diff --git a/concepts/arithmetization.html b/concepts/arithmetization.html index 378ebd53..2cad20f7 100644 --- a/concepts/arithmetization.html +++ b/concepts/arithmetization.html @@ -182,8 +182,10 @@ elements of
A subset of the columns that can participate in equality constraints.
diff --git a/concepts/chips.html b/concepts/chips.html index cefaf2bd..0e1ae724 100644 --- a/concepts/chips.html +++ b/concepts/chips.html @@ -166,7 +166,7 @@In order to combine functionality from several cores, we use a chip. To implement a -chip, we define a set of fixed, advice, and auxiliary columns, and then specify how they +chip, we define a set of fixed, advice, and instance columns, and then specify how they should be distributed between cores.
In the simplest case, each core will use columns disjoint from the other cores. However, it is allowed to share a column between cores. It is important to optimize the number of advice diff --git a/design/proving-system/circuit-commitments.html b/design/proving-system/circuit-commitments.html index ee276bf8..04b62bf3 100644 --- a/design/proving-system/circuit-commitments.html +++ b/design/proving-system/circuit-commitments.html @@ -168,13 +168,13 @@
At the start of proof creation, the prover has a table of cell assignments that it claims satisfy the constraint system. The table has rows, and is broken into advice, -auxiliary, and fixed columns. We define as the assignment in the th row of +instance, and fixed columns. We define as the assignment in the th row of the th fixed column. Without loss of generality, we'll similarly define to -represent the advice and auxiliary assignments.
+represent the advice and instance assignments.We separate fixed columns here because they are provided by the verifier, whereas the -advice and auxiliary columns are provided by the prover. In practice, the commitments to -auxiliary and fixed columns are computed by both the prover and verifier, and only the +advice and instance columns are provided by the prover. In practice, the commitments to +instance and fixed columns are computed by both the prover and verifier, and only the advice commitments are stored in the proof.
To commit to these assignments, we construct Lagrange polynomials of degree for diff --git a/design/proving-system/comparison.html b/design/proving-system/comparison.html index 71f467f8..68a725f4 100644 --- a/design/proving-system/comparison.html +++ b/design/proving-system/comparison.html @@ -195,7 +195,7 @@ equivalent objects in Halo 2 (which builds on the nomenclature from the Halo pap
Step 8 of the algorithm computes a "non-hiding" commitment prior to the inner product argument, which opens to the same value as but is a commitment to a randomly-drawn polynomial. The remainder of the protocol involves no blinding. By -contrast, in Halo 2 we blind every single commitment that we make (even for auxiliary +contrast, in Halo 2 we blind every single commitment that we make (even for instance and fixed polynomials, though using a blinding factor of 1 for the fixed polynomials); this makes the protocol simpler to reason about. As a consequence of this, the verifier needs to handle the cumulative blinding factor at the end of the protocol, and so there diff --git a/print.html b/print.html index 91c0570a..bb3ab6f2 100644 --- a/print.html +++ b/print.html @@ -290,8 +290,10 @@ elements of
A subset of the columns that can participate in equality constraints.
@@ -400,7 +402,7 @@ bound).In order to combine functionality from several cores, we use a chip. To implement a -chip, we define a set of fixed, advice, and auxiliary columns, and then specify how they +chip, we define a set of fixed, advice, and instance columns, and then specify how they should be distributed between cores.
In the simplest case, each core will use columns disjoint from the other cores. However, it is allowed to share a column between cores. It is important to optimize the number of advice @@ -536,7 +538,7 @@ impl<F: FieldExt> FieldChip<F> { fn configure( meta: &mut ConstraintSystem<F>, advice: [Column<Advice>; 2], - aux: Column<Aux>, + instance: Column<Instance>, ) -> FieldConfig { let perm = Permutation::new(meta, &advice); let s_mul = meta.fixed_column(); @@ -574,10 +576,10 @@ impl<F: FieldExt> FieldChip<F> { // We choose somewhat-arbitrarily that we will use the second advice // column for exposing numbers as public inputs. let a = meta.query_advice(advice[1], Rotation::cur()); - let p = meta.query_aux(aux, Rotation::cur()); + let p = meta.query_instance(instance, Rotation::cur()); let s = meta.query_fixed(s_pub, Rotation::cur()); - // We simply constrain the advice cell to be equal to the aux cell, + // We simply constrain the advice cell to be equal to the instance cell, // when the selector is enabled. s * (p + a * -F::one()) }); @@ -694,7 +696,7 @@ impl<F: FieldExt> NumericInstructions for FieldChip<F> { )?; region.constrain_equal(&config.perm, num.cell, out)?; - // We don't assign to the auxiliary column inside the circuit; + // We don't assign to the instance column inside the circuit; // the mapping of public inputs to cells is provided to the prover. Ok(()) }, @@ -723,10 +725,10 @@ impl<F: FieldExt> Circuit<F> for MyCircuit<F> { // We create the two advice columns that FieldChip uses for I/O. let advice = [meta.advice_column(), meta.advice_column()]; - // We also need an auxiliary column to store public inputs. - let aux = meta.aux_column(); + // We also need an instance column to store public inputs. + let instance = meta.instance_column(); - FieldChip::configure(meta, advice, aux) + FieldChip::configure(meta, advice, instance) } fn synthesize(&self, cs: &mut impl Assignment<F>, config: Self::Config) -> Result<(), Error> { @@ -774,7 +776,7 @@ in the circuit, and tell us exactly what is failing (if anything).
}; // Arrange the public input. We expose the multiplication result in row 6 - // of the aux column, so we position it there in our public inputs. + // of the instance column, so we position it there in our public inputs. let mut public_inputs = vec![Fp::zero(); 1 << k]; public_inputs[6] = c; @@ -1162,13 +1164,13 @@ constrained by the ruleAt the start of proof creation, the prover has a table of cell assignments that it claims satisfy the constraint system. The table has rows, and is broken into advice, -auxiliary, and fixed columns. We define as the assignment in the th row of +instance, and fixed columns. We define as the assignment in the th row of the th fixed column. Without loss of generality, we'll similarly define to -represent the advice and auxiliary assignments.
+represent the advice and instance assignments.We separate fixed columns here because they are provided by the verifier, whereas the -advice and auxiliary columns are provided by the prover. In practice, the commitments to -auxiliary and fixed columns are computed by both the prover and verifier, and only the +advice and instance columns are provided by the prover. In practice, the commitments to +instance and fixed columns are computed by both the prover and verifier, and only the advice commitments are stored in the proof.
To commit to these assignments, we construct Lagrange polynomials of degree for @@ -1419,7 +1421,7 @@ equivalent objects in Halo 2 (which builds on the nomenclature from the Halo pap
Step 8 of the algorithm computes a "non-hiding" commitment prior to
the inner product argument, which opens to the same value as but is a commitment to
a randomly-drawn polynomial. The remainder of the protocol involves no blinding. By
-contrast, in Halo 2 we blind every single commitment that we make (even for auxiliary
+contrast, in Halo 2 we blind every single commitment that we make (even for instance
and fixed polynomials, though using a blinding factor of 1 for the fixed polynomials);
this makes the protocol simpler to reason about. As a consequence of this, the verifier
needs to handle the cumulative blinding factor at the end of the protocol, and so there
diff --git a/searchindex.js b/searchindex.js
index 35fdd06c..aec7e1ba 100644
--- a/searchindex.js
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@@ -1 +1 @@
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: This library is being actively developed and should not be used in production software.","breadcrumbs":"halo2 Crates.io","id":"0","title":"halo2 Crates.io"},"1":{"body":"","breadcrumbs":"Documentation","id":"1","title":"Documentation"},"10":{"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","id":"10","title":"A simple example"},"100":{"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)={10if 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−1aiLi(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)=n1X−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−xjX−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−xjxi−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 » Lagrange basis functions","id":"100","title":"Lagrange basis functions"},"101":{"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−1A(xi)Li(X), where X∈{x0,x1,⋯,x1−n}.","breadcrumbs":"Background Material » Lagrange interpolation","id":"101","title":"Lagrange interpolation"},"102":{"body":"Dasgupta, S., Papadimitriou, C. H., & Vazirani, U. V. (2008). \"Algorithms\" (ch. 2). New York: McGraw-Hill Higher Education. https://people.eecs.berkeley.edu/~vazirani/algorithms/chap2.pdf http://www.cs.ust.hk/mjg_lib/Classes/COMP3711H_Fall16/lectures/FFT_Slides.pdf Berrut, J. and Trefethen, L. (2004). \"Barycentric Lagrange Interpolation.\" https://people.maths.ox.ac.uk/trefethen/barycentric.pdf","breadcrumbs":"Background Material » References","id":"102","title":"References"},"103":{"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","id":"103","title":"Cryptographic groups"},"104":{"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 » Pedersen commitment","id":"104","title":"Pedersen commitment"},"105":{"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 » Vector Pedersen commitment","id":"105","title":"Vector Pedersen commitment"},"106":{"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 » Diffie--Hellman","id":"106","title":"Diffie--Hellman"},"107":{"body":"","breadcrumbs":"Background Material » Multiscalar multiplication","id":"107","title":"Multiscalar multiplication"},"108":{"body":"Reference: https://jbootle.github.io/Misc/pippenger.pdf","breadcrumbs":"Background Material » TODO: Pippenger's algorithm","id":"108","title":"TODO: Pippenger's algorithm"},"109":{"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. 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: This library is being actively developed and should not be used in production software.","breadcrumbs":"halo2 Crates.io","id":"0","title":"halo2 Crates.io"},"1":{"body":"","breadcrumbs":"Documentation","id":"1","title":"Documentation"},"10":{"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","id":"10","title":"A simple example"},"100":{"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)={10if 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−1aiLi(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)=n1X−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−xjX−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−xjxi−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 » Lagrange basis functions","id":"100","title":"Lagrange basis functions"},"101":{"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−1A(xi)Li(X), where X∈{x0,x1,⋯,x1−n}.","breadcrumbs":"Background Material » Lagrange interpolation","id":"101","title":"Lagrange interpolation"},"102":{"body":"Dasgupta, S., Papadimitriou, C. H., & Vazirani, U. V. (2008). \"Algorithms\" (ch. 2). New York: McGraw-Hill Higher Education. https://people.eecs.berkeley.edu/~vazirani/algorithms/chap2.pdf http://www.cs.ust.hk/mjg_lib/Classes/COMP3711H_Fall16/lectures/FFT_Slides.pdf Berrut, J. and Trefethen, L. (2004). \"Barycentric Lagrange Interpolation.\" https://people.maths.ox.ac.uk/trefethen/barycentric.pdf","breadcrumbs":"Background Material » References","id":"102","title":"References"},"103":{"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","id":"103","title":"Cryptographic groups"},"104":{"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 » Pedersen commitment","id":"104","title":"Pedersen commitment"},"105":{"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 » Vector Pedersen commitment","id":"105","title":"Vector Pedersen commitment"},"106":{"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 » Diffie--Hellman","id":"106","title":"Diffie--Hellman"},"107":{"body":"","breadcrumbs":"Background Material » Multiscalar multiplication","id":"107","title":"Multiscalar multiplication"},"108":{"body":"Reference: https://jbootle.github.io/Misc/pippenger.pdf","breadcrumbs":"Background Material » TODO: Pippenger's algorithm","id":"108","title":"TODO: Pippenger's algorithm"},"109":{"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","id":"109","title":"Elliptic curves"},"11":{"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. 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: This library is being actively developed and should not be used in production software.","breadcrumbs":"halo2 Crates.io","id":"0","title":"halo2 Crates.io"},"1":{"body":"","breadcrumbs":"Documentation","id":"1","title":"Documentation"},"10":{"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","id":"10","title":"A simple example"},"100":{"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)={10if 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−1aiLi(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)=n1X−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−xjX−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−xjxi−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 » Lagrange basis functions","id":"100","title":"Lagrange basis functions"},"101":{"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−1A(xi)Li(X), where X∈{x0,x1,⋯,x1−n}.","breadcrumbs":"Background Material » Lagrange interpolation","id":"101","title":"Lagrange interpolation"},"102":{"body":"Dasgupta, S., Papadimitriou, C. H., & Vazirani, U. V. (2008). \"Algorithms\" (ch. 2). New York: McGraw-Hill Higher Education. https://people.eecs.berkeley.edu/~vazirani/algorithms/chap2.pdf http://www.cs.ust.hk/mjg_lib/Classes/COMP3711H_Fall16/lectures/FFT_Slides.pdf Berrut, J. and Trefethen, L. (2004). \"Barycentric Lagrange Interpolation.\" https://people.maths.ox.ac.uk/trefethen/barycentric.pdf","breadcrumbs":"Background Material » References","id":"102","title":"References"},"103":{"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","id":"103","title":"Cryptographic groups"},"104":{"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 » Pedersen commitment","id":"104","title":"Pedersen commitment"},"105":{"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 » Vector Pedersen commitment","id":"105","title":"Vector Pedersen commitment"},"106":{"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 » Diffie--Hellman","id":"106","title":"Diffie--Hellman"},"107":{"body":"","breadcrumbs":"Background Material » Multiscalar multiplication","id":"107","title":"Multiscalar multiplication"},"108":{"body":"Reference: https://jbootle.github.io/Misc/pippenger.pdf","breadcrumbs":"Background Material » TODO: Pippenger's algorithm","id":"108","title":"TODO: Pippenger's algorithm"},"109":{"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","id":"109","title":"Elliptic curves"},"11":{"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. 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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","id":"10","title":"A simple example"},"100":{"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)={10if 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−1aiLi(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)=n1X−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−xjX−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−xjxi−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 » Lagrange basis functions","id":"100","title":"Lagrange basis functions"},"101":{"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−1A(xi)Li(X), where X∈{x0,x1,⋯,x1−n}.","breadcrumbs":"Background Material » Lagrange interpolation","id":"101","title":"Lagrange interpolation"},"102":{"body":"Dasgupta, S., Papadimitriou, C. H., & Vazirani, U. V. (2008). \"Algorithms\" (ch. 2). New York: McGraw-Hill Higher Education. https://people.eecs.berkeley.edu/~vazirani/algorithms/chap2.pdf http://www.cs.ust.hk/mjg_lib/Classes/COMP3711H_Fall16/lectures/FFT_Slides.pdf Berrut, J. and Trefethen, L. 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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 » Pedersen commitment","id":"104","title":"Pedersen commitment"},"105":{"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 » Vector Pedersen commitment","id":"105","title":"Vector Pedersen commitment"},"106":{"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×. 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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. 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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. 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index a2f366b0..b40f251b 100644
--- a/user/simple-example.html
+++ b/user/simple-example.html
@@ -261,7 +261,7 @@ impl<F: FieldExt> FieldChip<F> {
fn configure(
meta: &mut ConstraintSystem<F>,
advice: [Column<Advice>; 2],
- aux: Column<Aux>,
+ instance: Column<Instance>,
) -> FieldConfig {
let perm = Permutation::new(meta, &advice);
let s_mul = meta.fixed_column();
@@ -299,10 +299,10 @@ impl<F: FieldExt> FieldChip<F> {
// We choose somewhat-arbitrarily that we will use the second advice
// column for exposing numbers as public inputs.
let a = meta.query_advice(advice[1], Rotation::cur());
- let p = meta.query_aux(aux, Rotation::cur());
+ let p = meta.query_instance(instance, Rotation::cur());
let s = meta.query_fixed(s_pub, Rotation::cur());
- // We simply constrain the advice cell to be equal to the aux cell,
+ // We simply constrain the advice cell to be equal to the instance cell,
// when the selector is enabled.
s * (p + a * -F::one())
});
@@ -419,7 +419,7 @@ impl<F: FieldExt> NumericInstructions for FieldChip<F> {
)?;
region.constrain_equal(&config.perm, num.cell, out)?;
- // We don't assign to the auxiliary column inside the circuit;
+ // We don't assign to the instance column inside the circuit;
// the mapping of public inputs to cells is provided to the prover.
Ok(())
},
@@ -448,10 +448,10 @@ impl<F: FieldExt> Circuit<F> for MyCircuit<F> {
// We create the two advice columns that FieldChip uses for I/O.
let advice = [meta.advice_column(), meta.advice_column()];
- // We also need an auxiliary column to store public inputs.
- let aux = meta.aux_column();
+ // We also need an instance column to store public inputs.
+ let instance = meta.instance_column();
- FieldChip::configure(meta, advice, aux)
+ FieldChip::configure(meta, advice, instance)
}
fn synthesize(&self, cs: &mut impl Assignment<F>, config: Self::Config) -> Result<(), Error> {
@@ -499,7 +499,7 @@ in the circuit, and tell us exactly what is failing (if anything).