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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":"The halo2 crate includes several utilities to help you design and implement your circuits.","breadcrumbs":"User Documentation » Developer tools","id":"10","title":"Developer tools"},"100":{"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 » Square roots","id":"100","title":"Square roots"},"101":{"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,α2,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 2nth roots of unity). There is a two-to-one mapping between the elements and their squares.","breadcrumbs":"Background Material » Roots of unity","id":"101","title":"Roots of unity"},"102":{"body":"Friedman, R. (n.d.) \"Cyclic Groups and Elementary Number Theory II\" (p. 5).","breadcrumbs":"Background Material » References","id":"102","title":"References"},"103":{"body":"Let A(X) be a polynomial over Fp with formal indeterminate X. As an example, A(X)=a0+a1X+a2X2+a3X3 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+a1X+a2X2+⋯+an−1Xn−1=a0+X(a1+X(a2+⋯+X(an−2+Xan−1))),","breadcrumbs":"Background Material » Polynomials","id":"103","title":"Polynomials"},"104":{"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":"104","title":"Fast Fourier Transform (FFT)"},"105":{"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":"105","title":"Motivation: Fast polynomial multiplication"},"106":{"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":"106","title":"The (radix-2) Cooley-Tukey algorithm"},"107":{"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":"107","title":"Inverse FFT"},"108":{"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":"108","title":"The Schwartz-Zippel lemma"},"109":{"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":"109","title":"Vanishing polynomial"},"11":{"body":"halo2::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 » Mock prover","id":"11","title":"Mock prover"},"110":{"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":"110","title":"Lagrange basis functions"},"111":{"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":"111","title":"Lagrange interpolation"},"112":{"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":"112","title":"References"},"113":{"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":"113","title":"Cryptographic groups"},"114":{"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":"114","title":"Pedersen commitment"},"115":{"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":"115","title":"Vector Pedersen commitment"},"116":{"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":"116","title":"Diffie--Hellman"},"117":{"body":"","breadcrumbs":"Background Material » Multiscalar multiplication","id":"117","title":"Multiscalar multiplication"},"118":{"body":"Reference: https://jbootle.github.io/Misc/pippenger.pdf","breadcrumbs":"Background Material » TODO: Pippenger's algorithm","id":"118","title":"TODO: Pippenger's algorithm"},"119":{"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":"119","title":"Elliptic curves"},"12":{"body":"The dev-graph feature flag exposes several helper methods for creating graphical representations of circuits. halo2::dev::circuit_layout renders the circuit layout as a grid: Columns are layed out from left to right as advice, instance, and fixed. The order of columns is otherwise without meaning. Advice columns have a red background. Instance columns have a white 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. halo2::dev::circuit_dot_graph builds a DOT graph string representing the given circuit, which can then be rendered witha variety of layout programs . The graph is built from calls to Layouter::namespace both within the circuit, and inside the gadgets and chips that it uses.","breadcrumbs":"User Documentation » Circuit visualizations","id":"12","title":"Circuit visualizations"},"120":{"body":"","breadcrumbs":"Background Material » Curve arithmetic","id":"120","title":"Curve arithmetic"},"121":{"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 » Point doubling","id":"121","title":"Point doubling"},"122":{"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 » Projective coordinates","id":"122","title":"Projective coordinates"},"123":{"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−y0x1−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 X1Z2=X2Z1 and Y1Z2=Y2Z1.","breadcrumbs":"Background Material » Point addition","id":"123","title":"Point addition"},"124":{"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: (ζpx,ζp2x), because the computation x3 effectively kills the ζ component of the x-coordinate. Applying the map (x,y)↦(ζpx,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 » Curve endomorphisms","id":"124","title":"Curve endomorphisms"},"125":{"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 » Curve point compression","id":"125","title":"Curve point compression"},"126":{"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 » Serialization","id":"126","title":"Serialization"},"127":{"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 » Deserialization","id":"127","title":"Deserialization"},"128":{"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 » Cycles of curves","id":"128","title":"Cycles of curves"},"129":{"body":"Reference: https://github.com/zcash/pasta","breadcrumbs":"Background Material » TODO: Pallas-Vesta curves","id":"129","title":"TODO: Pallas-Vesta curves"},"13":{"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:

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","breadcrumbs":"User Documentation » Cost estimator","id":"13","title":"Cost estimator"},"130":{"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 » Hashing to curves","id":"130","title":"Hashing to curves"},"131":{"body":"Reference: https://eprint.iacr.org/2019/403.pdf","breadcrumbs":"Background Material » TODO: Simplified SWU","id":"131","title":"TODO: Simplified SWU"},"132":{"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 » References","id":"132","title":"References"},"133":{"body":"We call the field over which the circuit is defined F=Fp. Let n=2k, and assume that ω is a primitive root of unity of order n in F×, so that F× has a multiplicative subgroup H={1,ω,ω2,⋯,ωn−1}. This forms a Lagrange basis corresponding to the elements in the subgroup.","breadcrumbs":"Background Material » [WIP] UltraPLONK arithmetisation","id":"133","title":"[WIP] UltraPLONK arithmetisation"},"134":{"body":"A polynomial rule defines a constraint that must hold between its specified columns at every row (i.e. at every element in the multiplicative subgroup). e.g. a * sa + b * sb + a * b * sm + c * sc + PI = 0","breadcrumbs":"Background Material » Polynomial rules","id":"134","title":"Polynomial rules"},"135":{"body":"fixed columns : fixed for all instances of a particular circuit. These include selector columns, which toggle parts of a polynomial rule \"on\" or \"off\" to form a \"custom gate\". They can also include any other fixed data. advice columns : variable values assigned in each instance of the circuit. Corresponds to the prover's secret witness. public input : like advice columns, but publicly known values. Each column is a vector of n values, e.g. a=[a0,a1,⋯,an−1]. We can think of the vector as the evaluation form of the column polynomial a(X),X∈H. To recover the coefficient form, we can use Lagrange interpolation , such that a(ωi)=ai.","breadcrumbs":"Background Material » Columns","id":"135","title":"Columns"},"136":{"body":"Define permutation between a set of columns, e.g. σ(a,b,c) Assert equalities between specific cells in these columns, e.g. b1=c0 Construct permuted columns which should evaluate to same value as original columns","breadcrumbs":"Background Material » Equality constraints","id":"136","title":"Equality constraints"},"137":{"body":"Z(ωi):=0≤j≤i∏Ck(ωj)+βSk(ωj)+γCk(ωj)+βδkωj+γ, where i=0,⋯,n−1 indexes over the size of the multiplicative subgroup, and k=0,⋯,m−1 indexes over the advice columns involved in the permutation. This is a running product, where each term includes the cumulative product of the terms before it. TODO: what is δ? keep columns linearly independent Check the constraints: First term is equal to one L0(X)⋅(1−Z(X))=0 Running product is well-constructed. For each row, we check that this holds: Z(ωi)⋅(C(ωi)+βSk(ωi)+γ)−Z(ωi−1)⋅(C(ωi)+δkβωi+γ)=0 Rearranging gives Z(ωi)=Z(ωi−1)C(ωi)+βSk(ωi)+γC(ωi)+βδkωi+γ, which is how we defined the grand product polynomial in the first place.","breadcrumbs":"Background Material » Permutation grand product","id":"137","title":"Permutation grand product"},"138":{"body":"Reference: Generic Lookups with PLONK (DRAFT)","breadcrumbs":"Background Material » Lookup","id":"138","title":"Lookup"},"139":{"body":"We want to check that the expressions defined by the gate constraints, permutation constraints and loookup constraints evaluate to zero at all elements in the multiplicative subgroup. To do this, the prover collapses all the expressions into one polynomial H(X)=i=0∑eyiEi(X), where e is the number of expressions and y is a random challenge used to keep the constraints linearly independent. The prover then divides this by the vanishing polynomial (see section: Vanishing polynomial ) and commits to the resulting quotient Commit(Q(X)),where Q(X)=ZH(X)H(X). The verifier responds with a random evaluation point x, to which the prover replies with the claimed evaluations q=Q(x),{ei}i=0e={Ei(x)}i=0e. Now, all that remains for the verifier to check is that the evaluations satisfy q=?ZH(x)∑i=0eyiei. Notice that we have yet to check that the committed polynomials indeed evaluate to the claimed values at x,q=?Q(x),{ei}i=0e=?{Ei(x)}i=0e. This check is handled by the polynomial commitment scheme (described in the next section).","breadcrumbs":"Background Material » Vanishing argument","id":"139","title":"Vanishing argument"},"14":{"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":"14","title":"A simple example"},"140":{"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","id":"140","title":"Polynomial commitment using inner product argument"},"141":{"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 » Setup","id":"141","title":"Setup"},"142":{"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 » Commit","id":"142","title":"Commit"},"143":{"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=log2d 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 argment, the verifier selects a random group element U and sends it to the prover. We initialise 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: LjRj=⟨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 » Open (prover) and OpenVerify (verifier)","id":"143","title":"Open (prover) and OpenVerify (verifier)"},"144":{"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−1X2i−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","id":"144","title":"Recursion"},"15":{"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; /// 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) -> Result<(), Error>;\n}","breadcrumbs":"User Documentation » Define instructions","id":"15","title":"Define instructions"},"16":{"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 » Define a chip implementation","id":"16","title":"Define a chip implementation"},"17":{"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], // We need to create a permutation between our advice columns. This allows us to // copy numbers within these columns from arbitrary rows, which we can use to load // inputs into our instruction regions. perm: Permutation, // 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, // The selector for the public-input gate, which uses one of the advice columns. s_pub: Selector,\n} impl FieldChip { fn construct(config: >::Config) -> Self { Self { config, _marker: PhantomData, } } fn configure( meta: &mut ConstraintSystem, advice: [Column; 2], instance: Column, ) -> >::Config { let perm = Permutation::new( meta, &advice .iter() .map(|column| (*column).into()) .collect::>(), ); let s_mul = meta.selector(); let s_pub = 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, Rotation::cur()); // The polynomial expression 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. s_mul * (lhs * rhs + out * -F::one()) }); // Define our public-input gate! meta.create_gate(\"public input\", |meta| { // 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_instance(instance, Rotation::cur()); let s = meta.query_selector(s_pub, Rotation::cur()); // We simply constrain the advice cell to be equal to the instance cell, // when the selector is enabled. s * (p + a * -F::one()) }); FieldConfig { advice, perm, s_mul, s_pub, } }\n}","breadcrumbs":"User Documentation » Configure the chip","id":"17","title":"Configure the chip"},"18":{"body":"/// A variable representing a number.\n#[derive(Clone)]\nstruct Number { cell: Cell, value: Option,\n} impl NumericInstructions for FieldChip { type Num = Number; fn load_private( &self, mut layouter: impl Layouter, value: Option, ) -> Result { let config = self.config(); let mut num = None; layouter.assign_region( || \"load private\", |mut region| { let cell = region.assign_advice( || \"private input\", config.advice[0], 0, || value.ok_or(Error::SynthesisError), )?; num = Some(Number { cell, value }); Ok(()) }, )?; Ok(num.unwrap()) } fn mul( &self, mut layouter: impl Layouter, a: Self::Num, b: Self::Num, ) -> Result { let config = self.config(); let mut out = None; 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. let lhs = region.assign_advice( || \"lhs\", config.advice[0], 0, || a.value.ok_or(Error::SynthesisError), )?; let rhs = region.assign_advice( || \"rhs\", config.advice[1], 0, || b.value.ok_or(Error::SynthesisError), )?; region.constrain_equal(&config.perm, a.cell, lhs)?; region.constrain_equal(&config.perm, b.cell, rhs)?; // Now we can assign the multiplication result into the output position. let value = a.value.and_then(|a| b.value.map(|b| a * b)); let cell = region.assign_advice( || \"lhs * rhs\", config.advice[0], 1, || value.ok_or(Error::SynthesisError), )?; // Finally, we return a variable representing the output, // to be used in another part of the circuit. out = Some(Number { cell, value }); Ok(()) }, )?; Ok(out.unwrap()) } fn expose_public(&self, mut layouter: impl Layouter, num: Self::Num) -> Result<(), Error> { let config = self.config(); layouter.assign_region( || \"expose public\", |mut region: Region<'_, F>| { // Enable the public-input gate. config.s_pub.enable(&mut region, 0)?; // Load the output into the correct advice column. let out = region.assign_advice( || \"public advice\", config.advice[1], 0, || num.value.ok_or(Error::SynthesisError), )?; region.constrain_equal(&config.perm, num.cell, out)?; // We don't assign to the instance column inside the circuit; // the mapping of public inputs to cells is provided to the prover. Ok(()) }, ) }\n}","breadcrumbs":"User Documentation » Implement chip traits","id":"18","title":"Implement chip traits"},"19":{"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.\nstruct MyCircuit { 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; 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(); FieldChip::configure(meta, advice, instance) } fn synthesize(&self, cs: &mut impl Assignment, config: Self::Config) -> Result<(), Error> { let mut layouter = SingleChipLayouter::new(cs)?; 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)?; // We only have access to plain multiplication. // We could implement our circuit as: // asq = a*a // bsq = b*b // c = asq*bsq // // but it's more efficient to implement it as: // ab = a*b // c = ab^2 let ab = field_chip.mul(layouter.namespace(|| \"a * b\"), a, b)?; let c = field_chip.mul(layouter.namespace(|| \"ab * ab\"), ab.clone(), ab)?; // Expose the result as a public input to the circuit. field_chip.expose_public(layouter.namespace(|| \"expose c\"), c) }\n}","breadcrumbs":"User Documentation » Build the circuit","id":"19","title":"Build the circuit"},"2":{"body":"Copyright 2020 The Electric Coin Company. You may use this package under the Bootstrap Open Source Licence, version 1.0, or at your option, any later version. See the file LICENSE-BOSL for the terms of the Bootstrap Open Source Licence, version 1.0. The purpose of the BOSL is to allow commercial improvements to the package while ensuring that all improvements are open source. See here for why the BOSL exists.","breadcrumbs":"License","id":"2","title":"License"},"20":{"body":"halo2::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 = 3; // Prepare the private and public inputs to the circuit! let a = Fp::from(2); let b = Fp::from(3); let c = a.square() * b.square(); // Instantiate the circuit with the private inputs. let circuit = MyCircuit { a: Some(a), b: Some(b), }; // Arrange the public input. We expose the multiplication result in row 6 // 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; // 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[6] += Fp::one(); let prover = MockProver::run(k, &circuit, vec![public_inputs]).unwrap(); assert_eq!( prover.verify(), Err(vec![VerifyFailure::Gate { gate_index: 1, gate_name: \"public input\", row: 6, }]) );","breadcrumbs":"User Documentation » Testing the circuit","id":"20","title":"Testing the circuit"},"21":{"body":"You can find the source code for this example here .","breadcrumbs":"User Documentation » Full example","id":"21","title":"Full example"},"22":{"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","id":"22","title":"Lookup tables"},"23":{"body":"","breadcrumbs":"User Documentation » Gadgets","id":"23","title":"Gadgets"},"24":{"body":"This section contains various ideas and snippets that you might find useful while writing halo2 circuits.","breadcrumbs":"User Documentation » Tips and tricks","id":"24","title":"Tips and tricks"},"25":{"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 » Small range constraints","id":"25","title":"Small range constraints"},"26":{"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 » Algorithm","id":"41","title":"Algorithm"},"42":{"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 » Broken alternatives","id":"42","title":"Broken alternatives"},"43":{"body":"We need to check a permutation of cells in m columns, represented in Lagrange basis by polynomials p0,…,pm−1. We first label each cell in those m columns with a unique element of F×. 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. If we have a permutation σ(column:i,row:j)=(column:i′,row:j′), we can represent it as 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(X) such that IDi(X)=δi⋅X. Now given our permutation represented by s0,…,sm−1 over columns represented by p0,…,pm−1, we want to ensure that: i=0∏m−1j=0∏n−1(pi(ωj)+β⋅si(ωj)+γpi(ωj)+β⋅δi⋅ωj+γ)=1 Let ZP be such that ZP(ω0)=ZP(ωn)=1 and for 0≤j {}\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 » Chip instructions","id":"69","title":"Chip instructions"},"7":{"body":"In order to combine functionality from several chips, we compose them in a tree. The top-level chip defines a set of fixed, advice, and instance columns, and then specifies how they should be distributed between lower-level chips. In the simplest case, each lower-level chips will use columns disjoint from the other chips. However, it is allowed to share a column between chips. It is important to optimize the number of advice columns in particular, because that affects proof size. The result (possibly after optimization) is a UPA configuration. Our circuit implementation will be parameterized on a chip, and can use any features of the supported lower-level chips via the top-level chip. Our hope is that less expert users will normally be able to find an existing chip that supports the operations they need, or only have to make minor modifications to an existing chip. Expert users will have full control to do the kind of circuit optimizations that ECC is famous for 🙂.","breadcrumbs":"Concepts » Composing chips","id":"7","title":"Composing chips"},"70":{"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 for SHA-256","id":"70","title":"16-bit table chip for SHA-256"},"71":{"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′EnewAnew========(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+Wtreduce6(H′+D)reduce7(H′+Maj(A,B,C)+Σ0(A)) where reducei must handle a carry 0≤carry1 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","id":"95","title":"Fields"},"96":{"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∏kgiai. 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 » Groups","id":"96","title":"Groups"},"97":{"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. 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