diff --git a/background/upa.html b/background/upa.html index 1f10f484..14840385 100644 --- a/background/upa.html +++ b/background/upa.html @@ -178,8 +178,9 @@ every row (i.e. at every element in the multiplicative subgroup).
and we tried to implement adding an equality constraint just using step 4 of the above +
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 , rather than the correct .
We need to represent permutations over columns, represented by polynomials .
-We first assign a unique element of as an "extended domain" element for each cell -that can participate in the permutation argument.
+We need to check a permutation of cells in columns, represented in Lagrange basis by +polynomials .
+We first label each cell in those columns with a unique element of .
Let be a root of unity and let be a root of unity, where with odd and . -We will use as the extended domain element for the +We will use as the label for the cell in the th row of the th column of the permutation argument.
If we have a permutation , we can represent it as a vector of polynomials such that .
Notice that the identity permutation can be represented by the vector of polynomials such that .
-Now given our permutation represented by , over advice columns represented by +
Now given our permutation represented by over columns represented by , we want to ensure that:
Let be such that and for : diff --git a/print.html b/print.html index e1735878..078f3566 100644 --- a/print.html +++ b/print.html @@ -1122,8 +1122,9 @@ nothing to do.
and we tried to implement adding an equality constraint just using step 4 of the above +
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 , rather than the correct .
We need to represent permutations over columns, represented by polynomials .
-We first assign a unique element of as an "extended domain" element for each cell -that can participate in the permutation argument.
+We need to check a permutation of cells in columns, represented in Lagrange basis by +polynomials .
+We first label each cell in those columns with a unique element of .
Let be a root of unity and let be a root of unity, where with odd and . -We will use as the extended domain element for the +We will use as the label for the cell in the th row of the th column of the permutation argument.
If we have a permutation , we can represent it as a vector of polynomials such that .
Notice that the identity permutation can be represented by the vector of polynomials such that .
-Now given our permutation represented by , over advice columns represented by +
Now given our permutation represented by over columns represented by , we want to ensure that:
Let be such that and for : @@ -3050,8 +3051,9 @@ every row (i.e. at every element in the multiplicative subgroup).
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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":"This crate currently has a Minimum Supported Rust Version of 1.49.0. When const generics are stabilized in Rust 1.51.0 , we plan to require that version.","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":"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 » Fast Fourier Transform (FFT)","id":"100","title":"Fast Fourier Transform (FFT)"},"101":{"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+a1X+a2X2+⋯+an−1Xn−1,=b0+b1X+b2X2+⋯+bn−1Xn−1,=a0⋅(b0+b1X+b2X2+⋯+bn−1Xn−1)+a1X⋅(b0+b1X+b2X2+⋯+bn−1Xn−1)+⋯+an−1Xn−1⋅(b0+b1X+b2X2+⋯+bn−1Xn−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⋮11ωω2⋮ωn−11ω2ω2⋅2⋮ω2(n−1)………⋯1ωn−1ω2⋅(n−1)⋮ω(n−1)2⎦⎥⎥⎥⎥⎥⎥⎤⋅⎣⎢⎢⎢⎢⎢⎢⎡a0a1a2⋮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 » Motivation: Fast polynomial multiplication","id":"101","title":"Motivation: Fast polynomial multiplication"},"102":{"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+a1X+a2X2+⋯+an−1Xn−1, we split it up into even and odd terms: AevenAodd=a0+a2X+⋯+an−2X2n−1,=a1+a3X+⋯+an−1X2n−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)+ωniAodd((ωni)2),=Aeven((ωn2n+i)2)+ωn2n+iAodd((ωn2n+i)2)=Aeven((−ωni)2)−ωniAodd((−ωni)2)←(negation lemma)=Aeven((ωni)2)−ωniAodd((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(nlog2n) operations, also known as the Fast Fourier Transform (FFT).","breadcrumbs":"Background Material » The (radix-2) Cooley-Tukey algorithm","id":"102","title":"The (radix-2) Cooley-Tukey algorithm"},"103":{"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=n1Vω−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 » Inverse FFT","id":"103","title":"Inverse FFT"},"104":{"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 » The Schwartz-Zippel lemma","id":"104","title":"The Schwartz-Zippel lemma"},"105":{"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 » Vanishing polynomial","id":"105","title":"Vanishing polynomial"},"106":{"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":"106","title":"Lagrange basis functions"},"107":{"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":"107","title":"Lagrange interpolation"},"108":{"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 » References","id":"108","title":"References"},"109":{"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":"109","title":"Cryptographic groups"},"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. 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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":"This crate currently has a Minimum Supported Rust Version of 1.49.0. When const generics are stabilized in Rust 1.51.0 , we plan to require that version.","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":"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 » Fast Fourier Transform (FFT)","id":"100","title":"Fast Fourier Transform (FFT)"},"101":{"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+a1X+a2X2+⋯+an−1Xn−1,=b0+b1X+b2X2+⋯+bn−1Xn−1,=a0⋅(b0+b1X+b2X2+⋯+bn−1Xn−1)+a1X⋅(b0+b1X+b2X2+⋯+bn−1Xn−1)+⋯+an−1Xn−1⋅(b0+b1X+b2X2+⋯+bn−1Xn−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⋮11ωω2⋮ωn−11ω2ω2⋅2⋮ω2(n−1)………⋯1ωn−1ω2⋅(n−1)⋮ω(n−1)2⎦⎥⎥⎥⎥⎥⎥⎤⋅⎣⎢⎢⎢⎢⎢⎢⎡a0a1a2⋮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 » Motivation: Fast polynomial multiplication","id":"101","title":"Motivation: Fast polynomial multiplication"},"102":{"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+a1X+a2X2+⋯+an−1Xn−1, we split it up into even and odd terms: AevenAodd=a0+a2X+⋯+an−2X2n−1,=a1+a3X+⋯+an−1X2n−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)+ωniAodd((ωni)2),=Aeven((ωn2n+i)2)+ωn2n+iAodd((ωn2n+i)2)=Aeven((−ωni)2)−ωniAodd((−ωni)2)←(negation lemma)=Aeven((ωni)2)−ωniAodd((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(nlog2n) operations, also known as the Fast Fourier Transform (FFT).","breadcrumbs":"Background Material » The (radix-2) Cooley-Tukey algorithm","id":"102","title":"The (radix-2) Cooley-Tukey algorithm"},"103":{"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=n1Vω−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 » Inverse FFT","id":"103","title":"Inverse FFT"},"104":{"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 » The Schwartz-Zippel lemma","id":"104","title":"The Schwartz-Zippel lemma"},"105":{"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 » Vanishing polynomial","id":"105","title":"Vanishing polynomial"},"106":{"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":"106","title":"Lagrange basis functions"},"107":{"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":"107","title":"Lagrange interpolation"},"108":{"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 » References","id":"108","title":"References"},"109":{"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":"109","title":"Cryptographic groups"},"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 below image was also taken from [2] .)","breadcrumbs":"Background Material » Inverse FFT","id":"103","title":"Inverse FFT"},"104":{"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 » The Schwartz-Zippel lemma","id":"104","title":"The Schwartz-Zippel lemma"},"105":{"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 » Vanishing polynomial","id":"105","title":"Vanishing polynomial"},"106":{"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":"106","title":"Lagrange basis functions"},"107":{"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":"107","title":"Lagrange interpolation"},"108":{"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 » References","id":"108","title":"References"},"109":{"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":"109","title":"Cryptographic groups"},"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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