diff --git a/background/fields.html b/background/fields.html
index 834affa9..77b73a01 100644
--- a/background/fields.html
+++ b/background/fields.html
@@ -209,12 +209,13 @@ which is the unique element (aside) Additive vs multiplicative notation
-
If ⋅ is written as + and the identity as 0 or O, then we say the
-group is "written additively". If ⋅ is written as × or omitted (i.e.
-a⋅b written as ab) and the identity as 1, then we say it is "written
-multiplicatively". 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
+
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 times
If the order n of a finite group is prime, then the group is cyclic, and every
diff --git a/print.html b/print.html
index 37750551..b71d81ae 100644
--- a/print.html
+++ b/print.html
@@ -1689,12 +1689,13 @@ which is the unique element (aside) Additive vs multiplicative notation
-
If ⋅ is written as + and the identity as 0 or O, then we say the
-group is "written additively". If ⋅ is written as × or omitted (i.e.
-a⋅b written as ab) and the identity as 1, then we say it is "written
-multiplicatively". 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
+
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 times
If the order n of a finite group is prime, then the group is cyclic, and every
diff --git a/searchindex.js b/searchindex.js
index 03937a0e..7fe987bb 100644
--- a/searchindex.js
+++ b/searchindex.js
@@ -1 +1 @@
-Object.assign(window.search, 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: This library is being actively developed and should not be used in production software.","breadcrumbs":"halo2 Crates.io","id":"0","title":"halo2 Crates.io"},"1":{"body":"","breadcrumbs":"Documentation","id":"1","title":"Documentation"},"10":{"body":"Let's start with a simple circuit, to introduce you to the common APIs and how they are used. The circuit will take a public input c, and will prove knowledge of two private inputs a and b such that a⋅b=c. # extern crate halo2;\nuse halo2::arithmetic::FieldExt; struct MyCircuit { a: F, b: F,\n} TODO","breadcrumbs":"User Documentation » A simple example","id":"10","title":"A simple example"},"100":{"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 points in the subgroup.","breadcrumbs":"Background Material » [WIP] UltraPLONK arithmetisation","id":"100","title":"[WIP] UltraPLONK arithmetisation"},"101":{"body":"A polynomial rule defines a constraint that must hold between its specified columns at every row (i.e. at every point in the multiplicative subgroup). e.g. a * sa + b * sb + a * b * sm + c * sc + PI = 0","breadcrumbs":"Background Material » Polynomial rules","id":"101","title":"Polynomial rules"},"102":{"body":"fixed (i.e. \"selector\") columns : fixed for all instances of a particular circuit. These columns toggle parts of a polynomial rule \"on\" or \"off\" to form a \"custom gate\". 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":"102","title":"Columns"},"103":{"body":"Define permutation between a set of columns, e.g. σ(a,b,c) Copy specific cells between these columns, e.g. b1=c0 Construct permuted columns which should evaluate to same value as original columns","breadcrumbs":"Background Material » Copy constraints","id":"103","title":"Copy constraints"},"104":{"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":"104","title":"Permutation grand product"},"105":{"body":"Reference: Generic Lookups with PLONK (DRAFT)","breadcrumbs":"Background Material » Lookup","id":"105","title":"Lookup"},"106":{"body":"We want to check that the expressions defined by the gate constraints, permutation constraints and loookup constraints evaluate to zero at all points 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":"106","title":"Vanishing argument"},"107":{"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":"107","title":"Polynomial commitment using inner product argument"},"108":{"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":"108","title":"Setup"},"109":{"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":"109","title":"Commit"},"11":{"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":"11","title":"Lookup tables"},"110":{"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":"110","title":"Open (prover) and OpenVerify (verifier)"},"111":{"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":"111","title":"Recursion"},"12":{"body":"","breadcrumbs":"User Documentation » Gadgets","id":"12","title":"Gadgets"},"13":{"body":"This section contains various ideas and snippets that you might find useful while writing halo2 circuits.","breadcrumbs":"User Documentation » Tips and tricks","id":"13","title":"Tips and tricks"},"14":{"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":"14","title":"Small range constraints"},"15":{"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: 00011011→0000⟹0→0→0001⟹1→1→0100⟹2→4→0101⟹3→5 Then, we construct the Lagrange basis polynomial for each point using the identity: lj(X)=0≤m≤k,m=j∏xj−xmx−xm, where k+1 is the number of data points. (k=3 in our example above.) Recall that the Lagrange basis polynomial lj(X) evaluates to 1 at X=xj and 0 at all other xi,j=i. Continuing our example, we get four Lagrange basis polynomials: l0(X)l1(X)l2(X)l3(X)====(−3)(−2)(−1)(X−3)(X−2)(X−1)(−2)(−1)(1)(X−3)(X−2)(X)(−1)(1)(2)(X−3)(X−1)(X)(1)(2)(3)(X−2)(X−1)(X) Our polynomial constraint is then ⟹f(0)l0(X)0⋅l0(X)++f(1)l1(X)1⋅l1(X)++f(2)l2(X)4⋅l2(X)++f(3)l3(X)−f(X)5⋅l3(X)−f(X)==00.","breadcrumbs":"User Documentation » Small set interpolation","id":"15","title":"Small set interpolation"},"16":{"body":"","breadcrumbs":"Design","id":"16","title":"Design"},"17":{"body":"We use slightly different language than others to describe PLONK concepts. Here's the overview: We like to think of PLONK-like arguments as tables, where each column corresponds to a \"wire\". We refer to entries in this table as \"cells\". We like to call \"selector polynomials\" and so on \"fixed columns\" instead. We then refer specifically to a \"selector constraint\" when a cell in a fixed column is being used to control whether a particular constraint is enabled in that row. We call the other polynomials \"advice columns\" usually, when they're populated by the prover. We use the term \"rule\" to refer to a \"gate\" like A(X)⋅qA(X)+B(X)⋅qB(X)+A(X)⋅B(X)⋅qM(X)+C(X)⋅qC(X)=0. TODO: Check how consistent we are with this, and update the code and docs to match.","breadcrumbs":"Note on Language","id":"17","title":"Note on Language"},"18":{"body":"","breadcrumbs":"Design » Proving system","id":"18","title":"Proving system"},"19":{"body":"Consider the commitments A,B,C,D to polynomials a(X),b(X),c(X),d(X). Let's say that a and b were queried at the point x, while c and d were queried at both points x and ωx. (Here, ω is the primitive root of unity in the multiplicative subgroup over which we constructed the polynomials). We can group the commitments in terms of the sets of points at which they were queried: {x}AB{x,ωx}CD The multipoint opening optimisation proceeds as such: Sample random x3, at which we evaluate a(X),b(X),c(X),d(X). The prover provides evaluations of each polynomial at each point of interest: a(x3),b(x3),c(x3),d(x3),c(ωx3),d(ωx3) Sample random x4, to keep a,b,c,d linearly independent. Accumulate polynomials and their corresponding evaluations according to the point set at which they were queried: q_polys: q1(X)q2(X)==a(X)c(X)++x4b(X)x4d(X) q_eval_sets: [ [a(x_3) + x_4 b(x_3)], [ c(x_3) + x_4 d(x_3), c(\\omega x_3) + x_4 d(\\omega x_3) ] ] NB: q_eval_sets is a vector of sets of evaluations, where the outer vector goes over the point sets, and the inner vector goes over the points in each set. Interpolate each set of values in q_eval_sets: r_polys: r1(X)s.t.r2(X)s.t.r1(x3)r2(x3)r2(ωx3)===a(x3)+x4b(x3)c(x3)+x4d(x3)c(ωx3)+x4d(ωx3) Construct f_polys which check the correctness of q_polys: f_polys f1(X)f2(X)==X−x3q1(X)−r1(X)(X−x3)(X−ωx3)q2(X)−r2(X) If q1(x3)=r1(x3), then f1(X) should be a polynomial. If q2(x3)=r2(x3) and q2(ωx3)=r2(ωx3) then f2(X) should be a polynomial. Sample random x5 to keep the f_polys linearly independent. Construct f(X)=f1(X)+x5f2(X). Sample random x6, at which we evaluate f(X): f(x6)==f1(x6)x6−x3q1(x6)−r1(x6)++x5f2(x6)x5(x6−x3)(x6−ωx3)q2(x6)−r2(x6) Sample random x7 to keep f(X) and q_polys linearly independent. Construct final_poly, final_poly(X)=f(X)+x7q1(X)+x72q2(X), which is the polynomial we commit to in the inner product argument.","breadcrumbs":"Design » Proving system » Multipoint opening argument","id":"19","title":"Multipoint opening argument"},"2":{"body":"Copyright 2020 The Electric Coin Company. You may use this package under the Transitive Grace Period Public Licence, version 1.0, or at your option, any later version. See the file LICENSE-TGPPL for the terms of the Transitive Grace Period Public Licence, version 1.0. The purpose of the TGPPL is to allow commercial improvements to the package while ensuring that all improvements are eventually open source. See here for why the TGPPL exists, graphically illustrated on three slides.","breadcrumbs":"License","id":"2","title":"License"},"20":{"body":"Given that gates in halo2 circuits operate \"locally\" (on cells in the current row or defined relative rows), it is common to need to copy a value from some arbitrary cell into the current row for use in a gate. This is performed with an equality constraint, which enforces that the source and destination cells contain the same value. We implement these equality constraints by constructing a permutation that represents the constraints, and then using a permutation argument within the proof to enforce them.","breadcrumbs":"Design » Proving system » Permutation argument","id":"20","title":"Permutation argument"},"21":{"body":"A permutation is a one-to-one and onto mapping of a set onto itself. A permutation can be factored uniquely into a composition of cycles (up to ordering of cycles, and rotation of each cycle). We sometimes use cycle notation to write permutations. Let (a b c) denote a cycle where a maps to b, b maps to c, and c maps to a (with the obvious generalisation to arbitrary-sized cycles). Writing two or more cycles next to each other denotes a composition of the corresponding permutations. For example, (a b) (c d) denotes the permutation that maps a to b, b to a, c to d, and d to c.","breadcrumbs":"Design » Proving system » Notation","id":"21","title":"Notation"},"22":{"body":"","breadcrumbs":"Design » Proving system » Constructing the permutation","id":"22","title":"Constructing the permutation"},"23":{"body":"We want to construct a permutation in which each subset of variables that are in a equality-constraint set form a cycle. For example, suppose that we have a circuit that defines the following equality constraints: a≡b a≡c d≡e From this we have the equality-constraint sets {a,b,c} and {d,e}. We want to construct the permutation: (a b c) (d e) which defines the mapping of [a,b,c,d,e] to [b,c,a,e,d].","breadcrumbs":"Design » Proving system » Goal","id":"23","title":"Goal"},"24":{"body":"We need to keep track of the set of cycles, which is a set of disjoint sets . Efficient data structures for this problem are known; for the sake of simplicity we choose one that is not asymptotically optimal but is easy to implement. We represent the current state as: an array mapping for the permutation itself; an auxiliary array aux that keeps track of a distinguished element of each cycle; another array sizes that keeps track of the size of each cycle. We have the invariant that for each element x in a given cycle C, aux(x) points to the same element c∈C. This allows us to quickly decide whether two given elements x and y are in the same cycle, by checking whether aux(x)=aux(y). Also, sizes(aux(x)) gives the size of the cycle containing x. (This is guaranteed only for sizes(aux(x))), not for sizes(x).) The algorithm starts with a representation of the identity permutation: for all x, we set mapping(x)=x, aux(x)=x, and sizes(x)=1. To add an equality constraint left≡right: Check whether left and right are already in the same cycle, i.e. whether aux(left)=aux(right). If so, there is nothing to do. Otherwise, left and right belong to different cycles. Make left the larger cycle and right the smaller one, by swapping them iff sizes(aux(left)) 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":"24","title":"Algorithm"},"25":{"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 4 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":"25","title":"Broken alternatives"},"26":{"body":"TODO: Document what we do with the permutation once we have it.","breadcrumbs":"Design » Proving system » Argument specification","id":"26","title":"Argument specification"},"27":{"body":"halo2 uses the following lookup technique, which allows for lookups in arbitrary sets, and is arguably simpler than Plookup.","breadcrumbs":"Design » Proving system » Lookup argument","id":"27","title":"Lookup argument"},"28":{"body":"In addition to the general notes on language : We call the Z(X) polynomial (the grand product argument polynomial for the permutation argument) the \"permutation product\" column.","breadcrumbs":"Design » Proving system » Note on Language","id":"28","title":"Note on Language"},"29":{"body":"We express lookups in terms of a \"subset argument\" over a table with 2k rows (numbered from 0), and columns A and S. The goal of the subset argument is to enforce that every cell in A is equal to some cell in S. This means that more than one cell in A can be equal to the same cell in S, and some cells in S don't need to be equal to any of the cells in A. S might be fixed, but it doesn't need to be. That is, we can support looking up values in either fixed or variable tables (where the latter includes advice columns). A and S can contain duplicates. If the sets represented by A and/or S are not naturally of size 2k, we extend S with duplicates and A with dummy values known to be in S. Alternatively we could add a \"lookup selector\" that controls which elements of the A column participate in lookups. This would modify the occurrence of A(X) in the permutation rule below to replace A with, say, S0 if a lookup is not selected. Let ℓi be the Lagrange basis polynomial that evaluates to 1 at row i, and 0 otherwise. We start by allowing the prover to supply permutation columns of A and S. Let's call these A′ and S′, respectively. We can enforce that they are permutations using a permutation argument with product column Z with the rules: Z(X)(A(X)+β)(S(X)+γ)−Z(ω−1X)(A′(X)+β)(S′(X)+γ)=0ℓ0(X)(Z(X)−1)=0 This is a version of the permutation argument which allows A′ and S′ to be permutations of A and S, respectively, but doesn't specify the exact permutations. β and γ are separate challenges so that we can combine these two permutation arguments into one without worrying that they might interfere with each other. The goal of these permutations is to allow A′ and S′ to be arranged by the prover in a particular way: All the cells of column A′ are arranged so that like-valued cells are vertically adjacent to each other. This could be done by some kind of sorting algorithm, but all that matters is that like-valued cells are on consecutive rows in column A′, and that A′ is a permutation of A. The first row in a sequence of like values in A′ is the row that has the corresponding value in S′. Apart from this constraint, S′ is any arbitrary permutation of S. Now, we'll enforce that either Ai′=Si′ or that Ai′=Ai−1′, using the rule (A′(X)−S′(X))⋅(A′(X)−A′(ω−1X))=0 In addition, we enforce A0′=S0′ using the rule ℓ0(X)⋅(A′(X)−S′(X))=0 Together these constraints effectively force every element in A′ (and thus A) to equal at least one element in S′ (and thus S). Proof: by induction on prefixes of the rows.","breadcrumbs":"Design » Proving system » Technique Description","id":"29","title":"Technique Description"},"3":{"body":"First we'll describe the concepts behind zero-knowledge proof systems; the arithmetization (kind of circuit description) used by Halo 2; and the abstractions we use to build circuit implementations.","breadcrumbs":"Concepts","id":"3","title":"Concepts"},"30":{"body":"There is the original column A and the fixed column S. There is a permutation product column Z. There are the two permutations A′ and S′. The gates are all of low degree.","breadcrumbs":"Design » Proving system » Cost","id":"30","title":"Cost"},"31":{"body":"halo2's lookup argument implementation generalizes the above technique in the following ways: A and S can be extended to multiple columns, combined using a random challenge. A′ and S′ stay as single columns. The commitments to the columns of S can be precomputed, then combined cheaply once the challenge is known by taking advantage of the homomorphic property of Pedersen commitments. Then, a lookup argument for an arbitrary-width relation can be implemented in terms of a subset argument, i.e. to constrain R(x,y,...) in each row, consider R as a set of tuples S (using the method of the previous point), and check that (x,y,...)∈R. In the case where R represents a function, this implicitly also checks that the inputs are in the domain. This is typically what we want, and often saves an additional range check. We can support multiple tables in the same circuit, by combining them into a single table that includes a tag column to identify the original table. The tag column could be merged with the \"lookup selector\" mentioned earlier, if this were implemented. These generalizations are similar to those in sections 4 and 5 of the Plookup paper That is, the differences from Plookup are in the subset argument. This argument can then be used in all the same ways; for instance, the optimized range check technique in section 5 of the Plookup paper can also be used with this subset argument.","breadcrumbs":"Design » Proving system » Generalizations","id":"31","title":"Generalizations"},"32":{"body":"","breadcrumbs":"Design » Implementation","id":"32","title":"Implementation"},"33":{"body":"In proving system implementations like bellman, there is a concrete Proof struct that encapsulates the proof data, is returned by a prover, and can be passed to a verifier. halo2 does not contain any proof-like structures, for several reasons: The Proof structures would contain vectors of (vectors of) curve points and scalars. This complicates serialization/deserialization of proofs because the lengths of these vectors depend on the configuration of the circuit. However, we didn't want to encode the lengths of vectors inside of proofs, because at runtime the circuit is fixed, and thus so are the proof sizes. It's easy to accidentally put stuff into a Proof structure that isn't also placed in the transcript, which is a hazard when developing and implementing a proving system. We needed to be able to create multiple PLONK proofs at the same time; these proofs share many different substructures when they are for the same circuit. Instead, halo2 treats proof objects as opaque byte streams. Creation and consumption of these byte streams happens via the transcript: The TranscriptWrite trait represents something that we can write proof components to (at proving time). The TranscriptRead trait represents something that we can read proof components from (at verifying time). Crucially, implementations of TranscriptWrite are responsible for simultaneously writing to some std::io::Write buffer at the same time that they hash things into the transcript, and similarly for TranscriptRead/std::io::Read. As a bonus, treating proofs as opaque byte streams ensures that verification accounts for the cost of deserialization, which isn't negligible due to point compression.","breadcrumbs":"Design » Proofs as opaque byte streams","id":"33","title":"Proofs as opaque byte streams"},"34":{"body":"In this section we document the gadgets and chip designs provided by halo2.","breadcrumbs":"Design » Gadgets","id":"34","title":"Gadgets"},"35":{"body":"","breadcrumbs":"Design » Gadgets » SHA-256","id":"35","title":"SHA-256"},"36":{"body":"SHA-256 is specified in NIST FIPS PUB 180-4 . Unlike the specification, we use ⊞ for addition modulo 232, and + for field addition. ⊕ is used for XOR.","breadcrumbs":"Design » Gadgets » Specification","id":"36","title":"Specification"},"37":{"body":"SHA-256 maintains state in eight 32-bit variables. It processes input as 512-bit blocks, but internally splits these blocks into 32-bit chunks. We therefore designed the SHA-256 gadget to consume input in 32-bit chunks.","breadcrumbs":"Design » Gadgets » Gadget interface","id":"37","title":"Gadget interface"},"38":{"body":"The SHA-256 gadget requires a chip with the following instructions: # extern crate halo2;\n# use halo2::plonk::Error;\n# use std::fmt;\n#\n# trait Chip: Sized {}\n# trait Layouter {}\nconst BLOCK_SIZE: usize = 16;\nconst DIGEST_SIZE: usize = 8; pub trait Sha256Instructions: Chip { /// Variable representing the SHA-256 internal state. type State: Clone + fmt::Debug; /// Variable representing a 32-bit word of the input block to the SHA-256 compression /// function. type BlockWord: Copy + fmt::Debug; /// Places the SHA-256 IV in the circuit, returning the initial state variable. fn initialization_vector(layouter: &mut impl Layouter) -> Result; /// Starting from the given initial state, processes a block of input and returns the /// final state. fn compress( layouter: &mut impl Layouter, initial_state: &Self::State, input: [Self::BlockWord; BLOCK_SIZE], ) -> Result; /// Converts the given state into a message digest. fn digest( layouter: &mut impl Layouter, state: &Self::State, ) -> Result<[Self::BlockWord; DIGEST_SIZE], Error>;\n} TODO: Add instruction for computing padding. This set of instructions was chosen to strike a balance between the reusability of the instructions, and the scope for chips to internally optimise them. In particular, we considered splitting the compression function into its constituent parts (Ch, Maj etc), and providing a compression function gadget that implemented the round logic. However, this would prevent chips from using relative references between the various parts of a compression round. Having an instruction that implements all compression rounds is also similar to the Intel SHA extensions, which provide an instruction that performs multiple compression rounds.","breadcrumbs":"Design » Gadgets » Chip instructions","id":"38","title":"Chip instructions"},"39":{"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":"39","title":"16-bit table chip for SHA-256"},"4":{"body":"The aim of any proof system is to be able to prove interesting mathematical or cryptographic statements . Typically, in a given protocol we will want to prove families of statements that differ in their public inputs . The prover will also need to show that they know some private inputs that make the statement hold. To do this we write down a relation , R, that specifies which combinations of public and private inputs are valid. The terminology above is intended to be aligned with the ZKProof Community Reference . To be precise, we should distinguish between the relation R, and its implementation to be used in a proof system. We call the latter a circuit . The language that we use to express circuits for a particular proof system is called an arithmetization . Usually, an arithmetization will define circuits in terms of polynomial constraints on variables over a field. The process of expressing a particular relation as a circuit is also sometimes called \"arithmetization\", but we'll avoid that usage. To create a proof of a statement, the prover will need to know the private inputs, and also intermediate values, called advice values, that are used by the circuit. We assume that we can compute advice values efficiently from the private and public inputs. The particular advice values will depend on how we write the circuit, not only on the high-level statement. The private inputs and advice values are collectively called a witness . Some authors use \"witness\" as just a synonym for private inputs. But in our usage, a witness includes advice, i.e. it includes all values that the prover supplies to the circuit. For example, suppose that we want to prove knowledge of a preimage x of a hash function H for a digest y: The private input would be the preimage x. The public input would be the digest y. The relation would be {(x,y):H(x)=y}. For a particular public input Y, the statement would be: {(x):H(x)=Y}. The advice would be all of the intermediate values in the circuit implementing the hash function. The witness would be x and the advice. A Non-interactive Argument allows a prover to create a proof for a given statement and witness. The proof is data that can be used to convince a verifier that there exists a witness for which the statement holds. The security property that such proofs cannot falsely convince a verifier is called soundness . A Non-interactive Argument of Knowledge ( NARK ) further convinces the verifier that the prover knew a witness for which the statement holds. This security property is called knowledge soundness , and it implies soundness. In practice knowledge soundness is more useful for cryptographic protocols than soundness: if we are interested in whether Alice holds a secret key in some protocol, say, we need Alice to prove that she knows the key, not just that it exists. Knowledge soundness is formalized by saying that an extractor , which can observe precisely how the proof is generated, must be able to compute the witness. This property is subtle given that proofs can be malleable . That is, depending on the proof system it may be possible to take an existing proof (or set of proofs) and, without knowing the witness(es), modify it/them to produce a distinct proof of the same or a related statement. Higher-level protocols that use malleable proof systems need to take this into account. Even without malleability, proofs can also potentially be replayed . For instance, we would not want Alice in our example to be able to present a proof generated by someone else, and have that be taken as a demonstration that she knew the key. If a proof yields no information about the witness (other than that a witness exists and was known to the prover), then we say that the proof system is zero knowledge . If a proof system produces short proofs ---i.e. of length polylogarithmic in the circuit size--- then we say that it is succinct . A succinct NARK is called a SNARK ( Succinct Non-Interactive Argument of Knowledge ). By this definition, a SNARK need not have verification time polylogarithmic in the circuit size. Some papers use the term efficient to describe a SNARK with that property, but we'll avoid that term since it's ambiguous for SNARKs that support amortized or recursive verification, which we'll get to later. A zk-SNARK is a zero-knowledge SNARK.","breadcrumbs":"Concepts » Proof systems","id":"4","title":"Proof systems"},"40":{"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":"64","title":"Fields"},"65":{"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 + and the identity as 0 or O, then we say the group is \"written additively\". If ⋅ is written as × or omitted (i.e. a⋅b written as ab) and the identity as 1, then we say it is \"written multiplicatively\". 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 (in multiplicative notation) as the smallest positive integer k such that ak=1. The order of the group is the number of elements, which (for finite groups) is also the maximum order of any element. Groups always have generators which are elements that, when the group operation is applied repeatedly with the same element some number of times, produce every other element of the group. That is, a generator has maximal order, which we also call the order of the group. There can be many different generators. A group is called cyclic if the whole group can be generated by a (not necessarily unique) single element. 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 O∈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↦[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":"65","title":"Groups"},"66":{"body":"We use the notation Fp× for the multiplicative group (i.e. the group operation is multiplication in Fp) over the set Fp−{0}. A quick way of obtaining the inverse in Fp× is a−1=ap−2. The reason for this stems from Fermat's little theorem , which states that ap=a(modp) for any integer a. If a is nonzero, we can divide by a twice to get ap−2=a−1. Let's assume that α is a generator of Fp×, so it has order p−1 (equal to the number of elements in Fp×). Therefore, for any element in a∈Fp× there is a unique integer i∈{0..p−2} such that a=αi. Notice that a×b where a,b∈Fp× can really be interpreted as αi×αj where a=αi and b=αj. Indeed, it holds that αi×αj=αi+j for all 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: This library is being actively developed and should not be used in production software.","breadcrumbs":"halo2 Crates.io","id":"0","title":"halo2 Crates.io"},"1":{"body":"","breadcrumbs":"Documentation","id":"1","title":"Documentation"},"10":{"body":"Let's start with a simple circuit, to introduce you to the common APIs and how they are used. The circuit will take a public input c, and will prove knowledge of two private inputs a and b such that a⋅b=c. # extern crate halo2;\nuse halo2::arithmetic::FieldExt; struct MyCircuit { a: F, b: F,\n} TODO","breadcrumbs":"User Documentation » A simple example","id":"10","title":"A simple example"},"100":{"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 points in the subgroup.","breadcrumbs":"Background Material » [WIP] UltraPLONK arithmetisation","id":"100","title":"[WIP] UltraPLONK arithmetisation"},"101":{"body":"A polynomial rule defines a constraint that must hold between its specified columns at every row (i.e. at every point in the multiplicative subgroup). e.g. a * sa + b * sb + a * b * sm + c * sc + PI = 0","breadcrumbs":"Background Material » Polynomial rules","id":"101","title":"Polynomial rules"},"102":{"body":"fixed (i.e. \"selector\") columns : fixed for all instances of a particular circuit. These columns toggle parts of a polynomial rule \"on\" or \"off\" to form a \"custom gate\". 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":"102","title":"Columns"},"103":{"body":"Define permutation between a set of columns, e.g. σ(a,b,c) Copy specific cells between these columns, e.g. b1=c0 Construct permuted columns which should evaluate to same value as original columns","breadcrumbs":"Background Material » Copy constraints","id":"103","title":"Copy constraints"},"104":{"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":"104","title":"Permutation grand product"},"105":{"body":"Reference: Generic Lookups with PLONK (DRAFT)","breadcrumbs":"Background Material » Lookup","id":"105","title":"Lookup"},"106":{"body":"We want to check that the expressions defined by the gate constraints, permutation constraints and loookup constraints evaluate to zero at all points 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":"106","title":"Vanishing argument"},"107":{"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":"107","title":"Polynomial commitment using inner product argument"},"108":{"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":"108","title":"Setup"},"109":{"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":"109","title":"Commit"},"11":{"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":"11","title":"Lookup tables"},"110":{"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":"110","title":"Open (prover) and OpenVerify (verifier)"},"111":{"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":"111","title":"Recursion"},"12":{"body":"","breadcrumbs":"User Documentation » Gadgets","id":"12","title":"Gadgets"},"13":{"body":"This section contains various ideas and snippets that you might find useful while writing halo2 circuits.","breadcrumbs":"User Documentation » Tips and tricks","id":"13","title":"Tips and tricks"},"14":{"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":"14","title":"Small range constraints"},"15":{"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: 00011011→0000⟹0→0→0001⟹1→1→0100⟹2→4→0101⟹3→5 Then, we construct the Lagrange basis polynomial for each point using the identity: lj(X)=0≤m≤k,m=j∏xj−xmx−xm, where k+1 is the number of data points. (k=3 in our example above.) Recall that the Lagrange basis polynomial lj(X) evaluates to 1 at X=xj and 0 at all other xi,j=i. Continuing our example, we get four Lagrange basis polynomials: l0(X)l1(X)l2(X)l3(X)====(−3)(−2)(−1)(X−3)(X−2)(X−1)(−2)(−1)(1)(X−3)(X−2)(X)(−1)(1)(2)(X−3)(X−1)(X)(1)(2)(3)(X−2)(X−1)(X) Our polynomial constraint is then ⟹f(0)l0(X)0⋅l0(X)++f(1)l1(X)1⋅l1(X)++f(2)l2(X)4⋅l2(X)++f(3)l3(X)−f(X)5⋅l3(X)−f(X)==00.","breadcrumbs":"User Documentation » Small set interpolation","id":"15","title":"Small set interpolation"},"16":{"body":"","breadcrumbs":"Design","id":"16","title":"Design"},"17":{"body":"We use slightly different language than others to describe PLONK concepts. Here's the overview: We like to think of PLONK-like arguments as tables, where each column corresponds to a \"wire\". We refer to entries in this table as \"cells\". We like to call \"selector polynomials\" and so on \"fixed columns\" instead. We then refer specifically to a \"selector constraint\" when a cell in a fixed column is being used to control whether a particular constraint is enabled in that row. We call the other polynomials \"advice columns\" usually, when they're populated by the prover. We use the term \"rule\" to refer to a \"gate\" like A(X)⋅qA(X)+B(X)⋅qB(X)+A(X)⋅B(X)⋅qM(X)+C(X)⋅qC(X)=0. TODO: Check how consistent we are with this, and update the code and docs to match.","breadcrumbs":"Note on Language","id":"17","title":"Note on Language"},"18":{"body":"","breadcrumbs":"Design » Proving system","id":"18","title":"Proving system"},"19":{"body":"Consider the commitments A,B,C,D to polynomials a(X),b(X),c(X),d(X). Let's say that a and b were queried at the point x, while c and d were queried at both points x and ωx. (Here, ω is the primitive root of unity in the multiplicative subgroup over which we constructed the polynomials). We can group the commitments in terms of the sets of points at which they were queried: {x}AB{x,ωx}CD The multipoint opening optimisation proceeds as such: Sample random x3, at which we evaluate a(X),b(X),c(X),d(X). The prover provides evaluations of each polynomial at each point of interest: a(x3),b(x3),c(x3),d(x3),c(ωx3),d(ωx3) Sample random x4, to keep a,b,c,d linearly independent. Accumulate polynomials and their corresponding evaluations according to the point set at which they were queried: q_polys: q1(X)q2(X)==a(X)c(X)++x4b(X)x4d(X) q_eval_sets: [ [a(x_3) + x_4 b(x_3)], [ c(x_3) + x_4 d(x_3), c(\\omega x_3) + x_4 d(\\omega x_3) ] ] NB: q_eval_sets is a vector of sets of evaluations, where the outer vector goes over the point sets, and the inner vector goes over the points in each set. Interpolate each set of values in q_eval_sets: r_polys: r1(X)s.t.r2(X)s.t.r1(x3)r2(x3)r2(ωx3)===a(x3)+x4b(x3)c(x3)+x4d(x3)c(ωx3)+x4d(ωx3) Construct f_polys which check the correctness of q_polys: f_polys f1(X)f2(X)==X−x3q1(X)−r1(X)(X−x3)(X−ωx3)q2(X)−r2(X) If q1(x3)=r1(x3), then f1(X) should be a polynomial. If q2(x3)=r2(x3) and q2(ωx3)=r2(ωx3) then f2(X) should be a polynomial. Sample random x5 to keep the f_polys linearly independent. Construct f(X)=f1(X)+x5f2(X). Sample random x6, at which we evaluate f(X): f(x6)==f1(x6)x6−x3q1(x6)−r1(x6)++x5f2(x6)x5(x6−x3)(x6−ωx3)q2(x6)−r2(x6) Sample random x7 to keep f(X) and q_polys linearly independent. Construct final_poly, final_poly(X)=f(X)+x7q1(X)+x72q2(X), which is the polynomial we commit to in the inner product argument.","breadcrumbs":"Design » Proving system » Multipoint opening argument","id":"19","title":"Multipoint opening argument"},"2":{"body":"Copyright 2020 The Electric Coin Company. You may use this package under the Transitive Grace Period Public Licence, version 1.0, or at your option, any later version. See the file LICENSE-TGPPL for the terms of the Transitive Grace Period Public Licence, version 1.0. The purpose of the TGPPL is to allow commercial improvements to the package while ensuring that all improvements are eventually open source. See here for why the TGPPL exists, graphically illustrated on three slides.","breadcrumbs":"License","id":"2","title":"License"},"20":{"body":"Given that gates in halo2 circuits operate \"locally\" (on cells in the current row or defined relative rows), it is common to need to copy a value from some arbitrary cell into the current row for use in a gate. This is performed with an equality constraint, which enforces that the source and destination cells contain the same value. We implement these equality constraints by constructing a permutation that represents the constraints, and then using a permutation argument within the proof to enforce them.","breadcrumbs":"Design » Proving system » Permutation argument","id":"20","title":"Permutation argument"},"21":{"body":"A permutation is a one-to-one and onto mapping of a set onto itself. A permutation can be factored uniquely into a composition of cycles (up to ordering of cycles, and rotation of each cycle). We sometimes use cycle notation to write permutations. Let (a b c) denote a cycle where a maps to b, b maps to c, and c maps to a (with the obvious generalisation to arbitrary-sized cycles). Writing two or more cycles next to each other denotes a composition of the corresponding permutations. For example, (a b) (c d) denotes the permutation that maps a to b, b to a, c to d, and d to c.","breadcrumbs":"Design » Proving system » Notation","id":"21","title":"Notation"},"22":{"body":"","breadcrumbs":"Design » Proving system » Constructing the permutation","id":"22","title":"Constructing the permutation"},"23":{"body":"We want to construct a permutation in which each subset of variables that are in a equality-constraint set form a cycle. For example, suppose that we have a circuit that defines the following equality constraints: a≡b a≡c d≡e From this we have the equality-constraint sets {a,b,c} and {d,e}. We want to construct the permutation: (a b c) (d e) which defines the mapping of [a,b,c,d,e] to [b,c,a,e,d].","breadcrumbs":"Design » Proving system » Goal","id":"23","title":"Goal"},"24":{"body":"We need to keep track of the set of cycles, which is a set of disjoint sets . Efficient data structures for this problem are known; for the sake of simplicity we choose one that is not asymptotically optimal but is easy to implement. We represent the current state as: an array mapping for the permutation itself; an auxiliary array aux that keeps track of a distinguished element of each cycle; another array sizes that keeps track of the size of each cycle. We have the invariant that for each element x in a given cycle C, aux(x) points to the same element c∈C. This allows us to quickly decide whether two given elements x and y are in the same cycle, by checking whether aux(x)=aux(y). Also, sizes(aux(x)) gives the size of the cycle containing x. (This is guaranteed only for sizes(aux(x))), not for sizes(x).) The algorithm starts with a representation of the identity permutation: for all x, we set mapping(x)=x, aux(x)=x, and sizes(x)=1. To add an equality constraint left≡right: Check whether left and right are already in the same cycle, i.e. whether aux(left)=aux(right). If so, there is nothing to do. Otherwise, left and right belong to different cycles. Make left the larger cycle and right the smaller one, by swapping them iff sizes(aux(left)) 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":"24","title":"Algorithm"},"25":{"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 4 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":"25","title":"Broken alternatives"},"26":{"body":"TODO: Document what we do with the permutation once we have it.","breadcrumbs":"Design » Proving system » Argument specification","id":"26","title":"Argument specification"},"27":{"body":"halo2 uses the following lookup technique, which allows for lookups in arbitrary sets, and is arguably simpler than Plookup.","breadcrumbs":"Design » Proving system » Lookup argument","id":"27","title":"Lookup argument"},"28":{"body":"In addition to the general notes on language : We call the Z(X) polynomial (the grand product argument polynomial for the permutation argument) the \"permutation product\" column.","breadcrumbs":"Design » Proving system » Note on Language","id":"28","title":"Note on Language"},"29":{"body":"We express lookups in terms of a \"subset argument\" over a table with 2k rows (numbered from 0), and columns A and S. The goal of the subset argument is to enforce that every cell in A is equal to some cell in S. This means that more than one cell in A can be equal to the same cell in S, and some cells in S don't need to be equal to any of the cells in A. S might be fixed, but it doesn't need to be. That is, we can support looking up values in either fixed or variable tables (where the latter includes advice columns). A and S can contain duplicates. If the sets represented by A and/or S are not naturally of size 2k, we extend S with duplicates and A with dummy values known to be in S. Alternatively we could add a \"lookup selector\" that controls which elements of the A column participate in lookups. This would modify the occurrence of A(X) in the permutation rule below to replace A with, say, S0 if a lookup is not selected. Let ℓi be the Lagrange basis polynomial that evaluates to 1 at row i, and 0 otherwise. We start by allowing the prover to supply permutation columns of A and S. Let's call these A′ and S′, respectively. We can enforce that they are permutations using a permutation argument with product column Z with the rules: Z(X)(A(X)+β)(S(X)+γ)−Z(ω−1X)(A′(X)+β)(S′(X)+γ)=0ℓ0(X)(Z(X)−1)=0 This is a version of the permutation argument which allows A′ and S′ to be permutations of A and S, respectively, but doesn't specify the exact permutations. β and γ are separate challenges so that we can combine these two permutation arguments into one without worrying that they might interfere with each other. The goal of these permutations is to allow A′ and S′ to be arranged by the prover in a particular way: All the cells of column A′ are arranged so that like-valued cells are vertically adjacent to each other. This could be done by some kind of sorting algorithm, but all that matters is that like-valued cells are on consecutive rows in column A′, and that A′ is a permutation of A. The first row in a sequence of like values in A′ is the row that has the corresponding value in S′. Apart from this constraint, S′ is any arbitrary permutation of S. Now, we'll enforce that either Ai′=Si′ or that Ai′=Ai−1′, using the rule (A′(X)−S′(X))⋅(A′(X)−A′(ω−1X))=0 In addition, we enforce A0′=S0′ using the rule ℓ0(X)⋅(A′(X)−S′(X))=0 Together these constraints effectively force every element in A′ (and thus A) to equal at least one element in S′ (and thus S). Proof: by induction on prefixes of the rows.","breadcrumbs":"Design » Proving system » Technique Description","id":"29","title":"Technique Description"},"3":{"body":"First we'll describe the concepts behind zero-knowledge proof systems; the arithmetization (kind of circuit description) used by Halo 2; and the abstractions we use to build circuit implementations.","breadcrumbs":"Concepts","id":"3","title":"Concepts"},"30":{"body":"There is the original column A and the fixed column S. There is a permutation product column Z. There are the two permutations A′ and S′. The gates are all of low degree.","breadcrumbs":"Design » Proving system » Cost","id":"30","title":"Cost"},"31":{"body":"halo2's lookup argument implementation generalizes the above technique in the following ways: A and S can be extended to multiple columns, combined using a random challenge. A′ and S′ stay as single columns. The commitments to the columns of S can be precomputed, then combined cheaply once the challenge is known by taking advantage of the homomorphic property of Pedersen commitments. Then, a lookup argument for an arbitrary-width relation can be implemented in terms of a subset argument, i.e. to constrain R(x,y,...) in each row, consider R as a set of tuples S (using the method of the previous point), and check that (x,y,...)∈R. In the case where R represents a function, this implicitly also checks that the inputs are in the domain. This is typically what we want, and often saves an additional range check. We can support multiple tables in the same circuit, by combining them into a single table that includes a tag column to identify the original table. The tag column could be merged with the \"lookup selector\" mentioned earlier, if this were implemented. These generalizations are similar to those in sections 4 and 5 of the Plookup paper That is, the differences from Plookup are in the subset argument. This argument can then be used in all the same ways; for instance, the optimized range check technique in section 5 of the Plookup paper can also be used with this subset argument.","breadcrumbs":"Design » Proving system » Generalizations","id":"31","title":"Generalizations"},"32":{"body":"","breadcrumbs":"Design » Implementation","id":"32","title":"Implementation"},"33":{"body":"In proving system implementations like bellman, there is a concrete Proof struct that encapsulates the proof data, is returned by a prover, and can be passed to a verifier. halo2 does not contain any proof-like structures, for several reasons: The Proof structures would contain vectors of (vectors of) curve points and scalars. This complicates serialization/deserialization of proofs because the lengths of these vectors depend on the configuration of the circuit. However, we didn't want to encode the lengths of vectors inside of proofs, because at runtime the circuit is fixed, and thus so are the proof sizes. It's easy to accidentally put stuff into a Proof structure that isn't also placed in the transcript, which is a hazard when developing and implementing a proving system. We needed to be able to create multiple PLONK proofs at the same time; these proofs share many different substructures when they are for the same circuit. Instead, halo2 treats proof objects as opaque byte streams. Creation and consumption of these byte streams happens via the transcript: The TranscriptWrite trait represents something that we can write proof components to (at proving time). The TranscriptRead trait represents something that we can read proof components from (at verifying time). Crucially, implementations of TranscriptWrite are responsible for simultaneously writing to some std::io::Write buffer at the same time that they hash things into the transcript, and similarly for TranscriptRead/std::io::Read. As a bonus, treating proofs as opaque byte streams ensures that verification accounts for the cost of deserialization, which isn't negligible due to point compression.","breadcrumbs":"Design » Proofs as opaque byte streams","id":"33","title":"Proofs as opaque byte streams"},"34":{"body":"In this section we document the gadgets and chip designs provided by halo2.","breadcrumbs":"Design » Gadgets","id":"34","title":"Gadgets"},"35":{"body":"","breadcrumbs":"Design » Gadgets » SHA-256","id":"35","title":"SHA-256"},"36":{"body":"SHA-256 is specified in NIST FIPS PUB 180-4 . Unlike the specification, we use ⊞ for addition modulo 232, and + for field addition. ⊕ is used for XOR.","breadcrumbs":"Design » Gadgets » Specification","id":"36","title":"Specification"},"37":{"body":"SHA-256 maintains state in eight 32-bit variables. It processes input as 512-bit blocks, but internally splits these blocks into 32-bit chunks. We therefore designed the SHA-256 gadget to consume input in 32-bit chunks.","breadcrumbs":"Design » Gadgets » Gadget interface","id":"37","title":"Gadget interface"},"38":{"body":"The SHA-256 gadget requires a chip with the following instructions: # extern crate halo2;\n# use halo2::plonk::Error;\n# use std::fmt;\n#\n# trait Chip: Sized {}\n# trait Layouter {}\nconst BLOCK_SIZE: usize = 16;\nconst DIGEST_SIZE: usize = 8; pub trait Sha256Instructions: Chip { /// Variable representing the SHA-256 internal state. type State: Clone + fmt::Debug; /// Variable representing a 32-bit word of the input block to the SHA-256 compression /// function. type BlockWord: Copy + fmt::Debug; /// Places the SHA-256 IV in the circuit, returning the initial state variable. fn initialization_vector(layouter: &mut impl Layouter) -> Result; /// Starting from the given initial state, processes a block of input and returns the /// final state. fn compress( layouter: &mut impl Layouter, initial_state: &Self::State, input: [Self::BlockWord; BLOCK_SIZE], ) -> Result; /// Converts the given state into a message digest. fn digest( layouter: &mut impl Layouter, state: &Self::State, ) -> Result<[Self::BlockWord; DIGEST_SIZE], Error>;\n} TODO: Add instruction for computing padding. This set of instructions was chosen to strike a balance between the reusability of the instructions, and the scope for chips to internally optimise them. In particular, we considered splitting the compression function into its constituent parts (Ch, Maj etc), and providing a compression function gadget that implemented the round logic. However, this would prevent chips from using relative references between the various parts of a compression round. Having an instruction that implements all compression rounds is also similar to the Intel SHA extensions, which provide an instruction that performs multiple compression rounds.","breadcrumbs":"Design » Gadgets » Chip instructions","id":"38","title":"Chip instructions"},"39":{"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":"39","title":"16-bit table chip for SHA-256"},"4":{"body":"The aim of any proof system is to be able to prove interesting mathematical or cryptographic statements . Typically, in a given protocol we will want to prove families of statements that differ in their public inputs . The prover will also need to show that they know some private inputs that make the statement hold. To do this we write down a relation , R, that specifies which combinations of public and private inputs are valid. The terminology above is intended to be aligned with the ZKProof Community Reference . To be precise, we should distinguish between the relation R, and its implementation to be used in a proof system. We call the latter a circuit . The language that we use to express circuits for a particular proof system is called an arithmetization . Usually, an arithmetization will define circuits in terms of polynomial constraints on variables over a field. The process of expressing a particular relation as a circuit is also sometimes called \"arithmetization\", but we'll avoid that usage. To create a proof of a statement, the prover will need to know the private inputs, and also intermediate values, called advice values, that are used by the circuit. We assume that we can compute advice values efficiently from the private and public inputs. The particular advice values will depend on how we write the circuit, not only on the high-level statement. The private inputs and advice values are collectively called a witness . Some authors use \"witness\" as just a synonym for private inputs. But in our usage, a witness includes advice, i.e. it includes all values that the prover supplies to the circuit. For example, suppose that we want to prove knowledge of a preimage x of a hash function H for a digest y: The private input would be the preimage x. The public input would be the digest y. The relation would be {(x,y):H(x)=y}. For a particular public input Y, the statement would be: {(x):H(x)=Y}. The advice would be all of the intermediate values in the circuit implementing the hash function. The witness would be x and the advice. A Non-interactive Argument allows a prover to create a proof for a given statement and witness. The proof is data that can be used to convince a verifier that there exists a witness for which the statement holds. The security property that such proofs cannot falsely convince a verifier is called soundness . A Non-interactive Argument of Knowledge ( NARK ) further convinces the verifier that the prover knew a witness for which the statement holds. This security property is called knowledge soundness , and it implies soundness. In practice knowledge soundness is more useful for cryptographic protocols than soundness: if we are interested in whether Alice holds a secret key in some protocol, say, we need Alice to prove that she knows the key, not just that it exists. Knowledge soundness is formalized by saying that an extractor , which can observe precisely how the proof is generated, must be able to compute the witness. This property is subtle given that proofs can be malleable . That is, depending on the proof system it may be possible to take an existing proof (or set of proofs) and, without knowing the witness(es), modify it/them to produce a distinct proof of the same or a related statement. Higher-level protocols that use malleable proof systems need to take this into account. Even without malleability, proofs can also potentially be replayed . For instance, we would not want Alice in our example to be able to present a proof generated by someone else, and have that be taken as a demonstration that she knew the key. If a proof yields no information about the witness (other than that a witness exists and was known to the prover), then we say that the proof system is zero knowledge . If a proof system produces short proofs ---i.e. of length polylogarithmic in the circuit size--- then we say that it is succinct . A succinct NARK is called a SNARK ( Succinct Non-Interactive Argument of Knowledge ). By this definition, a SNARK need not have verification time polylogarithmic in the circuit size. Some papers use the term efficient to describe a SNARK with that property, but we'll avoid that term since it's ambiguous for SNARKs that support amortized or recursive verification, which we'll get to later. A zk-SNARK is a zero-knowledge SNARK.","breadcrumbs":"Concepts » Proof systems","id":"4","title":"Proof systems"},"40":{"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":"64","title":"Fields"},"65":{"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":"65","title":"Groups"},"66":{"body":"We use the notation Fp× for the multiplicative group (i.e. the group operation is multiplication in Fp) over the set Fp−{0}. A quick way of obtaining the inverse in Fp× is a−1=ap−2. The reason for this stems from Fermat's little theorem , which states that ap=a(modp) for any integer a. If a is nonzero, we can divide by a twice to get ap−2=a−1. Let's assume that α is a generator of Fp×, so it has order p−1 (equal to the number of elements in Fp×). Therefore, for any element in a∈Fp× there is a unique integer i∈{0..p−2} such that a=αi. Notice that a×b where a,b∈Fp× can really be interpreted as αi×αj where a=αi and b=αj. Indeed, it holds that αi×αj=αi+j for all 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: This library is being actively developed and should not be used in production software.","breadcrumbs":"halo2 Crates.io","id":"0","title":"halo2 Crates.io"},"1":{"body":"","breadcrumbs":"Documentation","id":"1","title":"Documentation"},"10":{"body":"Let's start with a simple circuit, to introduce you to the common APIs and how they are used. The circuit will take a public input c, and will prove knowledge of two private inputs a and b such that a⋅b=c. # extern crate halo2;\nuse halo2::arithmetic::FieldExt; struct MyCircuit { a: F, b: F,\n} TODO","breadcrumbs":"User Documentation » A simple example","id":"10","title":"A simple example"},"100":{"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 points in the subgroup.","breadcrumbs":"Background Material » [WIP] UltraPLONK arithmetisation","id":"100","title":"[WIP] UltraPLONK arithmetisation"},"101":{"body":"A polynomial rule defines a constraint that must hold between its specified columns at every row (i.e. at every point in the multiplicative subgroup). e.g. a * sa + b * sb + a * b * sm + c * sc + PI = 0","breadcrumbs":"Background Material » Polynomial rules","id":"101","title":"Polynomial rules"},"102":{"body":"fixed (i.e. \"selector\") columns : fixed for all instances of a particular circuit. These columns toggle parts of a polynomial rule \"on\" or \"off\" to form a \"custom gate\". 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":"102","title":"Columns"},"103":{"body":"Define permutation between a set of columns, e.g. σ(a,b,c) Copy specific cells between these columns, e.g. b1=c0 Construct permuted columns which should evaluate to same value as original columns","breadcrumbs":"Background Material » Copy constraints","id":"103","title":"Copy constraints"},"104":{"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":"104","title":"Permutation grand product"},"105":{"body":"Reference: Generic Lookups with PLONK (DRAFT)","breadcrumbs":"Background Material » Lookup","id":"105","title":"Lookup"},"106":{"body":"We want to check that the expressions defined by the gate constraints, permutation constraints and loookup constraints evaluate to zero at all points 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":"106","title":"Vanishing argument"},"107":{"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":"107","title":"Polynomial commitment using inner product argument"},"108":{"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":"108","title":"Setup"},"109":{"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":"109","title":"Commit"},"11":{"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":"11","title":"Lookup tables"},"110":{"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":"110","title":"Open (prover) and OpenVerify (verifier)"},"111":{"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":"111","title":"Recursion"},"12":{"body":"","breadcrumbs":"User Documentation » Gadgets","id":"12","title":"Gadgets"},"13":{"body":"This section contains various ideas and snippets that you might find useful while writing halo2 circuits.","breadcrumbs":"User Documentation » Tips and tricks","id":"13","title":"Tips and tricks"},"14":{"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":"14","title":"Small range constraints"},"15":{"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: 00011011→0000⟹0→0→0001⟹1→1→0100⟹2→4→0101⟹3→5 Then, we construct the Lagrange basis polynomial for each point using the identity: lj(X)=0≤m≤k,m=j∏xj−xmx−xm, where k+1 is the number of data points. (k=3 in our example above.) Recall that the Lagrange basis polynomial lj(X) evaluates to 1 at X=xj and 0 at all other xi,j=i. Continuing our example, we get four Lagrange basis polynomials: l0(X)l1(X)l2(X)l3(X)====(−3)(−2)(−1)(X−3)(X−2)(X−1)(−2)(−1)(1)(X−3)(X−2)(X)(−1)(1)(2)(X−3)(X−1)(X)(1)(2)(3)(X−2)(X−1)(X) Our polynomial constraint is then ⟹f(0)l0(X)0⋅l0(X)++f(1)l1(X)1⋅l1(X)++f(2)l2(X)4⋅l2(X)++f(3)l3(X)−f(X)5⋅l3(X)−f(X)==00.","breadcrumbs":"User Documentation » Small set interpolation","id":"15","title":"Small set interpolation"},"16":{"body":"","breadcrumbs":"Design","id":"16","title":"Design"},"17":{"body":"We use slightly different language than others to describe PLONK concepts. Here's the overview: We like to think of PLONK-like arguments as tables, where each column corresponds to a \"wire\". We refer to entries in this table as \"cells\". We like to call \"selector polynomials\" and so on \"fixed columns\" instead. We then refer specifically to a \"selector constraint\" when a cell in a fixed column is being used to control whether a particular constraint is enabled in that row. We call the other polynomials \"advice columns\" usually, when they're populated by the prover. We use the term \"rule\" to refer to a \"gate\" like A(X)⋅qA(X)+B(X)⋅qB(X)+A(X)⋅B(X)⋅qM(X)+C(X)⋅qC(X)=0. TODO: Check how consistent we are with this, and update the code and docs to match.","breadcrumbs":"Note on Language","id":"17","title":"Note on Language"},"18":{"body":"","breadcrumbs":"Design » Proving system","id":"18","title":"Proving system"},"19":{"body":"Consider the commitments A,B,C,D to polynomials a(X),b(X),c(X),d(X). Let's say that a and b were queried at the point x, while c and d were queried at both points x and ωx. (Here, ω is the primitive root of unity in the multiplicative subgroup over which we constructed the polynomials). We can group the commitments in terms of the sets of points at which they were queried: {x}AB{x,ωx}CD The multipoint opening optimisation proceeds as such: Sample random x3, at which we evaluate a(X),b(X),c(X),d(X). The prover provides evaluations of each polynomial at each point of interest: a(x3),b(x3),c(x3),d(x3),c(ωx3),d(ωx3) Sample random x4, to keep a,b,c,d linearly independent. Accumulate polynomials and their corresponding evaluations according to the point set at which they were queried: q_polys: q1(X)q2(X)==a(X)c(X)++x4b(X)x4d(X) q_eval_sets: [ [a(x_3) + x_4 b(x_3)], [ c(x_3) + x_4 d(x_3), c(\\omega x_3) + x_4 d(\\omega x_3) ] ] NB: q_eval_sets is a vector of sets of evaluations, where the outer vector goes over the point sets, and the inner vector goes over the points in each set. Interpolate each set of values in q_eval_sets: r_polys: r1(X)s.t.r2(X)s.t.r1(x3)r2(x3)r2(ωx3)===a(x3)+x4b(x3)c(x3)+x4d(x3)c(ωx3)+x4d(ωx3) Construct f_polys which check the correctness of q_polys: f_polys f1(X)f2(X)==X−x3q1(X)−r1(X)(X−x3)(X−ωx3)q2(X)−r2(X) If q1(x3)=r1(x3), then f1(X) should be a polynomial. If q2(x3)=r2(x3) and q2(ωx3)=r2(ωx3) then f2(X) should be a polynomial. Sample random x5 to keep the f_polys linearly independent. Construct f(X)=f1(X)+x5f2(X). Sample random x6, at which we evaluate f(X): f(x6)==f1(x6)x6−x3q1(x6)−r1(x6)++x5f2(x6)x5(x6−x3)(x6−ωx3)q2(x6)−r2(x6) Sample random x7 to keep f(X) and q_polys linearly independent. Construct final_poly, final_poly(X)=f(X)+x7q1(X)+x72q2(X), which is the polynomial we commit to in the inner product argument.","breadcrumbs":"Design » Proving system » Multipoint opening argument","id":"19","title":"Multipoint opening argument"},"2":{"body":"Copyright 2020 The Electric Coin Company. You may use this package under the Transitive Grace Period Public Licence, version 1.0, or at your option, any later version. See the file LICENSE-TGPPL for the terms of the Transitive Grace Period Public Licence, version 1.0. The purpose of the TGPPL is to allow commercial improvements to the package while ensuring that all improvements are eventually open source. See here for why the TGPPL exists, graphically illustrated on three slides.","breadcrumbs":"License","id":"2","title":"License"},"20":{"body":"Given that gates in halo2 circuits operate \"locally\" (on cells in the current row or defined relative rows), it is common to need to copy a value from some arbitrary cell into the current row for use in a gate. This is performed with an equality constraint, which enforces that the source and destination cells contain the same value. We implement these equality constraints by constructing a permutation that represents the constraints, and then using a permutation argument within the proof to enforce them.","breadcrumbs":"Design » Proving system » Permutation argument","id":"20","title":"Permutation argument"},"21":{"body":"A permutation is a one-to-one and onto mapping of a set onto itself. A permutation can be factored uniquely into a composition of cycles (up to ordering of cycles, and rotation of each cycle). We sometimes use cycle notation to write permutations. Let (a b c) denote a cycle where a maps to b, b maps to c, and c maps to a (with the obvious generalisation to arbitrary-sized cycles). Writing two or more cycles next to each other denotes a composition of the corresponding permutations. For example, (a b) (c d) denotes the permutation that maps a to b, b to a, c to d, and d to c.","breadcrumbs":"Design » Proving system » Notation","id":"21","title":"Notation"},"22":{"body":"","breadcrumbs":"Design » Proving system » Constructing the permutation","id":"22","title":"Constructing the permutation"},"23":{"body":"We want to construct a permutation in which each subset of variables that are in a equality-constraint set form a cycle. For example, suppose that we have a circuit that defines the following equality constraints: a≡b a≡c d≡e From this we have the equality-constraint sets {a,b,c} and {d,e}. We want to construct the permutation: (a b c) (d e) which defines the mapping of [a,b,c,d,e] to [b,c,a,e,d].","breadcrumbs":"Design » Proving system » Goal","id":"23","title":"Goal"},"24":{"body":"We need to keep track of the set of cycles, which is a set of disjoint sets . Efficient data structures for this problem are known; for the sake of simplicity we choose one that is not asymptotically optimal but is easy to implement. We represent the current state as: an array mapping for the permutation itself; an auxiliary array aux that keeps track of a distinguished element of each cycle; another array sizes that keeps track of the size of each cycle. We have the invariant that for each element x in a given cycle C, aux(x) points to the same element c∈C. This allows us to quickly decide whether two given elements x and y are in the same cycle, by checking whether aux(x)=aux(y). Also, sizes(aux(x)) gives the size of the cycle containing x. (This is guaranteed only for sizes(aux(x))), not for sizes(x).) The algorithm starts with a representation of the identity permutation: for all x, we set mapping(x)=x, aux(x)=x, and sizes(x)=1. To add an equality constraint left≡right: Check whether left and right are already in the same cycle, i.e. whether aux(left)=aux(right). If so, there is nothing to do. Otherwise, left and right belong to different cycles. Make left the larger cycle and right the smaller one, by swapping them iff sizes(aux(left)) 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":"24","title":"Algorithm"},"25":{"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 4 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":"25","title":"Broken alternatives"},"26":{"body":"TODO: Document what we do with the permutation once we have it.","breadcrumbs":"Design » Proving system » Argument specification","id":"26","title":"Argument specification"},"27":{"body":"halo2 uses the following lookup technique, which allows for lookups in arbitrary sets, and is arguably simpler than Plookup.","breadcrumbs":"Design » Proving system » Lookup argument","id":"27","title":"Lookup argument"},"28":{"body":"In addition to the general notes on language : We call the Z(X) polynomial (the grand product argument polynomial for the permutation argument) the \"permutation product\" column.","breadcrumbs":"Design » Proving system » Note on Language","id":"28","title":"Note on Language"},"29":{"body":"We express lookups in terms of a \"subset argument\" over a table with 2k rows (numbered from 0), and columns A and S. The goal of the subset argument is to enforce that every cell in A is equal to some cell in S. This means that more than one cell in A can be equal to the same cell in S, and some cells in S don't need to be equal to any of the cells in A. S might be fixed, but it doesn't need to be. That is, we can support looking up values in either fixed or variable tables (where the latter includes advice columns). A and S can contain duplicates. If the sets represented by A and/or S are not naturally of size 2k, we extend S with duplicates and A with dummy values known to be in S. Alternatively we could add a \"lookup selector\" that controls which elements of the A column participate in lookups. This would modify the occurrence of A(X) in the permutation rule below to replace A with, say, S0 if a lookup is not selected. Let ℓi be the Lagrange basis polynomial that evaluates to 1 at row i, and 0 otherwise. We start by allowing the prover to supply permutation columns of A and S. Let's call these A′ and S′, respectively. We can enforce that they are permutations using a permutation argument with product column Z with the rules: Z(X)(A(X)+β)(S(X)+γ)−Z(ω−1X)(A′(X)+β)(S′(X)+γ)=0ℓ0(X)(Z(X)−1)=0 This is a version of the permutation argument which allows A′ and S′ to be permutations of A and S, respectively, but doesn't specify the exact permutations. β and γ are separate challenges so that we can combine these two permutation arguments into one without worrying that they might interfere with each other. The goal of these permutations is to allow A′ and S′ to be arranged by the prover in a particular way: All the cells of column A′ are arranged so that like-valued cells are vertically adjacent to each other. This could be done by some kind of sorting algorithm, but all that matters is that like-valued cells are on consecutive rows in column A′, and that A′ is a permutation of A. The first row in a sequence of like values in A′ is the row that has the corresponding value in S′. Apart from this constraint, S′ is any arbitrary permutation of S. Now, we'll enforce that either Ai′=Si′ or that Ai′=Ai−1′, using the rule (A′(X)−S′(X))⋅(A′(X)−A′(ω−1X))=0 In addition, we enforce A0′=S0′ using the rule ℓ0(X)⋅(A′(X)−S′(X))=0 Together these constraints effectively force every element in A′ (and thus A) to equal at least one element in S′ (and thus S). Proof: by induction on prefixes of the rows.","breadcrumbs":"Design » Proving system » Technique Description","id":"29","title":"Technique Description"},"3":{"body":"First we'll describe the concepts behind zero-knowledge proof systems; the arithmetization (kind of circuit description) used by Halo 2; and the abstractions we use to build circuit implementations.","breadcrumbs":"Concepts","id":"3","title":"Concepts"},"30":{"body":"There is the original column A and the fixed column S. There is a permutation product column Z. There are the two permutations A′ and S′. The gates are all of low degree.","breadcrumbs":"Design » Proving system » Cost","id":"30","title":"Cost"},"31":{"body":"halo2's lookup argument implementation generalizes the above technique in the following ways: A and S can be extended to multiple columns, combined using a random challenge. A′ and S′ stay as single columns. The commitments to the columns of S can be precomputed, then combined cheaply once the challenge is known by taking advantage of the homomorphic property of Pedersen commitments. Then, a lookup argument for an arbitrary-width relation can be implemented in terms of a subset argument, i.e. to constrain R(x,y,...) in each row, consider R as a set of tuples S (using the method of the previous point), and check that (x,y,...)∈R. In the case where R represents a function, this implicitly also checks that the inputs are in the domain. This is typically what we want, and often saves an additional range check. We can support multiple tables in the same circuit, by combining them into a single table that includes a tag column to identify the original table. The tag column could be merged with the \"lookup selector\" mentioned earlier, if this were implemented. These generalizations are similar to those in sections 4 and 5 of the Plookup paper That is, the differences from Plookup are in the subset argument. This argument can then be used in all the same ways; for instance, the optimized range check technique in section 5 of the Plookup paper can also be used with this subset argument.","breadcrumbs":"Design » Proving system » Generalizations","id":"31","title":"Generalizations"},"32":{"body":"","breadcrumbs":"Design » Implementation","id":"32","title":"Implementation"},"33":{"body":"In proving system implementations like bellman, there is a concrete Proof struct that encapsulates the proof data, is returned by a prover, and can be passed to a verifier. halo2 does not contain any proof-like structures, for several reasons: The Proof structures would contain vectors of (vectors of) curve points and scalars. This complicates serialization/deserialization of proofs because the lengths of these vectors depend on the configuration of the circuit. However, we didn't want to encode the lengths of vectors inside of proofs, because at runtime the circuit is fixed, and thus so are the proof sizes. It's easy to accidentally put stuff into a Proof structure that isn't also placed in the transcript, which is a hazard when developing and implementing a proving system. We needed to be able to create multiple PLONK proofs at the same time; these proofs share many different substructures when they are for the same circuit. Instead, halo2 treats proof objects as opaque byte streams. Creation and consumption of these byte streams happens via the transcript: The TranscriptWrite trait represents something that we can write proof components to (at proving time). The TranscriptRead trait represents something that we can read proof components from (at verifying time). Crucially, implementations of TranscriptWrite are responsible for simultaneously writing to some std::io::Write buffer at the same time that they hash things into the transcript, and similarly for TranscriptRead/std::io::Read. As a bonus, treating proofs as opaque byte streams ensures that verification accounts for the cost of deserialization, which isn't negligible due to point compression.","breadcrumbs":"Design » Proofs as opaque byte streams","id":"33","title":"Proofs as opaque byte streams"},"34":{"body":"In this section we document the gadgets and chip designs provided by halo2.","breadcrumbs":"Design » Gadgets","id":"34","title":"Gadgets"},"35":{"body":"","breadcrumbs":"Design » Gadgets » SHA-256","id":"35","title":"SHA-256"},"36":{"body":"SHA-256 is specified in NIST FIPS PUB 180-4 . Unlike the specification, we use ⊞ for addition modulo 232, and + for field addition. ⊕ is used for XOR.","breadcrumbs":"Design » Gadgets » Specification","id":"36","title":"Specification"},"37":{"body":"SHA-256 maintains state in eight 32-bit variables. It processes input as 512-bit blocks, but internally splits these blocks into 32-bit chunks. We therefore designed the SHA-256 gadget to consume input in 32-bit chunks.","breadcrumbs":"Design » Gadgets » Gadget interface","id":"37","title":"Gadget interface"},"38":{"body":"The SHA-256 gadget requires a chip with the following instructions: # extern crate halo2;\n# use halo2::plonk::Error;\n# use std::fmt;\n#\n# trait Chip: Sized {}\n# trait Layouter {}\nconst BLOCK_SIZE: usize = 16;\nconst DIGEST_SIZE: usize = 8; pub trait Sha256Instructions: Chip { /// Variable representing the SHA-256 internal state. type State: Clone + fmt::Debug; /// Variable representing a 32-bit word of the input block to the SHA-256 compression /// function. type BlockWord: Copy + fmt::Debug; /// Places the SHA-256 IV in the circuit, returning the initial state variable. fn initialization_vector(layouter: &mut impl Layouter) -> Result; /// Starting from the given initial state, processes a block of input and returns the /// final state. fn compress( layouter: &mut impl Layouter, initial_state: &Self::State, input: [Self::BlockWord; BLOCK_SIZE], ) -> Result; /// Converts the given state into a message digest. fn digest( layouter: &mut impl Layouter, state: &Self::State, ) -> Result<[Self::BlockWord; DIGEST_SIZE], Error>;\n} TODO: Add instruction for computing padding. This set of instructions was chosen to strike a balance between the reusability of the instructions, and the scope for chips to internally optimise them. In particular, we considered splitting the compression function into its constituent parts (Ch, Maj etc), and providing a compression function gadget that implemented the round logic. However, this would prevent chips from using relative references between the various parts of a compression round. Having an instruction that implements all compression rounds is also similar to the Intel SHA extensions, which provide an instruction that performs multiple compression rounds.","breadcrumbs":"Design » Gadgets » Chip instructions","id":"38","title":"Chip instructions"},"39":{"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":"39","title":"16-bit table chip for SHA-256"},"4":{"body":"The aim of any proof system is to be able to prove interesting mathematical or cryptographic statements . Typically, in a given protocol we will want to prove families of statements that differ in their public inputs . The prover will also need to show that they know some private inputs that make the statement hold. To do this we write down a relation , R, that specifies which combinations of public and private inputs are valid. The terminology above is intended to be aligned with the ZKProof Community Reference . To be precise, we should distinguish between the relation R, and its implementation to be used in a proof system. We call the latter a circuit . The language that we use to express circuits for a particular proof system is called an arithmetization . Usually, an arithmetization will define circuits in terms of polynomial constraints on variables over a field. The process of expressing a particular relation as a circuit is also sometimes called \"arithmetization\", but we'll avoid that usage. To create a proof of a statement, the prover will need to know the private inputs, and also intermediate values, called advice values, that are used by the circuit. We assume that we can compute advice values efficiently from the private and public inputs. The particular advice values will depend on how we write the circuit, not only on the high-level statement. The private inputs and advice values are collectively called a witness . Some authors use \"witness\" as just a synonym for private inputs. But in our usage, a witness includes advice, i.e. it includes all values that the prover supplies to the circuit. For example, suppose that we want to prove knowledge of a preimage x of a hash function H for a digest y: The private input would be the preimage x. The public input would be the digest y. The relation would be {(x,y):H(x)=y}. For a particular public input Y, the statement would be: {(x):H(x)=Y}. The advice would be all of the intermediate values in the circuit implementing the hash function. The witness would be x and the advice. A Non-interactive Argument allows a prover to create a proof for a given statement and witness. The proof is data that can be used to convince a verifier that there exists a witness for which the statement holds. The security property that such proofs cannot falsely convince a verifier is called soundness . A Non-interactive Argument of Knowledge ( NARK ) further convinces the verifier that the prover knew a witness for which the statement holds. This security property is called knowledge soundness , and it implies soundness. In practice knowledge soundness is more useful for cryptographic protocols than soundness: if we are interested in whether Alice holds a secret key in some protocol, say, we need Alice to prove that she knows the key, not just that it exists. Knowledge soundness is formalized by saying that an extractor , which can observe precisely how the proof is generated, must be able to compute the witness. This property is subtle given that proofs can be malleable . That is, depending on the proof system it may be possible to take an existing proof (or set of proofs) and, without knowing the witness(es), modify it/them to produce a distinct proof of the same or a related statement. Higher-level protocols that use malleable proof systems need to take this into account. Even without malleability, proofs can also potentially be replayed . For instance, we would not want Alice in our example to be able to present a proof generated by someone else, and have that be taken as a demonstration that she knew the key. If a proof yields no information about the witness (other than that a witness exists and was known to the prover), then we say that the proof system is zero knowledge . If a proof system produces short proofs ---i.e. of length polylogarithmic in the circuit size--- then we say that it is succinct . A succinct NARK is called a SNARK ( Succinct Non-Interactive Argument of Knowledge ). By this definition, a SNARK need not have verification time polylogarithmic in the circuit size. Some papers use the term efficient to describe a SNARK with that property, but we'll avoid that term since it's ambiguous for SNARKs that support amortized or recursive verification, which we'll get to later. A zk-SNARK is a zero-knowledge SNARK.","breadcrumbs":"Concepts » Proof systems","id":"4","title":"Proof systems"},"40":{"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":"64","title":"Fields"},"65":{"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 + and the identity as 0 or O, then we say the group is \"written additively\". If ⋅ is written as × or omitted (i.e. a⋅b written as ab) and the identity as 1, then we say it is \"written multiplicatively\". 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 (in multiplicative notation) as the smallest positive integer k such that ak=1. The order of the group is the number of elements, which (for finite groups) is also the maximum order of any element. Groups always have generators which are elements that, when the group operation is applied repeatedly with the same element some number of times, produce every other element of the group. That is, a generator has maximal order, which we also call the order of the group. There can be many different generators. A group is called cyclic if the whole group can be generated by a (not necessarily unique) single element. 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 O∈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↦[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":"65","title":"Groups"},"66":{"body":"We use the notation Fp× for the multiplicative group (i.e. the group operation is multiplication in Fp) over the set Fp−{0}. A quick way of obtaining the inverse in Fp× is a−1=ap−2. The reason for this stems from Fermat's little theorem , which states that ap=a(modp) for any integer a. If a is nonzero, we can divide by a twice to get ap−2=a−1. Let's assume that α is a generator of Fp×, so it has order p−1 (equal to the number of elements in Fp×). Therefore, for any element in a∈Fp× there is a unique integer i∈{0..p−2} such that a=αi. Notice that a×b where a,b∈Fp× can really be interpreted as αi×αj where a=αi and b=αj. Indeed, it holds that αi×αj=αi+j for all 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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":"109","title":"Commit"},"11":{"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":"11","title":"Lookup tables"},"110":{"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":"110","title":"Open (prover) and OpenVerify (verifier)"},"111":{"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":"111","title":"Recursion"},"12":{"body":"","breadcrumbs":"User Documentation » Gadgets","id":"12","title":"Gadgets"},"13":{"body":"This section contains various ideas and snippets that you might find useful while writing halo2 circuits.","breadcrumbs":"User Documentation » Tips and tricks","id":"13","title":"Tips and tricks"},"14":{"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":"14","title":"Small range constraints"},"15":{"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: 00011011→0000⟹0→0→0001⟹1→1→0100⟹2→4→0101⟹3→5 Then, we construct the Lagrange basis polynomial for each point using the identity: lj(X)=0≤m≤k,m=j∏xj−xmx−xm, where k+1 is the number of data points. (k=3 in our example above.) Recall that the Lagrange basis polynomial lj(X) evaluates to 1 at X=xj and 0 at all other xi,j=i. Continuing our example, we get four Lagrange basis polynomials: l0(X)l1(X)l2(X)l3(X)====(−3)(−2)(−1)(X−3)(X−2)(X−1)(−2)(−1)(1)(X−3)(X−2)(X)(−1)(1)(2)(X−3)(X−1)(X)(1)(2)(3)(X−2)(X−1)(X) Our polynomial constraint is then ⟹f(0)l0(X)0⋅l0(X)++f(1)l1(X)1⋅l1(X)++f(2)l2(X)4⋅l2(X)++f(3)l3(X)−f(X)5⋅l3(X)−f(X)==00.","breadcrumbs":"User Documentation » Small set interpolation","id":"15","title":"Small set interpolation"},"16":{"body":"","breadcrumbs":"Design","id":"16","title":"Design"},"17":{"body":"We use slightly different language than others to describe PLONK concepts. Here's the overview: We like to think of PLONK-like arguments as tables, where each column corresponds to a \"wire\". We refer to entries in this table as \"cells\". We like to call \"selector polynomials\" and so on \"fixed columns\" instead. We then refer specifically to a \"selector constraint\" when a cell in a fixed column is being used to control whether a particular constraint is enabled in that row. We call the other polynomials \"advice columns\" usually, when they're populated by the prover. We use the term \"rule\" to refer to a \"gate\" like A(X)⋅qA(X)+B(X)⋅qB(X)+A(X)⋅B(X)⋅qM(X)+C(X)⋅qC(X)=0. TODO: Check how consistent we are with this, and update the code and docs to match.","breadcrumbs":"Note on Language","id":"17","title":"Note on Language"},"18":{"body":"","breadcrumbs":"Design » Proving system","id":"18","title":"Proving system"},"19":{"body":"Consider the commitments A,B,C,D to polynomials a(X),b(X),c(X),d(X). Let's say that a and b were queried at the point x, while c and d were queried at both points x and ωx. (Here, ω is the primitive root of unity in the multiplicative subgroup over which we constructed the polynomials). We can group the commitments in terms of the sets of points at which they were queried: {x}AB{x,ωx}CD The multipoint opening optimisation proceeds as such: Sample random x3, at which we evaluate a(X),b(X),c(X),d(X). The prover provides evaluations of each polynomial at each point of interest: a(x3),b(x3),c(x3),d(x3),c(ωx3),d(ωx3) Sample random x4, to keep a,b,c,d linearly independent. Accumulate polynomials and their corresponding evaluations according to the point set at which they were queried: q_polys: q1(X)q2(X)==a(X)c(X)++x4b(X)x4d(X) q_eval_sets: [ [a(x_3) + x_4 b(x_3)], [ c(x_3) + x_4 d(x_3), c(\\omega x_3) + x_4 d(\\omega x_3) ] ] NB: q_eval_sets is a vector of sets of evaluations, where the outer vector goes over the point sets, and the inner vector goes over the points in each set. Interpolate each set of values in q_eval_sets: r_polys: r1(X)s.t.r2(X)s.t.r1(x3)r2(x3)r2(ωx3)===a(x3)+x4b(x3)c(x3)+x4d(x3)c(ωx3)+x4d(ωx3) Construct f_polys which check the correctness of q_polys: f_polys f1(X)f2(X)==X−x3q1(X)−r1(X)(X−x3)(X−ωx3)q2(X)−r2(X) If q1(x3)=r1(x3), then f1(X) should be a polynomial. If q2(x3)=r2(x3) and q2(ωx3)=r2(ωx3) then f2(X) should be a polynomial. Sample random x5 to keep the f_polys linearly independent. Construct f(X)=f1(X)+x5f2(X). Sample random x6, at which we evaluate f(X): f(x6)==f1(x6)x6−x3q1(x6)−r1(x6)++x5f2(x6)x5(x6−x3)(x6−ωx3)q2(x6)−r2(x6) Sample random x7 to keep f(X) and q_polys linearly independent. Construct final_poly, final_poly(X)=f(X)+x7q1(X)+x72q2(X), which is the polynomial we commit to in the inner product argument.","breadcrumbs":"Design » Proving system » Multipoint opening argument","id":"19","title":"Multipoint opening argument"},"2":{"body":"Copyright 2020 The Electric Coin Company. You may use this package under the Transitive Grace Period Public Licence, version 1.0, or at your option, any later version. See the file LICENSE-TGPPL for the terms of the Transitive Grace Period Public Licence, version 1.0. The purpose of the TGPPL is to allow commercial improvements to the package while ensuring that all improvements are eventually open source. See here for why the TGPPL exists, graphically illustrated on three slides.","breadcrumbs":"License","id":"2","title":"License"},"20":{"body":"Given that gates in halo2 circuits operate \"locally\" (on cells in the current row or defined relative rows), it is common to need to copy a value from some arbitrary cell into the current row for use in a gate. This is performed with an equality constraint, which enforces that the source and destination cells contain the same value. We implement these equality constraints by constructing a permutation that represents the constraints, and then using a permutation argument within the proof to enforce them.","breadcrumbs":"Design » Proving system » Permutation argument","id":"20","title":"Permutation argument"},"21":{"body":"A permutation is a one-to-one and onto mapping of a set onto itself. A permutation can be factored uniquely into a composition of cycles (up to ordering of cycles, and rotation of each cycle). We sometimes use cycle notation to write permutations. Let (a b c) denote a cycle where a maps to b, b maps to c, and c maps to a (with the obvious generalisation to arbitrary-sized cycles). Writing two or more cycles next to each other denotes a composition of the corresponding permutations. For example, (a b) (c d) denotes the permutation that maps a to b, b to a, c to d, and d to c.","breadcrumbs":"Design » Proving system » Notation","id":"21","title":"Notation"},"22":{"body":"","breadcrumbs":"Design » Proving system » Constructing the permutation","id":"22","title":"Constructing the permutation"},"23":{"body":"We want to construct a permutation in which each subset of variables that are in a equality-constraint set form a cycle. For example, suppose that we have a circuit that defines the following equality constraints: a≡b a≡c d≡e From this we have the equality-constraint sets {a,b,c} and {d,e}. We want to construct the permutation: (a b c) (d e) which defines the mapping of [a,b,c,d,e] to [b,c,a,e,d].","breadcrumbs":"Design » Proving system » Goal","id":"23","title":"Goal"},"24":{"body":"We need to keep track of the set of cycles, which is a set of disjoint sets . Efficient data structures for this problem are known; for the sake of simplicity we choose one that is not asymptotically optimal but is easy to implement. We represent the current state as: an array mapping for the permutation itself; an auxiliary array aux that keeps track of a distinguished element of each cycle; another array sizes that keeps track of the size of each cycle. We have the invariant that for each element x in a given cycle C, aux(x) points to the same element c∈C. This allows us to quickly decide whether two given elements x and y are in the same cycle, by checking whether aux(x)=aux(y). Also, sizes(aux(x)) gives the size of the cycle containing x. (This is guaranteed only for sizes(aux(x))), not for sizes(x).) The algorithm starts with a representation of the identity permutation: for all x, we set mapping(x)=x, aux(x)=x, and sizes(x)=1. To add an equality constraint left≡right: Check whether left and right are already in the same cycle, i.e. whether aux(left)=aux(right). If so, there is nothing to do. Otherwise, left and right belong to different cycles. Make left the larger cycle and right the smaller one, by swapping them iff sizes(aux(left)) 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":"24","title":"Algorithm"},"25":{"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 4 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":"25","title":"Broken alternatives"},"26":{"body":"TODO: Document what we do with the permutation once we have it.","breadcrumbs":"Design » Proving system » Argument specification","id":"26","title":"Argument specification"},"27":{"body":"halo2 uses the following lookup technique, which allows for lookups in arbitrary sets, and is arguably simpler than Plookup.","breadcrumbs":"Design » Proving system » Lookup argument","id":"27","title":"Lookup argument"},"28":{"body":"In addition to the general notes on language : We call the Z(X) polynomial (the grand product argument polynomial for the permutation argument) the \"permutation product\" column.","breadcrumbs":"Design » Proving system » Note on Language","id":"28","title":"Note on Language"},"29":{"body":"We express lookups in terms of a \"subset argument\" over a table with 2k rows (numbered from 0), and columns A and S. The goal of the subset argument is to enforce that every cell in A is equal to some cell in S. This means that more than one cell in A can be equal to the same cell in S, and some cells in S don't need to be equal to any of the cells in A. S might be fixed, but it doesn't need to be. That is, we can support looking up values in either fixed or variable tables (where the latter includes advice columns). A and S can contain duplicates. If the sets represented by A and/or S are not naturally of size 2k, we extend S with duplicates and A with dummy values known to be in S. Alternatively we could add a \"lookup selector\" that controls which elements of the A column participate in lookups. This would modify the occurrence of A(X) in the permutation rule below to replace A with, say, S0 if a lookup is not selected. Let ℓi be the Lagrange basis polynomial that evaluates to 1 at row i, and 0 otherwise. We start by allowing the prover to supply permutation columns of A and S. Let's call these A′ and S′, respectively. We can enforce that they are permutations using a permutation argument with product column Z with the rules: Z(X)(A(X)+β)(S(X)+γ)−Z(ω−1X)(A′(X)+β)(S′(X)+γ)=0ℓ0(X)(Z(X)−1)=0 This is a version of the permutation argument which allows A′ and S′ to be permutations of A and S, respectively, but doesn't specify the exact permutations. β and γ are separate challenges so that we can combine these two permutation arguments into one without worrying that they might interfere with each other. The goal of these permutations is to allow A′ and S′ to be arranged by the prover in a particular way: All the cells of column A′ are arranged so that like-valued cells are vertically adjacent to each other. This could be done by some kind of sorting algorithm, but all that matters is that like-valued cells are on consecutive rows in column A′, and that A′ is a permutation of A. The first row in a sequence of like values in A′ is the row that has the corresponding value in S′. Apart from this constraint, S′ is any arbitrary permutation of S. Now, we'll enforce that either Ai′=Si′ or that Ai′=Ai−1′, using the rule (A′(X)−S′(X))⋅(A′(X)−A′(ω−1X))=0 In addition, we enforce A0′=S0′ using the rule ℓ0(X)⋅(A′(X)−S′(X))=0 Together these constraints effectively force every element in A′ (and thus A) to equal at least one element in S′ (and thus S). Proof: by induction on prefixes of the rows.","breadcrumbs":"Design » Proving system » Technique Description","id":"29","title":"Technique Description"},"3":{"body":"First we'll describe the concepts behind zero-knowledge proof systems; the arithmetization (kind of circuit description) used by Halo 2; and the abstractions we use to build circuit implementations.","breadcrumbs":"Concepts","id":"3","title":"Concepts"},"30":{"body":"There is the original column A and the fixed column S. There is a permutation product column Z. There are the two permutations A′ and S′. The gates are all of low degree.","breadcrumbs":"Design » Proving system » Cost","id":"30","title":"Cost"},"31":{"body":"halo2's lookup argument implementation generalizes the above technique in the following ways: A and S can be extended to multiple columns, combined using a random challenge. A′ and S′ stay as single columns. The commitments to the columns of S can be precomputed, then combined cheaply once the challenge is known by taking advantage of the homomorphic property of Pedersen commitments. Then, a lookup argument for an arbitrary-width relation can be implemented in terms of a subset argument, i.e. to constrain R(x,y,...) in each row, consider R as a set of tuples S (using the method of the previous point), and check that (x,y,...)∈R. In the case where R represents a function, this implicitly also checks that the inputs are in the domain. This is typically what we want, and often saves an additional range check. We can support multiple tables in the same circuit, by combining them into a single table that includes a tag column to identify the original table. The tag column could be merged with the \"lookup selector\" mentioned earlier, if this were implemented. These generalizations are similar to those in sections 4 and 5 of the Plookup paper That is, the differences from Plookup are in the subset argument. This argument can then be used in all the same ways; for instance, the optimized range check technique in section 5 of the Plookup paper can also be used with this subset argument.","breadcrumbs":"Design » Proving system » Generalizations","id":"31","title":"Generalizations"},"32":{"body":"","breadcrumbs":"Design » Implementation","id":"32","title":"Implementation"},"33":{"body":"In proving system implementations like bellman, there is a concrete Proof struct that encapsulates the proof data, is returned by a prover, and can be passed to a verifier. halo2 does not contain any proof-like structures, for several reasons: The Proof structures would contain vectors of (vectors of) curve points and scalars. This complicates serialization/deserialization of proofs because the lengths of these vectors depend on the configuration of the circuit. However, we didn't want to encode the lengths of vectors inside of proofs, because at runtime the circuit is fixed, and thus so are the proof sizes. It's easy to accidentally put stuff into a Proof structure that isn't also placed in the transcript, which is a hazard when developing and implementing a proving system. We needed to be able to create multiple PLONK proofs at the same time; these proofs share many different substructures when they are for the same circuit. Instead, halo2 treats proof objects as opaque byte streams. Creation and consumption of these byte streams happens via the transcript: The TranscriptWrite trait represents something that we can write proof components to (at proving time). The TranscriptRead trait represents something that we can read proof components from (at verifying time). Crucially, implementations of TranscriptWrite are responsible for simultaneously writing to some std::io::Write buffer at the same time that they hash things into the transcript, and similarly for TranscriptRead/std::io::Read. As a bonus, treating proofs as opaque byte streams ensures that verification accounts for the cost of deserialization, which isn't negligible due to point compression.","breadcrumbs":"Design » Proofs as opaque byte streams","id":"33","title":"Proofs as opaque byte streams"},"34":{"body":"In this section we document the gadgets and chip designs provided by halo2.","breadcrumbs":"Design » Gadgets","id":"34","title":"Gadgets"},"35":{"body":"","breadcrumbs":"Design » Gadgets » SHA-256","id":"35","title":"SHA-256"},"36":{"body":"SHA-256 is specified in NIST FIPS PUB 180-4 . Unlike the specification, we use ⊞ for addition modulo 232, and + for field addition. ⊕ is used for XOR.","breadcrumbs":"Design » Gadgets » Specification","id":"36","title":"Specification"},"37":{"body":"SHA-256 maintains state in eight 32-bit variables. It processes input as 512-bit blocks, but internally splits these blocks into 32-bit chunks. We therefore designed the SHA-256 gadget to consume input in 32-bit chunks.","breadcrumbs":"Design » Gadgets » Gadget interface","id":"37","title":"Gadget interface"},"38":{"body":"The SHA-256 gadget requires a chip with the following instructions: # extern crate halo2;\n# use halo2::plonk::Error;\n# use std::fmt;\n#\n# trait Chip: Sized {}\n# trait Layouter {}\nconst BLOCK_SIZE: usize = 16;\nconst DIGEST_SIZE: usize = 8; pub trait Sha256Instructions: Chip { /// Variable representing the SHA-256 internal state. type State: Clone + fmt::Debug; /// Variable representing a 32-bit word of the input block to the SHA-256 compression /// function. type BlockWord: Copy + fmt::Debug; /// Places the SHA-256 IV in the circuit, returning the initial state variable. fn initialization_vector(layouter: &mut impl Layouter) -> Result; /// Starting from the given initial state, processes a block of input and returns the /// final state. fn compress( layouter: &mut impl Layouter, initial_state: &Self::State, input: [Self::BlockWord; BLOCK_SIZE], ) -> Result; /// Converts the given state into a message digest. fn digest( layouter: &mut impl Layouter, state: &Self::State, ) -> Result<[Self::BlockWord; DIGEST_SIZE], Error>;\n} TODO: Add instruction for computing padding. This set of instructions was chosen to strike a balance between the reusability of the instructions, and the scope for chips to internally optimise them. In particular, we considered splitting the compression function into its constituent parts (Ch, Maj etc), and providing a compression function gadget that implemented the round logic. However, this would prevent chips from using relative references between the various parts of a compression round. Having an instruction that implements all compression rounds is also similar to the Intel SHA extensions, which provide an instruction that performs multiple compression rounds.","breadcrumbs":"Design » Gadgets » Chip instructions","id":"38","title":"Chip instructions"},"39":{"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":"39","title":"16-bit table chip for SHA-256"},"4":{"body":"The aim of any proof system is to be able to prove interesting mathematical or cryptographic statements . Typically, in a given protocol we will want to prove families of statements that differ in their public inputs . The prover will also need to show that they know some private inputs that make the statement hold. To do this we write down a relation , R, that specifies which combinations of public and private inputs are valid. The terminology above is intended to be aligned with the ZKProof Community Reference . To be precise, we should distinguish between the relation R, and its implementation to be used in a proof system. We call the latter a circuit . The language that we use to express circuits for a particular proof system is called an arithmetization . Usually, an arithmetization will define circuits in terms of polynomial constraints on variables over a field. The process of expressing a particular relation as a circuit is also sometimes called \"arithmetization\", but we'll avoid that usage. To create a proof of a statement, the prover will need to know the private inputs, and also intermediate values, called advice values, that are used by the circuit. We assume that we can compute advice values efficiently from the private and public inputs. The particular advice values will depend on how we write the circuit, not only on the high-level statement. The private inputs and advice values are collectively called a witness . Some authors use \"witness\" as just a synonym for private inputs. But in our usage, a witness includes advice, i.e. it includes all values that the prover supplies to the circuit. For example, suppose that we want to prove knowledge of a preimage x of a hash function H for a digest y: The private input would be the preimage x. The public input would be the digest y. The relation would be {(x,y):H(x)=y}. For a particular public input Y, the statement would be: {(x):H(x)=Y}. The advice would be all of the intermediate values in the circuit implementing the hash function. The witness would be x and the advice. A Non-interactive Argument allows a prover to create a proof for a given statement and witness. The proof is data that can be used to convince a verifier that there exists a witness for which the statement holds. The security property that such proofs cannot falsely convince a verifier is called soundness . A Non-interactive Argument of Knowledge ( NARK ) further convinces the verifier that the prover knew a witness for which the statement holds. This security property is called knowledge soundness , and it implies soundness. In practice knowledge soundness is more useful for cryptographic protocols than soundness: if we are interested in whether Alice holds a secret key in some protocol, say, we need Alice to prove that she knows the key, not just that it exists. Knowledge soundness is formalized by saying that an extractor , which can observe precisely how the proof is generated, must be able to compute the witness. This property is subtle given that proofs can be malleable . That is, depending on the proof system it may be possible to take an existing proof (or set of proofs) and, without knowing the witness(es), modify it/them to produce a distinct proof of the same or a related statement. Higher-level protocols that use malleable proof systems need to take this into account. Even without malleability, proofs can also potentially be replayed . For instance, we would not want Alice in our example to be able to present a proof generated by someone else, and have that be taken as a demonstration that she knew the key. If a proof yields no information about the witness (other than that a witness exists and was known to the prover), then we say that the proof system is zero knowledge . If a proof system produces short proofs ---i.e. of length polylogarithmic in the circuit size--- then we say that it is succinct . A succinct NARK is called a SNARK ( Succinct Non-Interactive Argument of Knowledge ). By this definition, a SNARK need not have verification time polylogarithmic in the circuit size. Some papers use the term efficient to describe a SNARK with that property, but we'll avoid that term since it's ambiguous for SNARKs that support amortized or recursive verification, which we'll get to later. A zk-SNARK is a zero-knowledge SNARK.","breadcrumbs":"Concepts » Proof systems","id":"4","title":"Proof systems"},"40":{"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":"64","title":"Fields"},"65":{"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":"65","title":"Groups"},"66":{"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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