(base * scalar).to_affine()
})
.collect::<ArrayVec<C, H>>()
.into_inner()
.unwrap(),
);
window_table
}
/// For each window, we interpolate the $x$-coordinate.
/// Here, we pre-compute and store the coefficients of the interpolation polynomial.
pub fn compute_lagrange_coeffs<C: CurveAffine>(base: C, num_windows: usize) -> Vec<[C::Base; H]> {
// We are interpolating over the 3-bit window, k \in [0..8)
let points: Vec<_> = (0..H).map(|i| C::Base::from(i as u64)).collect();
let window_table = compute_window_table(base, num_windows);
window_table
.iter()
.map(|window_points| {
let x_window_points: Vec<_> = window_points
.iter()
.map(|point| *point.coordinates().unwrap().x())
.collect();
lagrange_interpolate(&points, &x_window_points)
.into_iter()
.collect::<ArrayVec<C::Base, H>>()
.into_inner()
.unwrap()
})
.collect()
}
/// For each window, $z$ is a field element such that for each point $(x, y)$ in the window:
/// - $z + y = u^2$ (some square in the field); and
/// - $z - y$ is not a square.
/// If successful, return a vector of `(z: u64, us: [C::Base; H])` for each window.
///
/// This function was used to generate the `z`s and `u`s for the Orchard fixed
/// 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.
pub fn find_zs_and_us<C: CurveAffine>(
base: C,
num_windows: usize,
) -> Option<Vec<(u64, [C::Base; H])>> {
// Closure to find z and u's for one window
let find_z_and_us = |window_points: &[C]| {
assert_eq!(H, window_points.len());
let ys: Vec<_> = window_points
.iter()
.map(|point| *point.coordinates().unwrap().y())
.collect();
(0..(1000 * (1 << (2 * H)))).find_map(|z| {
ys.iter()
.map(|&y| {
if (-y + C::Base::from(z)).sqrt().is_none().into() {
(y + C::Base::from(z)).sqrt().into()
} else {
None
}
})
.collect::<Option<ArrayVec<C::Base, H>>>()
.map(|us| (z, us.into_inner().unwrap()))
})
};
let window_table = compute_window_table(base, num_windows);
window_table
.iter()
.map(|window_points| find_z_and_us(window_points))
.collect()
}
/// Test that the z-values and u-values satisfy the conditions:
/// 1. z + y = u^2,
/// 2. z - y is not a square
/// for the y-coordinate of each fixed-base multiple in each window.
#[cfg(any(test, feature = "test-dependencies"))]
#[cfg_attr(docsrs, doc(cfg(feature = "test-dependencies")))]
pub fn test_zs_and_us<C: CurveAffine>(base: C, z: &[u64], u: &[[[u8; 32]; H]], num_windows: usize) {
let window_table = compute_window_table(base, num_windows);
for ((u, z), window_points) in u.iter().zip(z.iter()).zip(window_table) {
for (u, point) in u.iter().zip(window_points.iter()) {
let y = *point.coordinates().unwrap().y();
let mut u_repr = <C::Base as PrimeField>::Repr::default();
u_repr.as_mut().copy_from_slice(u);
let u = C::Base::from_repr(u_repr).unwrap();
assert_eq!(C::Base::from(*z) + y, u * u); // allow either square root
assert!(bool::from((C::Base::from(*z) - y).sqrt().is_none()));
}
}
}
/// Test that Lagrange interpolation coefficients reproduce the correct x-coordinate
/// for each fixed-base multiple in each window.
#[cfg(any(test, feature = "test-dependencies"))]
#[cfg_attr(docsrs, doc(cfg(feature = "test-dependencies")))]
pub fn test_lagrange_coeffs<C: CurveAffine>(base: C, num_windows: usize) {
/// Evaluate y = f(x) given the coefficients of f(x)
fn evaluate<C: CurveAffine>(x: u8, coeffs: &[C::Base]) -> C::Base {
let x = C::Base::from(x as u64);
coeffs
.iter()
.rev()
.cloned()
.reduce(|acc, coeff| acc * x + coeff)
.unwrap_or(C::Base::ZERO)
}
let lagrange_coeffs = compute_lagrange_coeffs(base, num_windows);
// Check first 84 windows, i.e. `k_0, k_1, ..., k_83`
for (idx, coeffs) in lagrange_coeffs[0..(num_windows - 1)].iter().enumerate() {
// Test each three-bit chunk in this window.
for bits in 0..(H as u8) {
{
// Interpolate the x-coordinate using this window's coefficients
let interpolated_x = evaluate::<C>(bits, coeffs);
// Compute the actual x-coordinate of the multiple [(k+2)*(8^w)]B.
let point = base
* C::Scalar::from(bits as u64 + 2)
* C::Scalar::from(H as u64).pow(&[idx as u64, 0, 0, 0]);
let x = *point.to_affine().coordinates().unwrap().x();
// Check that the interpolated x-coordinate matches the actual one.
assert_eq!(x, interpolated_x);
}
}
}
// Check last window.
for bits in 0..(H as u8) {
// Interpolate the x-coordinate using the last window's coefficients
let interpolated_x = evaluate::<C>(bits, &lagrange_coeffs[num_windows - 1]);
// Compute the actual x-coordinate of the multiple [k * (8^84) - offset]B,
// where offset = \sum_{j = 0}^{83} 2^{3j+1}
let offset = (0..(num_windows - 1)).fold(C::Scalar::ZERO, |acc, w| {
acc + C::Scalar::from(2).pow(&[FIXED_BASE_WINDOW_SIZE as u64 * w as u64 + 1, 0, 0, 0])
});
let scalar = C::Scalar::from(bits as u64)
* C::Scalar::from(H as u64).pow(&[(num_windows - 1) as u64, 0, 0, 0])
- offset;
let point = base * scalar;
let x = *point.to_affine().coordinates().unwrap().x();
// Check that the interpolated x-coordinate matches the actual one.
assert_eq!(x, interpolated_x);
}
}
#[cfg(test)]
mod tests {
use ff::FromUniformBytes;
use group::{ff::Field, Curve, Group};
use pasta_curves::{arithmetic::CurveAffine, pallas};
use proptest::prelude::*;
use super::{compute_window_table, find_zs_and_us, test_lagrange_coeffs, H, NUM_WINDOWS};
prop_compose! {
/// Generate an arbitrary Pallas point.
pub fn arb_point()(bytes in prop::array::uniform32(0u8..)) -> pallas::Point {
// Instead of rejecting out-of-range bytes, let's reduce them.
let mut buf = [0; 64];
buf[..32].copy_from_slice(&bytes);
let scalar = pallas::Scalar::from_uniform_bytes(&buf);
pallas::Point::generator() * scalar
}
}
proptest! {
#[test]
fn lagrange_coeffs(
base in arb_point(),
) {
test_lagrange_coeffs(base.to_affine(), NUM_WINDOWS);
}
}
#[test]
fn zs_and_us() {
let base = pallas::Point::random(rand::rngs::OsRng);
let (z, u): (Vec<u64>, Vec<[pallas::Base; H]>) =
find_zs_and_us(base.to_affine(), NUM_WINDOWS)
.unwrap()
.into_iter()
.unzip();
let window_table = compute_window_table(base.to_affine(), NUM_WINDOWS);
for ((u, z), window_points) in u.iter().zip(z.iter()).zip(window_table) {
for (u, point) in u.iter().zip(window_points.iter()) {
let y = *point.coordinates().unwrap().y();
assert_eq!(pallas::Base::from(*z) + y, u * u); // allow either square root
assert!(bool::from((pallas::Base::from(*z) - y).sqrt().is_none()));
}
}
}
}