halo2/searchindex.json

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: This library is being actively developed and should not be used in production software.","breadcrumbs":"halo2 Crates.io","id":"0","title":"halo2 Crates.io"},"1":{"body":"","breadcrumbs":"Documentation","id":"1","title":"Documentation"},"10":{"body":"Let's start with a simple circuit, to introduce you to the common APIs and how they are used. The circuit will take a public input c, and will prove knowledge of two private inputs a and b such that a2⋅b2=c.","breadcrumbs":"User Documentation » A simple example","id":"10","title":"A simple example"},"100":{"body":"This unfortunately requires an expensive inversion of 2y. We can avoid this by arranging our equations to \"defer\" the computation of the inverse, since we often do not need the actual affine (x′,y′) coordinate of the resulting point immediately after an individual curve operation. Let's introduce a third coordinate Z and scale our curve equation by Z3 like so: Z3y2=Z3x3+Z3b Our original curve is just this curve at the restriction Z=1. If we allow the affine point (x,y) to be represented by X=xZ, Y=yZ and Z=0 then we have the homogenous projective curve Y2Z=X3+Z3b. Obtaining (x,y) from (X,Y,Z) is as simple as computing (X/Z,Y/Z) when Z=0. (When Z=0, we are dealing with the point at infinity O:=(0:1:0).) In this form, we now have a convenient way to defer the inversion required by doubling a point. The general strategy is to express x′,y′ as rational functions using x=X/Z and y=Y/Z, rearrange to make their denominators the same, and then take the resulting point (X,Y,Z) to have Z be the shared denominator and X=x′Z,Y=y′Z. Projective coordinates are often, but not always, more efficient than affine coordinates. There may be exceptions to this when either we have a different way to apply Montgomery's trick, or when we're in the circuit setting where multiplications and inversions are about equally as expensive (at least in terms of circuit size). The following shows an example of doubling a point (X,Y,Z)=(xZ,yZ,Z) without an inversion. Substituting with X,Y,Z gives us λ=2y3x2​=2(Y/Z)3(X/Z)2​=2YZ3X2​ and gives us x′y′​=λ2−2x=λ2−Z2X​=4Y2Z29X4​−Z2X​=4Y2Z29X4−8XY2Z​=8Y3Z318X4YZ−16XY3Z2​=λ(x−x′)−y=λ(ZX​−4Y2Z29X4−8XY2Z​)−ZY​=2YZ3X2​(ZX​−4Y2Z29X4−8XY2Z​)−ZY​=2YZ23X3​−8Y3Z327X6−24X3Y2Z​−ZY​=8Y3Z312X3Y2Z−8Y4Z2−27X6+24X3Y2Z​​ Notice how the denominators of x′ and y′ are the same. Thus, instead of computing (x′,y′) we can compute (X,Y,Z) with Z=8Y3Z3 and X,Y set to the corresponding numerators such that X/Z=x′ and Y/Z=y′. This completely avoids the need to perform an inversion when doubling, and something analogous to this can be done when adding two distinct points.","breadcrumbs":"Background Material » Projective coordinates","id":"100","title":"Projective coordinates"},"101":{"body":"P+Q(xp​,yp​)+(xq​,yq​)λxr​yr​​=R=(xr​,yr​)=xq​−xp​yq​−yp​​=λ2−xp​−xq​=λ(xp​−xr​)−yp​​ Important notes: There exist efficient formulae [1] for point addition that do not have edge cases (so-called \"complete\" formulae) and that unify the addition and doubling cases together. The result of adding a point to its negation using those formulae produces Z=0, which represents the point at infinity. In addition, there are other models like the Jacobian representation where (x,y)=(xZ2,yZ3,Z) where the curve is rescaled by Z6 instead of Z3, and this representation has even more efficient arithmetic but no unified/complete formulae. We can easily compare two curve points (X1​,Y1​,Z1​) and (X2​,Y2​,Z2​) for equality in the homogenous projective coordinate space by \"homogenizing\" their Z-coordinates; the checks become X1​Z2​=X2​Z1​ and Y1​Z2​=Y2​Z1​.","breadcrumbs":"Background Material » TODO: Point addition","id":"101","title":"TODO: Point addition"},"102":{"body":"Imagine that Fp​ has a primitive cube root of unity, or in other words that 3∣p−1 and so an element ζp​ generates a 3-order multiplicative subgroup. Notice that a point (x,y) on our example elliptic curve y2=x3+b has two cousin points: (ζp​x,ζp2​x), because the computation x3 effectively kills the ζ component of the x-coordinate. Applying the map (x,y)↦(ζp​x,y) is an application of an endomorphism over the curve. The exact mechanics involved are complicated, but when the curve has a prime q number of points (and thus a prime \"order\") the effect of the endomorphism is to multiply the point by a scalar in Fq​ which is also a primitive cube root ζq​ in the scalar field.","breadcrumbs":"Background Material » Curve endomorphisms","id":"102","title":"Curve endomorphisms"},"103":{"body":"TODO","breadcrumbs":"Background Material » Curve point compression","id":"103","title":"Curve point compression"},"104":{"body":"Let Ep​ be an elliptic curve over a finite field Fp​, where p is a prime. We denote this by Ep​/Fp​. and we denote the group of points of Ep​ over Fp​, with order q=#E(Fp​). For this curve, we call Fp​ the \"base field\" and Fq​ the \"scalar field\". We instantiate our proof system over the elliptic curve Ep​/Fp​. This allows us to prove statements about Fq​-arithmetic circuit satisfiability. (aside) If our curve Ep​ is over Fp​, why is the arithmetic circuit instead in Fq​? The proof system is basically working on encodings of the scalars in the circuit (or more precisely, commitments to polynomials whose coefficients are scalars). The scalars are in Fq​ when their encodings/commitments are elliptic curve points in Ep​/Fp​. However, most of the verifier's arithmetic computations are over the base field Fp​, and are thus efficiently expressed as an Fp​-arithmetic circuit. (aside) Why are the verifier's computations (mainly) over Fp​? The Halo 2 verifier actually has to perform group operations using information output by the circuit. Group operations like point doubling and addition use arithmetic in Fp​, because the coordinates of points are in Fp​. This motivates us to construct another curve with scalar field Fp​, which has an Fp​-arithmetic circuit that can efficiently verify proofs from the first curve. As a bonus, if this second curve had base field Eq​/Fq​, it would generate proofs that could be efficiently verified in the first curve's Fq​-arithmetic circuit. In other words, we instantiate a second proof system over Eq​/Fq​, forming a 2-cycle with the first:","breadcrumbs":"Background Material » Cycles of curves","id":"104","title":"Cycles of curves"},"105":{"body":"Reference: https://github.com/zcash/pasta","breadcrumbs":"Background Material » TODO: Pallas-Vesta curves","id":"105","title":"TODO: Pallas-Vesta curves"},"106":{"body":"Sometimes it is useful to be able to produce a random point on an elliptic curve Ep​/Fp​ corresponding to some input, in such a way that no-one will know its discrete logarithm (to any other base). This is described in detail in the Internet draft on Hashing to Elliptic Curves . Several algorithms can be used depending on efficiency and security requirements. The framework used in the Internet Draft makes use of several functions: hash_to_field: takes a byte sequence input and maps it to a element in the base field Fp​ map_to_curve: takes an Fp​ element and maps it to Ep​.","breadcrumbs":"Background Material » Hashing to curves","id":"106","title":"Hashing to curves"},"107":{"body":"Reference: https://eprint.iacr.org/2019/403.pdf","breadcrumbs":"Background Material » TODO: Simplified SWU","id":"107","title":"TODO: Simplified SWU"},"108":{"body":"Renes, J., Costello, C., & Batina, L. (2016, May). \"Complete addition formulas for prime order elliptic curves.\" In Annual International Conference on the Theory and Applications of Cryptographic Techniques (pp. 403-428). Springer, Berlin, Heidelberg. https://eprint.iacr.org/2015/1060.pdf","breadcrumbs":"Background Material » References","id":"108","title":"References"},"109":{"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":"109","title":"[WIP] UltraPLONK arithmetisation"},"11":{"body":"Firstly, we need to define the instructions that our circuit will rely on. Instructions are the boundary between high-level gadgets and the low-level circuit operations. Instructions may be as coarse or as granular as desired, but in practice you want to strike a balance between an instruction being large enough to effectively optimize its implementation, and small enough that it is meaningfully reusable. For our circuit, we will use three instructions: Load a private number into the circuit. Multiply two numbers. Expose a number as a public input to the circuit. We also need a type for a variable representing a number. Instruction interfaces provide associated types for their inputs and outputs, to allow the implementations to represent these in a way that makes the most sense for their optimization goals. trait NumericInstructions: Chip { /// Variable representing a number. type Num; /// Loads a number into the circuit as a private input. fn load_private( layouter: &mut impl Layouter<Self>, a: Option<Self::Field>, ) -> Result<Self::Num, Error>; /// Returns `c = a * b`. fn mul( layouter: &mut impl Layouter<Self>, a: Self::Num, b: Self::Num, ) -> Result<Self::Num, Error>; /// Exposes a number as a public input to the circuit. fn expose_public(layouter: &mut impl Layouter<Self>, num: Self::Num) -> Result<(), Error>;\n}","breadcrumbs":"User Documentation » Define instructions","id":"11","title":"Define instructions"},"110":{"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":"110","title":"Polynomial rules"},"111":{"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":"111","title":"Columns"},"112":{"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":"112","title":"Copy constraints"},"113":{"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":"113","title":"Permutation grand product"},"114":{"body":"Reference: Generic Lookups with PLONK (DRAFT)","breadcrumbs":"Background Material » Lookup","id":"114","title":"Lookup"},"115":{"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∑e​yiEi​(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=0e​yiei​​. 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":"115","title":"Vanishing argument"},"116":{"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":"116","title":"Polynomial commitment using inner product argument"},"117":{"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":"117","title":"Setup"},"118":{"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":"118","title":"Commit"},"119":{"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=log2​d 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: Lj​Rj​​=⟨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":"119","title":"Open (prover) and OpenVerify (verifier)"},"12":{"body":"For our circuit, we will build a chip that provides the above numeric instructions for a finite field. /// The chip that will implement our instructions! Chips do not store any persistent state\n/// themselves, and usually only contain type markers if necessary.\nstruct FieldChip<F: FieldExt> { _marker: PhantomData<F>,\n} Every chip needs to implement the Chip trait. This defines the properties of the chip that a Layouter may rely on when synthesizing a circuit, as well as enabling any initial state that the chip requires to be loaded into the circuit. impl<F: FieldExt> Chip for FieldChip<F> { type Config = FieldConfig; type Field = F; fn load(_layouter: &mut impl Layouter<Self>) -> Result<(), halo2::plonk::Error> { // None of the instructions implemented by this chip have any fixed state. But if // we required e.g. a lookup table, this is where we would load it. Ok(()) }\n}","breadcrumbs":"User Documentation » Define a chip implementation","id":"12","title":"Define a chip implementation"},"120":{"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−1​X2i−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":"120","title":"Recursion"},"13":{"body":"The chip needs to be configured with the columns, permutations, and gates that will be required to implement all of the desired instructions. /// Chip state is stored in a separate config struct. This is generated by the chip during\n/// configuration, and then handed to the `Layouter`, which makes it available to the chip\n/// when it needs to implement its instructions.\n#[derive(Clone, Debug)]\nstruct FieldConfig { /// For this chip, we will use two advice columns to implement our instructions. These /// are also the columns through which we communicate with other parts of the circuit. advice: [Column<Advice>; 2], // We need to create a permutation between our advice columns. This allows us to // copy numbers within these columns from arbitrary rows, which we can use to load // inputs into our instruction regions. perm: Permutation, // We need a selector to enable the multiplication gate, so that we aren't placing // any constraints on cells where `NumericInstructions::mul` is not being used. // This is important when building larger circuits, where columns are used by // multiple sets of instructions. s_mul: Column<Fixed>, // The selector for the public-input gate, which uses one of the advice columns. s_pub: Column<Fixed>,\n} impl<F: FieldExt> FieldChip<F> { fn configure( meta: &mut ConstraintSystem<F>, advice: [Column<Advice>; 2], aux: Column<Aux>, ) -> FieldConfig { let perm = Permutation::new(meta, &advice); let s_mul = meta.fixed_column(); let s_pub = meta.fixed_column(); // Define our multiplication gate! meta.create_gate(\"mul\", |meta| { // To implement multiplication, we need three advice cells and a selector // cell. We arrange them like so: // // | a0 | a1 | s_mul | // |-----|-----|-------| // | lhs | rhs | s_mul | // | out | | | // // Gates may refer to any relative offsets we want, but each distinct offset // adds a cost to the proof. The most common offsets are 0 (the current row), // 1 (the next row), and -1 (the previous row), for which `Rotation` has // specific constructors. let lhs = meta.query_advice(advice[0], Rotation::cur()); let rhs = meta.query_advice(advice[1], Rotation::cur()); let out = meta.query_advice(advice[0], Rotation::next()); let s_mul = meta.query_fixed(s_mul, Rotation::cur()); // The polynomial expression returned from `create_gate` will be constrained // by the proving system to equal zero. Our expression has the following // properties: // - When s_mul = 0, any value is allowed in lhs, rhs, and out. // - When s_mul != 0, this constrains lhs * rhs = out. s_mul * (lhs * rhs + out * -F::one()) }); // Define our public-input gate! meta.create_gate(\"public input\", |meta| { // We choose somewhat-arbitrarily that we will use the second advice column // for exposing numbers as public inputs. let a = meta.query_advice(advice[1], Rotation::cur()); let p = meta.query_aux(aux, Rotation::cur()); let s = meta.query_fixed(s_pub, Rotation::cur()); // We simply constrain the advice cell to be equal to the aux cell, when the // selector is enabled. s * (p + a * -F::one()) }); FieldConfig { advice, perm, s_mul, s_pub, } }\n}","breadcrumbs":"User Documentation » Configure the chip","id":"13","title":"Configure the chip"},"14":{"body":"/// A variable representing a number.\n#[derive(Clone)]\nstruct Number<F: FieldExt> { cell: Cell, value: Option<F>,\n} impl<F: FieldExt> NumericInstructions for FieldChip<F> { type Num = Number<F>; fn load_private( layouter: &mut impl Layouter<Self>, value: Option<Self::Field>, ) -> Result<Self::Num, Error> { let config = layouter.config().clone(); let mut num = None; layouter.assign_region( || \"load private\", |mut region| { let cell = region.assign_advice( || \"private input\", config.advice[0], 0, || value.ok_or(Error::SynthesisError), )?; num = Some(Number { cell, value }); Ok(()) }, )?; Ok(num.unwrap()) } fn mul( layouter: &mut impl Layouter<Self>, a: Self::Num, b: Self::Num, ) -> Result<Self::Num, Error> { let config = layouter.config().clone(); let mut out = None; layouter.assign_region( || \"mul\", |mut region| { // We only want to use a single multiplication gate in this region, so we // enable it at region offset 0; this means it will constrain cells at // offsets 0 and 1. region.assign_fixed(|| \"example mul\", config.s_mul, 0, || Ok(F::one()))?; // The inputs we've been given could be located anywhere in the circuit, // but we can only rely on relative offsets inside this region. So we // assign new cells inside the region and constrain them to have the same // values as the inputs. let lhs = region.assign_advice( || \"lhs\", config.advice[0], 0, || a.value.ok_or(Error::SynthesisError), )?; let rhs = region.assign_advice( || \"rhs\", config.advice[1], 0, || b.value.ok_or(Error::SynthesisError), )?; region.constrain_equal(&config.perm, a.cell, lhs)?; region.constrain_equal(&config.perm, b.cell, rhs)?; // Now we can assign the multiplication result into the output position. let value = a.value.and_then(|a| b.value.map(|b| a * b)); let cell = region.assign_advice( || \"lhs * rhs\", config.advice[0], 1, || value.ok_or(Error::SynthesisError), )?; // Finally, we return a variable representing the output, to be used in // another part of the circuit. out = Some(Number { cell, value }); Ok(()) }, )?; Ok(out.unwrap()) } fn expose_public(layouter: &mut impl Layouter<Self>, num: Self::Num) -> Result<(), Error> { let config = layouter.config().clone(); layouter.assign_region( || \"expose public\", |mut region| { // Enable the public-input gate. region.assign_fixed(|| \"public result\", config.s_pub, 0, || Ok(F::one()))?; // Load the output into the correct advice column. let out = region.assign_advice( || \"public advice\", config.advice[1], 0, || num.value.ok_or(Error::SynthesisError), )?; region.constrain_equal(&config.perm, num.cell, out)?; // We don't assign to the auxiliary column inside the circuit; the mapping // of public inputs to cells is provided to the prover. Ok(()) }, ) }\n}","breadcrumbs":"User Documentation » Implement chip traits","id":"14","title":"Implement chip traits"},"15":{"body":"Now that we have the instructions we need, and a chip that implements them, we can finally build our circuit! /// The full circuit implementation.\n///\n/// In this struct we store the private input variables. We use `Option<F>` because they\n/// won't have any value during key generation. During proving, if any of these were\n/// `None` we would get an error.\nstruct MyCircuit<F: FieldExt> { a: Option<F>, b: Option<F>,\n} impl<F: FieldExt> Circuit<F> for MyCircuit<F> { // Since we are using a single chip for everything, we can just reuse its config. type Config = FieldConfig; fn configure(meta: &mut ConstraintSystem<F>) -> Self::Config { // We create the two advice columns that FieldChip uses for I/O. let advice = [meta.advice_column(), meta.advice_column()]; // We also need an auxiliary column to store public inputs. let aux = meta.aux_column(); FieldChip::configure(meta, advice, aux) } fn synthesize(&self, cs: &mut impl Assignment<F>, config: Self::Config) -> Result<(), Error> { let mut layouter = SingleChip::new(cs, config); // Load our private values into the circuit. let a = FieldChip::load_private(&mut layouter, self.a)?; let b = FieldChip::load_private(&mut layouter, self.b)?; // We only have access to plain multiplication. We could implement our circuit as: // asq = a*a // bsq = b*b // c = asq*bsq // // but it's more efficient to implement it as: // ab = a*b // c = ab^2 let ab = FieldChip::mul(&mut layouter, a, b)?; let c = FieldChip::mul(&mut layouter, ab.clone(), ab)?; // Expose the result as a public input to the circuit. FieldChip::expose_public(&mut layouter, c) }\n}","breadcrumbs":"User Documentation » Build the circuit","id":"15","title":"Build the circuit"},"16":{"body":"halo2::dev::MockProver can be used to test that the circuit is working correctly. The private and public inputs to the circuit are constructed as we will do to create a proof, but by passing them to MockProver::run we get an object that can test every constraint in the circuit, and tell us exactly what is failing (if anything). // The number of rows in our circuit cannot exceed 2^k. Since our example circuit is // very small, we can pick a very small value here. let k = 3; // Prepare the private and public inputs to the circuit! let a = Fp::from(2); let b = Fp::from(3); let c = a.square() * b.square(); // Instantiate the circuit with the private inputs. let circuit = MyCircuit { a: Some(a), b: Some(b), }; // Arrange the public input. We expose the multiplication result in row 4 of the aux // column, so we position it there in our public inputs. let mut public_inputs = vec![Fp::zero(); 1 << k]; public_inputs[6] = c; // Given the correct public input, our circuit will verify. let prover = MockProver::run(k, &circuit, vec![public_inputs.clone()]).unwrap(); assert_eq!(prover.verify(), Ok(())); // If we try some other public input, the proof will fail! public_inputs[6] += Fp::one(); let prover = MockProver::run(k, &circuit, vec![public_inputs]).unwrap(); assert_eq!( prover.verify(), Err(VerifyFailure::Gate { gate_index: 1, gate_name: \"public input\", row: 6, }) );","breadcrumbs":"User Documentation » Testing the circuit","id":"16","title":"Testing the circuit"},"17":{"body":"You can find the source code for this example here .","breadcrumbs":"User Documentation » Full example","id":"17","title":"Full example"},"18":{"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":"18","title":"Lookup tables"},"19":{"body":"","breadcrumbs":"User Documentation » Gadgets","id":"19","title":"Gadgets"},"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":"This section contains various ideas and snippets that you might find useful while writing halo2 circuits.","breadcrumbs":"User Documentation » Tips and tricks","id":"20","title":"Tips and tricks"},"21":{"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":"21","title":"Small range constraints"},"22":{"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​−xm​x−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":"22","title":"Small set interpolation"},"23":{"body":"","breadcrumbs":"Design","id":"23","title":"Design"},"24":{"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":"24","title":"Note on Language"},"25":{"body":"","breadcrumbs":"Design » Proving system","id":"25","title":"Proving system"},"26":{"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)​++​x4​b(X)x4​d(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​)+x4​b(x3​)c(x3​)+x4​d(x3​)c(ωx3​)+x4​d(ωx3​)​ Construct f_polys which check the correctness of q_polys: f_polys f1​(X)f2​(X)​==​X−x3​q1​(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)+x5​f2​(X). Sample random x6​, at which we evaluate f(X): f(x6​)​==​f1​(x6​)x6​−x3​q1​(x6​)−r1​(x6​)​​++​x5​f2​(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)+x7​q1​(X)+x72​q2​(X), which is the polynomial we commit to in the inner product argument.","breadcrumbs":"Design » Proving system » Multipoint opening argument","id":"26","title":"Multipoint opening argument"},"27":{"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":"27","title":"Permutation argument"},"28":{"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":"28","title":"Notation"},"29":{"body":"","breadcrumbs":"Design » Proving system » Constructing the permutation","id":"29","title":"Constructing the permutation"},"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":"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":"30","title":"Goal"},"31":{"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))<sizes(aux(right)). Following the mapping around the right (smaller) cycle, for each element x set aux(x)=aux(left). Splice the smaller cycle into the larger one by swapping mapping(left) with mapping(right). For example, given two disjoint cycles (A B C D) and (E F G H): A +---> 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":"31","title":"Algorithm"},"32":{"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":"32","title":"Broken alternatives"},"33":{"body":"TODO: Document what we do with the permutation once we have it.","breadcrumbs":"Design » Proving system » Argument specification","id":"33","title":"Argument specification"},"34":{"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":"34","title":"Lookup argument"},"35":{"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":"35","title":"Note on Language"},"36":{"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":"36","title":"Technique Description"},"37":{"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":"37","title":"Cost"},"38":{"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":"38","title":"Generalizations"},"39":{"body":"","breadcrumbs":"Design » Proving system » Comparison to other work","id":"39","title":"Comparison to other work"},"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":"Appendix A.2 of BCMS20 describes a polynomial commitment scheme that is similar to the one described in BGH19 (BCMS20 being a generalization of the original Halo paper). Halo 2 builds on both of these works, and thus itself uses a polynomial commitment scheme that is very similar to the one in BCMS20. The following table provides a mapping between the variable names in BCMS20, and the equivalent objects in Halo 2 (which builds on the nomenclature from the Halo paper): BCMS20 Halo 2 S H H U C msm or P α ι ξ0​ z ξi​ challenge_i H′ [z]U pˉ​ s_poly ωˉ s_poly_blind Cˉ s_poly_commitment h(X) g(X) ω′ blind / ξ c a c a=a0​ v′ ab Halo 2's polynomial commitment scheme differs from Appendix A.2 of BCMS20 in two ways: Step 8 of the Open algorithm computes a \"non-hiding\" commitment C′ prior to the inner product argument, which opens to the same value as C but is a commitment to a randomly-drawn polynomial. The remainder of the protocol involves no blinding. By contrast, in Halo 2 we blind every single commitment that we make (even for auxiliary and fixed polynomials, though using a blinding factor of 1 for the fixed polynomials); this makes the protocol simpler to reason about. As a consequence of this, the verifier needs to handle the cumulative blinding factor at the end of the protocol, and so there is no need to derive an equivalent to C′ at the start of the protocol. C′ is also an input to the random oracle for ξ0​; in Halo 2 we utilize a transcript that has already committed to the equivalent components of C′ prior to sampling z. The PCDL​.SuccinctCheck subroutine (Figure 2 of BCMS20) computes the initial group element C0​ by adding [v]H′=[vϵ]H, which requires two scalar multiplications. Instead, we subtract [v]G0​ from the original commitment P, so that we're effectively opening the polynomial at the point to the value zero. The computation [v]G0​ is more efficient in the context of recursion because G0​ is a fixed base (so we can use lookup tables).","breadcrumbs":"Design » Proving system » BCMS20 Appendix A.2","id":"40","title":"BCMS20 Appendix A.2"},"41":{"body":"","breadcrumbs":"Design » Implementation","id":"41","title":"Implementation"},"42":{"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":"42","title":"Proofs as opaque byte streams"},"43":{"body":"In this section we document the gadgets and chip designs provided by halo2.","breadcrumbs":"Design » Gadgets","id":"43","title":"Gadgets"},"44":{"body":"","breadcrumbs":"Design » Gadgets » SHA-256","id":"44","title":"SHA-256"},"45":{"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":"45","title":"Specification"},"46":{"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":"46","title":"Gadget interface"},"47":{"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<C: Chip> {}\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<Self>) -> Result<Self::State, Error>; /// Starting from the given initial state, processes a block of input and returns the /// final state. fn compress( layouter: &mut impl Layouter<Self>, initial_state: &Self::State, input: [Self::BlockWord; BLOCK_SIZE], ) -> Result<Self::State, Error>; /// Converts the given state into a message digest. fn digest( layouter: &mut impl Layouter<Self>, 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":"47","title":"Chip instructions"},"48":{"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":"48","title":"16-bit table chip for SHA-256"},"49":{"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′Enew​Anew​​========​(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​+Wt​reduce6​(H′+D)reduce7​(H′+Maj(A,B,C)+Σ0​(A))​ where reducei​ must handle a carry 0≤carry<i. The SHA-256 compression function Define spread as a table mapping a 16-bit input to an output interleaved with zero bits. We do not require a separate table for range checks because spread can be used.","breadcrumbs":"Design » Gadgets » SHA-256 » Compression round","id":"49","title":"Compression round"},"5":{"body":"The arithmetization used by Halo 2 comes from PLONK , or more precisely its extension UltraPLONK that supports custom gates and lookup arguments. We'll call it UPA ( UltraPLONK arithmetization ). The term UPA and some of the other terms we use to describe it are not used in the PLONK paper. UPA circuits are defined in terms of a rectangular matrix of values. We refer to rows , columns , and cells of this matrix with the conventional meanings. A UPA circuit depends on a configuration : A finite field F, where cell values (for a given statement and witness) will be elements of F. The number of columns in the matrix, and a specification of each column as being fixed , advice , or auxiliary . Fixed columns are fixed by the circuit; advice columns correspond to witness values; and auxiliary columns are used for public inputs. A subset of the columns that can participate in equality constraints. A polynomial degree bound . A sequence of polynomial constraints . These are multivariate polynomials over F that must evaluate to zero for each row . The variables in a polynomial constraint may refer to a cell in a given column of the current row, or a given column of another row relative to this one (with wrap-around, i.e. taken modulo n). The maximum degree of each polynomial is given by the polynomial degree bound. A sequence of lookup arguments defined over tuples of input columns and table columns . A UPA circuit also defines: The number of rows n in the matrix. n must correspond to the size of a multiplicative subgroup of F×; typically a power of two. A sequence of equality constraints , which specify that two given cells must have equal values. The values of the fixed columns at each row. From a circuit description we can generate a proving key and a verification key , which are needed for the operations of proving and verification for that circuit. Note that we specify the ordering of columns, polynomial constraints, lookup arguments, and equality constraints, even though these do not affect the meaning of the circuit. This makes it easier to define the generation of proving and verification keys as a deterministic process. Typically, a configuration will define polynomial constraints that are switched off and on by selectors defined in fixed columns. For example, a constraint qi​⋅p(...)=0 can be switched off for a particular row i by setting qi​=0. In this case we sometimes refer to a set of constraints controlled by a set of selector columns that are designed to be used together, as a gate . Typically there will be a standard gate that supports generic operations like field multiplication and division, and possibly also custom gates that support more specialized operations.","breadcrumbs":"Concepts » UltraPLONK Arithmetization","id":"5","title":"UltraPLONK Arithmetization"},"50":{"body":"To implement addition modulo 232, we note that this is equivalent to adding the operands using field addition, and then masking away all but the lowest 32 bits of the result. For example, if we have two operands a and b: a⊞b=c, we decompose each operand (along with the result) into 16-bit chunks: (aL​:Z216​,aH​:Z216​)⊞(bL​:Z216​,bH​:Z216​)=(cL​:Z216​,cH​:Z216​), and then reformulate the constraint using field addition: carry⋅232+cH​⋅216+cL​=(aH​+bH​)⋅216+aL​+bL​. More generally, any bit-decomposition of the output can be used, not just a decomposition into 16-bit chunks. Note that this correctly handles the carry from aL​+bL​. This constraint requires that each chunk is correctly range-checked (or else an assignment could overflow the field). The operand and result chunks can be constrained using spread, by looking up each chunk in the \"dense\" column within a subset of the table. This way we additionally get the \"spread\" form of the output for free; in particular this is true for the output of the bottom-right ⊞ which becomes Anew​, and the output of the leftmost ⊞ which becomes Enew​. We will use this below to optimize Maj and Ch. carry must be constrained to the precise range of allowed carry values for the number of operands. We do this with a small range constraint .","breadcrumbs":"Design » Gadgets » SHA-256 » Modular addition","id":"50","title":"Modular addition"},"51":{"body":"Maj can be done in 4 lookups: 2spread∗2 chunks As mentioned above, after the first round we already have A in spread form A′. Similarly, B and C are equal to the A and B respectively of the previous round, and therefore in the steady state we already have them in spread form B′ and C′. In fact we can also assume we have them in spread form in the first round, either from the fixed IV or from the use of spread to reduce the output of the feedforward in the previous block. Add the spread forms in the field: M′=A′+B′+C′; We can add them as 32-bit words or in pieces; it's equivalent Witness the compressed even bits Mieven​ and the compressed odd bits Miodd​ for i={0..1}; Constrain M′=spread(M0even​)+2⋅spread(M0odd​)+232⋅spread(M1even​)+233⋅spread(M1odd​), where Miodd​ is the Maj function output. Note: by \"even\" bits we mean the bits of weight an even-power of 2, i.e. of weight 20,22,…. Similarly by \"odd\" bits we mean the bits of weight an odd-power of 2.","breadcrumbs":"Design » Gadgets » SHA-256 » Maj function","id":"51","title":"Maj function"},"52":{"body":"TODO: can probably be optimised to 4 or 5 lookups using an additional table. Ch can be done in 8 lookups: 4spread∗2 chunks As mentioned above, after the first round we already have E in spread form E′. Similarly, F and G are equal to the E and F respectively of the previous round, and therefore in the steady state we already have them in spread form F′ and G′. In fact we can also assume we have them in spread form in the first round, either from the fixed IV or from the use of spread to reduce the output of the feedforward in the previous block. Calculate P′=E′+F′ and Q′=(evens−E′)+G′, where evens=spread(232−1). We can add them as 32-bit words or in pieces; it's equivalent. evens−E′ works to compute the spread of ¬E even though negation and spread do not commute in general. It works because each spread bit in E′ is subtracted from 1, so there are no borrows. Witness Pieven​,Piodd​,Qieven​,Qiodd​ such that P′=spread(P0even​)+2⋅spread(P0odd​)+232⋅spread(P1even​)+233⋅spread(P1odd​), and similarly for Q′. {Piodd​+Qiodd​}i=0..1​ is the Ch function output.","breadcrumbs":"Design » Gadgets » SHA-256 » Ch function","id":"52","title":"Ch function"},"53":{"body":"Σ0​(A) can be done in 6 lookups. To achieve this we first split A into pieces (a,b,c,d), of lengths (2,11,9,10) bits respectively counting from the little end. At the same time we obtain the spread forms of these pieces. This can all be done in two PLONK rows, because the 10 and 11-bit pieces can be handled using spread lookups, and the 9-bit piece can be split into 3∗3-bit subpieces. The latter and the remaining 2-bit piece can be range-checked by polynomial constraints in parallel with the two lookups, two small pieces in each row. The spread forms of these small pieces are found by interpolation. Note that the splitting into pieces can be combined with the reduction of Anew​, i.e. no extra lookups are needed for the latter. In the last round we reduce Anew​ after adding the feedforward (requiring a carry of up to 7 which is fine). (A⋙2)⊕(A⋙13)⊕(A⋙22) is equivalent to (A⋙2)⊕(A⋙13)⊕(A⋘10): Then, using 4 more spread lookups we obtain the result as the even bits of a linear combination of the pieces: R′=​(a(b(c430a421b423c​∣∣∣∣∣∣+++​dab420d419a412b​∣∣∣∣∣∣⇓+++​cda411c49d410a​∣∣∣∣∣∣+++​b)c)d)bcd​⊕⊕++​ That is, we witness the compressed even bits Rieven​ and the compressed odd bits Riodd​, and constrain R′=spread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​) where {Rieven​}i=0..1​ is the Σ0​ function output.","breadcrumbs":"Design » Gadgets » SHA-256 » Σ_0 function","id":"53","title":"Σ_0 function"},"54":{"body":"Σ1​(E) can be done in 6 lookups. To achieve this we first split E into pieces (a,b,c,d), of lengths (6,5,14,7) bits respectively counting from the little end. At the same time we obtain the spread forms of these pieces. This can all be done in two PLONK rows, because the 7 and 14-bit pieces can be handled using spread lookups, the 5-bit piece can be split into 3 and 2-bit subpieces, and the 6-bit piece can be split into 2∗3-bit subpieces. The four small pieces can be range-checked by polynomial constraints in parallel with the two lookups, two small pieces in each row. The spread forms of these small pieces are found by interpolation. Note that the splitting into pieces can be combined with the reduction of Enew​, i.e. no extra lookups are needed for the latter. In the last round we reduce Enew​ after adding the feedforward (requiring a carry of up to 6 which is fine). (E⋙6)⊕(E⋙11)⊕(E⋙25) is equivalent to (E⋙6)⊕(E⋙11)⊕(E⋘7). Then, using 4 more spread lookups we obtain the result as the even bits of a linear combination of the pieces, in the same way we did for Σ0​: R′=​(a(b(c426a427b418c​∣∣∣∣∣∣+++​dab419d421a413b​∣∣∣∣∣∣⇓+++​cda45c414d47a​∣∣∣∣∣∣+++​b)c)d)bcd​⊕⊕++​ That is, we witness the compressed even bits Rieven​ and the compressed odd bits Riodd​, and constrain R′=spread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​) where {Rieven​}i=0..1​ is the Σ1​ function output.","breadcrumbs":"Design » Gadgets » SHA-256 » Σ_1 function","id":"54","title":"Σ_1 function"},"55":{"body":"For each block M∈{0,1}512 of the padded message, 64 words of 32 bits each are constructed as follows: The first 16 are obtained by splitting M into 32-bit blocks M=W0​∣∣W1​∣∣⋯∣∣W14​∣∣W15​; The remaining 48 words are constructed using the formula: Wi​=σ1​(Wi−2​)⊞Wi−7​⊞σ0​(Wi−15​)⊞Wi−16​, for 16≤i<64. Note: 0-based numbering is used for the W word indices. σ0​(X)σ1​(X)​==​(X⋙7)⊕(X⋙18)⊕(X≫3)(X⋙17)⊕(X⋙19)⊕(X≫10)​ Note: ≫ is a right- shift , not a rotation.","breadcrumbs":"Design » Gadgets » SHA-256 » Block decomposition","id":"55","title":"Block decomposition"},"56":{"body":"(X⋙7)⊕(X⋙18)⊕(X≫3) is equivalent to (X⋙7)⊕(X⋘14)⊕(X≫3). As above but with pieces (a,b,c,d) of lengths (3,4,11,14) counting from the little end. Split b into two 2-bit subpieces. R′=​(0[3](b(c428b421c​∣∣∣∣∣∣++​dab415d425a417b​∣∣∣∣∣∣⇓+++​cda44c411d414a​∣∣∣∣∣∣+++​b)c)d)bcd​⊕⊕++​","breadcrumbs":"Design » Gadgets » SHA-256 » σ_0 function","id":"56","title":"σ_0 function"},"57":{"body":"(X⋙17)⊕(X⋙19)⊕(X≫10) is equivalent to (X⋘15)⊕(X⋘13)⊕(X≫10). TODO: this diagram doesn't match the expression on the right. This is just for consistency with the other diagrams. As above but with pieces (a,b,c,d) of lengths (10,7,2,13) counting from the little end. Split b into (3,2,2)-bit subpieces. R′=​(0[10](b(c425b430c​∣∣∣∣∣∣++​dab49d415a423b​∣∣∣∣∣∣⇓+++​cda47c42d413a​∣∣∣∣∣∣+++​b)c)d)bcd​⊕⊕++​","breadcrumbs":"Design » Gadgets » SHA-256 » σ_1 function","id":"57","title":"σ_1 function"},"58":{"body":"We apply σ0​ to W1..48​, and σ1​ to W14..61​. In order to avoid redundant applications of spread, we can merge the splitting into pieces for σ0​ and σ1​ in the case of W14..48​. Merging the piece lengths (3,4,11,14) and (10,7,2,13) gives pieces of lengths (3,4,3,7,1,1,13). If we can do the merged split in 3 rows (as opposed to a total of 4 rows when splitting for σ0​ and σ1​ separately), we save 35 rows. These might even be doable in 2 rows; not sure. —Daira We can merge the reduction mod 232 of W16..61​ into their splitting when they are used to compute subsequent words, similarly to what we did for A and E in the round function. We will still need to reduce W62..63​ since they are not split. (Technically we could leave them unreduced since they will be reduced later when they are used to compute Anew​ and Enew​ -- but that would require handling a carry of up to 10 rather than 6, so it's not worth the complexity.) The resulting message schedule cost is: 2 rows to constrain W0​ to 32 bits This is technically optional, but let's do it for robustness, since the rest of the input is constrained for free. 13∗2 rows to split W1..13​ into (3,4,11,14)-bit pieces 35∗3 rows to split W14..48​ into (3,4,3,7,1,1,13)-bit pieces (merged with a reduction for W16..48​) 13∗2 rows to split W49..61​ into (10,7,2,13)-bit pieces (merged with a reduction) 4∗48 rows to extract the results of σ0​ for W1..48​ 4∗48 rows to extract the results of σ1​ for W14..61​ 2∗2 rows to reduce W62..63​ =547 rows.","breadcrumbs":"Design » Gadgets » SHA-256 » Message scheduling","id":"58","title":"Message scheduling"},"59":{"body":"For each round: 8 rows for Ch 4 rows for Maj 6 rows for Σ0​ 6 rows for Σ1​ reduce6​ and reduce7​ are always free =24 per round This gives 24∗64=1792 rows for all of \"step 3\", to which we need to add: 547 rows for message scheduling 2∗8 rows for 8 reductions mod 232 in \"step 4\" giving a total of 2099 rows.","breadcrumbs":"Design » Gadgets » SHA-256 » Overall cost","id":"59","title":"Overall cost"},"6":{"body":"The previous section gives a fairly low-level description of a circuit. When implementing circuits we will typically use a higher-level API which aims for the desirable characteristics of auditability, efficiency, modularity, and expressiveness. Some of the terminology and concepts used in this API are taken from an analogy with integrated circuit design and layout. As for integrated circuits , the above desirable characteristics are easier to obtain by composing cores that provide efficient pre-built implementations of particular functionality. For example, we might have cores that implement particular cryptographic primitives such as a hash function or cipher, or algorithms like scalar multiplication or pairings. In UPA, it is possible to build up arbitrary logic just from standard gates that do field multiplication and addition. However, very significant efficiency gains can be obtained by using custom gates. Using our API, we define cores that \"know\" how to use particular sets of custom gates. This creates an abstraction layer that isolates the implementation of a high-level circuit from the complexity of using custom gates directly. Even if we sometimes need to \"wear two hats\", by implementing both a high-level circuit and the cores that it uses, the intention is that this separation will result in code that is easier to understand, audit, and maintain/reuse. This is partly because some potential implementation errors are ruled out by construction. Gates in UPA refer to cells by relative references , i.e. to the cell in a given column, and the row at a given offset relative to the one in which the gate's selector is set. We call this an offset reference when the offset is nonzero (i.e. offset references are a subset of relative references). Relative references contrast with absolute references used in equality constraints, which can point to any cell. The motivation for offset references is to reduce the number of columns needed in the configuration, which reduces proof size. If we did not have offset references then we would need a column to hold each value referred to by a custom gate, and we would need to use equality constraints to copy values from other cells of the circuit into that column. With offset references, we not only need fewer columns; we also do not need equality constraints to be supported for all of those columns, which improves efficiency. In R1CS (which may be more familiar to some readers, but don't worry if it isn't), a circuit consists of a \"sea of gates\" with no semantically significant ordering. Because of offset references, the order of rows in a UPA circuit, on the other hand, is significant. We're going to make some simplifying assumptions and define some abstractions to tame the resulting complexity: the aim will be that, at the gadget level where we do most of our circuit construction, we will not have to deal with relative references or with gate layout explicitly. We will partition a circuit into regions , where each region contains a disjoint subset of cells, and relative references only ever point within a region. Part of the responsibility of a core implementation is to ensure that gates that make offset references are laid out in the correct positions in a region. Given the set of regions and their shapes , we will use a separate floor planner to decide where (i.e. at what starting row) each region is placed. There is a default floor planner that implements a very general algorithm, but you can write your own floor planner if you need to. Floor planning will in general leave gaps in the matrix, because the gates in a given row did not use all available columns. These are filled in ---as far as possible--- by gates that do not require offset references, which allows them to be placed on any row. Cores can also define lookup tables. If more than one table is defined for the same lookup argument, we can use a tag column to specify which table is used on each row. It is also possible to perform a lookup in the union of several tables (limited by the polynomial degree bound).","breadcrumbs":"Concepts » Cores","id":"6","title":"Cores"},"60":{"body":"We only require one table spread, with 216 rows and 3 columns. We need a tag column to allow selecting (7,10,11,13,14)-bit subsets of the table for Σ0..1​ and σ0..1​.","breadcrumbs":"Design » Gadgets » SHA-256 » Tables","id":"60","title":"Tables"},"61":{"body":"row tag table (16b) spread (32b) 0 0 0000000000000000 00000000000000000000000000000000 1 0 0000000000000001 00000000000000000000000000000001 2 0 0000000000000010 00000000000000000000000000000100 3 0 0000000000000011 00000000000000000000000000000101 ... 0 ... ... 27−1 0 0000000001111111 00000000000000000001010101010101 27 1 0000000010000000 00000000000000000100000000000000 ... 1 ... ... 210−1 1 0000001111111111 00000000000001010101010101010101 ... 2 ... ... 211−1 2 0000011111111111 00000000010101010101010101010101 ... 3 ... ... 213−1 3 0001111111111111 00000001010101010101010101010101 ... 4 ... ... 214−1 4 0011111111111111 00000101010101010101010101010101 ... 5 ... ... 216−1 5 1111111111111111 01010101010101010101010101010101 For example, to do an 11-bit spread lookup, we polynomial-constrain the tag to be in {0,1,2}. For the most common case of a 16-bit lookup, we don't need to constrain the tag. Note that we can fill any unused rows beyond 216 with a duplicate entry, e.g. all-zeroes.","breadcrumbs":"Design » Gadgets » SHA-256 » spread table","id":"61","title":"spread table"},"62":{"body":"","breadcrumbs":"Design » Gadgets » SHA-256 » Gates","id":"62","title":"Gates"},"63":{"body":"Input from previous operations: E′,F′,G′, 64-bit spread forms of 32-bit words E,F,G, assumed to be constrained by previous operations in practice, we'll have the spread forms of E′,F′,G′ after they've been decomposed into 16-bit subpieces evens is defined as spread(232−1) evens0​=evens1​=spread(216−1) E ∧ F s_ch a0​ a1​ a2​ a3​ a4​ 0 {0,1,2,3,4,5} P0even​ spread(P0even​) spread(Elo) spread(Ehi) 1 {0,1,2,3,4,5} P0odd​ spread(P0odd​) spread(P1odd​) 0 {0,1,2,3,4,5} P1even​ spread(P1even​) spread(Flo) spread(Fhi) 0 {0,1,2,3,4,5} P1odd​ spread(P1odd​) ¬E ∧ G s_ch_neg a0​ a1​ a2​ a3​ a4​ a5​ 0 {0,1,2,3,4,5} Q0even​ spread(Q0even​) spread(Eneglo​) spread(Eneghi​) spread(Elo) 1 {0,1,2,3,4,5} Q0odd​ spread(Q0odd​) spread(Q1odd​) spread(Ehi) 0 {0,1,2,3,4,5} Q1even​ spread(Q1even​) spread(Glo) spread(Ghi) 0 {0,1,2,3,4,5} Q1odd​ spread(Q1odd​) Constraints: s_ch (choice): LHS−RHS=0 LHS=a3​ω−1+a3​ω+232(a4​ω−1+a4​ω) RHS=a2​ω−1+2∗a2​+232(a2​ω+2∗a3​) s_ch_neg (negation): s_ch with an extra negation check spread lookup on (a0​,a1​,a2​) permutation between (a2​,a3​) Output: Ch(E,F,G)=Podd+Qodd=(P0odd​+Q0odd​)+216(P1odd​+Q1odd​)","breadcrumbs":"Design » Gadgets » SHA-256 » Choice gate","id":"63","title":"Choice gate"},"64":{"body":"Input from previous operations: A′,B′,C′, 64-bit spread forms of 32-bit words A,B,C, assumed to be constrained by previous operations in practice, we'll have the spread forms of A′,B′,C′ after they've been decomposed into 16-bit subpieces s_maj a0​ a1​ a2​ a3​ a4​ a5​ 0 {0,1,2,3,4,5} M0even​ spread(M0even​) spread(Alo) spread(Ahi) 1 {0,1,2,3,4,5} M0odd​ spread(M0odd​) spread(M1odd​) spread(Blo) spread(Bhi) 0 {0,1,2,3,4,5} M1even​ spread(M1even​) spread(Clo) spread(Chi) 0 {0,1,2,3,4,5} M1odd​ spread(M1odd​) Constraints: s_maj (majority): LHS−RHS=0 LHS=spread(M0even​)+2⋅spread(M0odd​)+232⋅spread(M1even​)+233⋅spread(M1odd​) RHS=A′+B′+C′ spread lookup on (a0​,a1​,a2​) permutation between (a2​,a3​) Output: Maj(A,B,C)=Modd=M0odd​+216M1odd​","breadcrumbs":"Design » Gadgets » SHA-256 » Majority gate","id":"64","title":"Majority gate"},"65":{"body":"A is a 32-bit word split into (2,11,9,10)-bit chunks, starting from the little end. We refer to these chunks as (a(2),b(11),c(9),d(10)) respectively, and further split c(9) into three 3-bit chunks c(9)lo,c(9)mid,c(9)hi. We witness the spread versions of the small chunks. Σ0​(A)​==​(A⋙2)⊕(A⋙13)⊕(A⋙22)(A⋙2)⊕(A⋙13)⊕(A⋘10)​ s_upp_sigma_0 a0​ a1​ a2​ a3​ a4​ a5​ a6​ 0 {0,1,2,3,4,5} R0even​ spread(R0even​) c(9)lo spread(c(9)lo) c(9)mid spread(c(9)mid) 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(10)) spread(b(11)) c(9) 0 {0,1,2,3,4,5} R1even​ spread(R1even​) a(2) spread(a(2)) c(9)hi spread(c(9)hi) 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) Constraints: s_upp_sigma_0 (Σ0​ constraint): LHS−RHS+tag+decompose=0 tagdecomposeLHS​===​constrain1​(a0​ω−1)+constrain2​(a0​ω)a(2)+22b(11)+213c(9)lo+216c(9)mid+219c(9)hi+222d(10)−Aspread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​)​ RHS=​430spread(a(2))421spread(b(11))429spread(c(9)hi)​+++​420spread(d(10))419spread(a(2))426spread(c(9)mid)​+++​417spread(c(9)hi)49spread(d(10))423spread(c(9)lo)​+++​414spread(c(9)mid)46spread(c(9)hi)412spread(b(11))​+++​411spread(c(9)lo)43spread(c(9)mid)410spread(a(2))​+++​spread(b(11))spread(c(9)lo)spread(d(10))​++​ spread lookup on a0​,a1​,a2​ 2-bit range check and 2-bit spread check on a(2) 3-bit range check and 3-bit spread check on c(9)lo,c(9)mid,c(9)hi (see section Helper gates ) Output: Σ0​(A)=Reven=R0even​+216R1even​","breadcrumbs":"Design » Gadgets » SHA-256 » Σ_0 gate","id":"65","title":"Σ_0 gate"},"66":{"body":"E is a 32-bit word split into (6,5,14,7)-bit chunks, starting from the little end. We refer to these chunks as (a(6),b(5),c(14),d(7)) respectively, and further split a(6) into two 3-bit chunks a(6)lo,a(6)hi and b into (2,3)-bit chunks b(5)lo,b(5)hi. We witness the spread versions of the small chunks. Σ1​(E)​==​(E⋙6)⊕(E⋙11)⊕(E⋙25)(E⋙6)⊕(E⋙11)⊕(E⋘7)​ s_upp_sigma_1 a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ 0 {0,1,2,3,4,5} R0even​ spread(R0even​) b(5)lo spread(b(5)lo) b(5)hi spread(b(5)hi) b(5) 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(7)) spread(c(14)) 0 {0,1,2,3,4,5} R1even​ spread(R1even​) a(6)lo spread(a(6)lo) a(6)hi spread(a(6)hi) a(6) 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) Constraints: s_upp_sigma_1 (Σ1​ constraint): LHS−RHS+tag+decompose=0 tagdecomposeLHS​===​a0​ω−1+constrain4​(a0​ω)a(6)lo+23a(6)hi+26b(5)lo+28b(5)hi+211c(14)+225d(7)−Espread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​)​ RHS=​429spread(a(6)hi)429spread(b(5)hi)418spread(c(14))​+++​426spread(a(6)lo)427spread(b(5)lo)415spread(b(5)hi)​+++​419spread(d(7))424spread(a(6)hi)413spread(b(5)lo)​+++​45spread(c(14))421spread(a(6)lo)410spread(a(6)hi)​+++​42spread(b(5)hi)414spread(d(7))47spread(a(6)lo)​+++​spread(b(5)lo)spread(c(14))spread(d(7))​++​ spread lookup on a0​,a1​,a2​ 2-bit range check and 2-bit spread check on b(5)lo 3-bit range check and 3-bit spread check on a(6)lo,a(6)hi,b(4)hi (see section Helper gates ) Output: Σ1​(E)=Reven=R0even​+216R1even​","breadcrumbs":"Design » Gadgets » SHA-256 » Σ_1 gate","id":"66","title":"Σ_1 gate"},"67":{"body":"v1 v1 of the σ0​ gate takes in a word that's split into (3,4,11,14)-bit chunks (already constrained by message scheduling). We refer to these chunks respectively as (a(3),b(4),c(11),d(14)). b(4 is further split into two 2-bit chunks b(4)lo,b(4)hi. We witness the spread versions of the small chunks. We already have spread(c(11)) and spread(d(14)) from the message scheduling. (X⋙7)⊕(X⋙18)⊕(X≫3) is equivalent to (X⋙7)⊕(X⋘14)⊕(X≫3). s_low_sigma_0 a0​ a1​ a2​ a3​ a4​ a5​ a6​ 0 {0,1,2,3,4,5} R0even​ spread(R0even​) b(4)lo spread(b(4)lo) b(4)hi spread(b(4)hi) 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(c) spread(d) b(4) 0 {0,1,2,3,4,5} R1even​ spread(R1even​) 0 0 a spread(a) 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) Constraints: s_low_sigma_0 (σ0​ v1 constraint): LHS−RHS=0 LHS​=​spread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​)​ RHS=​430b(4)hi421c(11)​++​415d(14)428b(4)lo419b(4)hi​+++​44c(11)425a(3)417b(4)lo​+++​42b(4)hi411d(14)414a(3)​+++​b(4)loc(11)d(14)​++​ check that b was properly split into subsections for 4-bit pieces. Wb(4)lo+22Wb(4)hi−W=0 2-bit range check and 2-bit spread check on b(4)lo,b(4)hi 3-bit range check and 3-bit spread check on a(3) v2 v2 of the σ0​ gate takes in a word that's split into (3,4,3,7,1,1,13)-bit chunks (already constrained by message scheduling). We refer to these chunks respectively as (a(3),b(4),c(3),d(7),e(1),f(1),g(13)). We already have spread(d(7)),spread(g(13)) from the message scheduling. The 1-bit e(1),f(1) remain unchanged by the spread operation and can be used directly. We further split b(4) into two 2-bit chunks b(4)lo,b(4)hi. We witness the spread versions of the small chunks. (X⋙7)⊕(X⋙18)⊕(X≫3) is equivalent to (X⋙7)⊕(X⋘14)⊕(X≫3). s_low_sigma_0_v2 a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ 0 {0,1,2,3,4,5} R0even​ spread(R0even​) b(4)lo spread(b(4)lo) b(4)hi spread(b(4)hi) 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(7)) spread(g(13)) b(4) e(1) 0 {0,1,2,3,4,5} R1even​ spread(R1even​) a(3) spread(a(3)) c(3) spread(c(3)) f(1) 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) Constraints: s_low_sigma_0_v2 (σ0​ v2 constraint): LHS−RHS=0 LHS​=​spread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​)​ RHS=​430b(4)hi431e(1)​++​416g(13)428b(4)lo424d(7)​+++​415f(1)425a(3)421c(3)​+++​414e(1)412g(13)419b(4)hi​+++​47d(7)411f(1)417b(4)lo​+++​44c(3)410e(1)414a(3)​+++​42b(4)hi43d(7)41g(13)​+++​b(4)loc(3)f(1)​++​ check that b was properly split into subsections for 4-bit pieces. Wb(4)lo+22Wb(4)hi−W=0 2-bit range check and 2-bit spread check on b(4)lo,b(4)hi 3-bit range check and 3-bit spread check on a(3),c(3)","breadcrumbs":"Design » Gadgets » SHA-256 » σ_0 gate","id":"67","title":"σ_0 gate"},"68":{"body":"v1 v1 of the σ1​ gate takes in a word that's split into (10,7,2,13)-bit chunks (already constrained by message scheduling). We refer to these chunks respectively as (a(10),b(7),c(2),d(13)). b(7) is further split into (2,2,3)-bit chunks b(7)lo,b(7)mid,b(7)hi. We witness the spread versions of the small chunks. We already have spread(a(10)) and spread(d(13)) from the message scheduling. (X⋙17)⊕(X⋙19)⊕(X≫10) is equivalent to (X⋘15)⊕(X⋘13)⊕(X≫10). s_low_sigma_1 a0​ a1​ a2​ a3​ a4​ a5​ a6​ 0 {0,1,2,3,4,5} R0even​ spread(R0even​) b(7)lo spread(b(7)lo) b(7)mid spread(b(7)mid) 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(a(10)) spread(d(13)) b(7) 0 {0,1,2,3,4,5} R1even​ spread(R1even​) c(2) spread(c(2)) b(7)hi spread(b(7)hi) 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) Constraints: s_low_sigma_1 (σ1​ v1 constraint): LHS−RHS=0 LHS​=​spread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​)​ RHS=​429b(7)hi430c(2)​++​49d(13)427b(7)mid427b(7)hi​+++​47c(2)425b(7)lo425b(7)mid​+++​44b(7)hi415a(10)423b(7)lo​+++​42b(7)mid42d(13)413a(10)​+++​b(7)loc(2)d(13)​++​ check that b was properly split into subsections for 7-bit pieces. Wb(7)lo+22Wb(7)mid+24Wb(7)hi−W=0 2-bit range check and 2-bit spread check on b(7)lo,b(7)mid,c(2) 3-bit range check and 3-bit spread check on b(7)hi v2 v2 of the σ1​ gate takes in a word that's split into (3,4,3,7,1,1,13)-bit chunks (already constrained by message scheduling). We refer to these chunks respectively as (a(3),b(4),c(3),d(7),e(1),f(1),g(13)). We already have spread(d(7)),spread(g(13)) from the message scheduling. The 1-bit e(1),f(1) remain unchanged by the spread operation and can be used directly. We further split b(4) into two 2-bit chunks b(4)lo,b(4)hi. We witness the spread versions of the small chunks. (X⋙17)⊕(X⋙19)⊕(X≫10) is equivalent to (X⋘15)⊕(X⋘13)⊕(X≫10). s_low_sigma_1_v2 a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ 0 {0,1,2,3,4,5} R0even​ spread(R0even​) b(4)lo spread(b(4)lo) b(4)hi spread(b(4)hi) 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(7)) spread(g(13)) b(4) e(1) 0 {0,1,2,3,4,5} R1even​ spread(R1even​) a(3) spread(a(3)) c(3) spread(c(3)) f(1) 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) Constraints: s_low_sigma_1_v2 (σ1​ v2 constraint): LHS−RHS=0 LHS​=​spread(R0even​)+2⋅spread(R0odd​)+232⋅spread(R1even​)+233⋅spread(R1odd​)​ RHS=​425d(7)431f(1)​++​422c(3)430e(1)​++​420b(4)hi423d(7)​++​49g(13)418b(4)lo420c(3)​+++​48f(1)415a418b(4)hi​+++​47e(1)42g(13)416b(4)lo​+++​d(7)41f(1)413a​+++​e(1)g(13)​+​ check that b was properly split into subsections for 4-bit pieces. Wb(4)lo+22Wb(4)hi−W=0 2-bit range check and 2-bit spread check on b(4)lo,b(4)hi 3-bit range check and 3-bit spread check on a(3),c(3)","breadcrumbs":"Design » Gadgets » SHA-256 » σ_1 gate","id":"68","title":"σ_1 gate"},"69":{"body":"Small range constraints Let constrainn​(x)=∏i=0n​(x−i). Constraining this expression to equal zero enforces that x is in [0..n]. 2-bit range check (a−3)(a−2)(a−1)(a)=0 sr2 a0​ 1 a 2-bit spread l1​(a)+4∗l2​(a)+5∗l3​(a)−a′=0 ss2 a0​ a1​ 1 a a' with interpolation polynomials: l0​(a)=(−3)(−2)(−1)(a−3)(a−2)(a−1)​ (spread(00)=0000) l1​(a)=(−2)(−1)(1)(a−3)(a−2)(a)​ (spread(01)=0001) l2​(a)=(−1)(1)(2)(a−3)(a−1)(a)​ (spread(10)=0100) l3​(a)=(1)(2)(3)(a−2)(a−1)(a)​ (spread(11)=0101) 3-bit range check (a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)(a)=0 sr3 a0​ 1 a 3-bit spread l1​(a)+4∗l2​(a)+5∗l3​(a)+16∗l4​(a)+17∗l5​(a)+20∗l6​(a)+21∗l7​(a)−a′=0 ss3 a0​ a1​ 1 a a' with interpolation polynomials: l0​(a)=(−7)(−6)(−5)(−4)(−3)(−2)(−1)(a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)​ (spread(000)=000000) l1​(a)=(−6)(−5)(−4)(−3)(−2)(−1)(1)(a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a)​ (spread(001)=000001) l2​(a)=(−5)(−4)(−3)(−2)(−1)(1)(2)(a−7)(a−6)(a−5)(a−4)(a−3)(a−1)(a)​ (spread(010)=000100) l3​(a)=(−4)(−3)(−2)(−1)(1)(2)(3)(a−7)(a−6)(a−5)(a−3)(a−2)(a−1)(a)​ (spread(011)=000101) l4​(a)=(−3)(−2)(−1)(1)(2)(3)(4)(a−7)(a−6)(a−5)(a−3)(a−2)(a−1)(a)​ (spread(100)=010000) l5​(a)=(−2)(−1)(1)(2)(3)(4)(5)(a−7)(a−6)(a−4)(a−3)(a−2)(a−1)(a)​ (spread(101)=010001) l6​(a)=(−1)(1)(2)(3)(4)(5)(6)(a−7)(a−5)(a−4)(a−3)(a−2)(a−1)(a)​ (spread(110)=010100) l7​(a)=(1)(2)(3)(4)(5)(6)(7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)(a)​ (spread(111)=010101) reduce_6 gate Addition (mod232) of 6 elements Input: E {eilo​,eihi​}i=05​ carry Check: E=e0​+e1​+e2​+e3​+e4​+e5​(mod32) Assume inputs are constrained to 16 bits. Addition gate (sa): a0​+a1​+a2​+a3​+a4​+a5​+a6​−a7​=0 Carry gate (sc): 216a6​ω−1+a6​+[(a6​−5)(a6​−4)(a6​−3)(a6​−2)(a6​−1)(a6​)]=0 sa sc a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ 1 0 e0lo​ e1lo​ e2lo​ e3lo​ e4lo​ e5lo​ −carry∗216 Elo 1 1 e0hi​ e1hi​ e2hi​ e3hi​ e4hi​ e5hi​ carry Ehi Assume inputs are constrained to 16 bits. Addition gate (sa): a0​ω−1+a1​ω−1+a2​ω−1+a0​+a1​+a2​+a3​ω−1−a3​=0 Carry gate (sc): 216a3​ω+a3​ω−1=0 sa sc a0​ a1​ a2​ a3​ 0 0 e0lo​ e1lo​ e2lo​ −carry∗216 1 1 e3lo​ e4lo​ e5lo​ Elo 0 0 e0hi​ e1hi​ e2hi​ carry 1 0 e3hi​ e4hi​ e5hi​ Ehi reduce_7 gate Addition (mod232) of 7 elements Input: E {eilo​,eihi​}i=06​ carry Check: E=e0​+e1​+e2​+e3​+e4​+e5​+e6​(mod32) Assume inputs are constrained to 16 bits. Addition gate (sa): a0​+a1​+a2​+a3​+a4​+a5​+a6​+a7​−a8​=0 Carry gate (sc): 216a7​ω−1+a7​+[(a7​−6)(a7​−5)(a7​−4)(a7​−3)(a7​−2)(a7​−1)(a7​)]=0 sa sc a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ 1 0 e0lo​ e1lo​ e2lo​ e3lo​ e4lo​ e5lo​ e6lo​ −carry∗216 Elo 1 1 e0hi​ e1hi​ e2hi​ e3hi​ e4hi​ e5hi​ e6hi​ carry Ehi","breadcrumbs":"Design » Gadgets » SHA-256 » Helper gates","id":"69","title":"Helper gates"},"7":{"body":"In order to combine functionality from several cores, we use a chip . To implement a chip, we define a set of fixed, advice, and auxiliary columns, and then specify how they should be distributed between cores. In the simplest case, each core will use columns disjoint from the other cores. However, it is allowed to share a column between cores. It is important to optimize the number of advice columns in particular, because that affects proof size. The result (possibly after optimization) is a UPA configuration. Our circuit implementation will be parameterized on a chip, and can use any features of the supported cores via the chip. Our hope is that less expert users will normally be able to find an existing chip that supports the operations they need, or only have to make minor modifications to an existing chip. Expert users will have full control to do the kind of circuit optimizations that ECC is famous for 🙂.","breadcrumbs":"Concepts » Chips","id":"7","title":"Chips"},"70":{"body":"For each block M∈{0,1}512 of the padded message, 64 words of 32 bits each are constructed as follows: the first 16 are obtained by splitting M into 32-bit blocks M=W0​∣∣W1​∣∣⋯∣∣W14​∣∣W15​; the remaining 48 words are constructed using the formula: Wi​=σ1​(Wi−2​)⊞Wi−7​⊞σ0​(Wi−15​)⊞Wi−16​, for 16≤i<64. sw sd0 sd1 sd2 sd3 ss0 ss0_v2 ss1 ss1_v2 a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W0lo​ spread(W0lo​) W0lo​ W0hi​ W0​ σ0​(W1​)lo σ1​(W14​)lo W9lo​ 1 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W0hi​ spread(W0hi​) W16​ σ0​(W1​)hi σ1​(W14​)hi W9hi​ carry16​ 0 1 1 0 0 0 0 0 0 {0,1,2,3,4} W1d(14)​ spread(W1d(14)​) W1lo​ W1hi​ W1​ σ0​(W2​)lo σ1​(W15​)lo W10lo​ 1 0 0 0 0 0 0 0 0 {0,1,2} W1c(11)​ spread(W1c(11)​) W1a(3)​ W1b(4)​ W17​ σ0​(W2​)hi σ1​(W15​)hi W10hi​ carry17​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W1b(4)lo​ spread(W1b(4)lo​) W1b(4)hi​ spread(W1b(4)hi​) 0 0 0 0 0 1 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R0odd​) spread(R1odd​) spread(W1c(11)​) spread(W1d(14)​) W1b(4)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R1even​) 0 0 W1a(3)​ spread(W1a(3)​) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1odd​) σ0​v1R0​ σ0​v1R1​ σ0​v1R0even​ σ0​v1R0odd​ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 0 0 0 0 0 0 0 0 {0,1,2,3} W14g(13)​ spread(W14g(13)​) W14a(3)​ W14c(3)​ 0 1 0 1 0 0 0 0 0 0 W14d(7)​ spread(W14d(7)​) W14lo​ W14hi​ W14​ σ0​(W15​)lo σ1​(W28​)lo W23lo​ 1 0 0 0 0 0 0 0 0 0 W14b(4)​ spread(W14b(4)​) W14e(1)​ W14f(1)​ W30​ σ0​(W15​)hi σ1​(W28​)hi W23hi​ carry30​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W14b(4)lo​ spread(W14b(4)lo​) W14b(4)hi​ spread(W14b(4)hi​) 0 0 0 0 0 0 1 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(W14d(7)​) spread(W14g(13)​) W1b(14)​ W14e(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) W14a(3)​ spread(W14a(3)​) W14c(3)​ spread(W14c(3)​) W14f(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) σ0​v2R0​ σ0​v2R1​ σ0​v2R0even​ σ0​v2R0odd​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W14b(4)lo​ spread(W14b(4)lo​) W14b(4)hi​ spread(W14b(4)hi​) 0 0 0 0 0 0 0 0 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d) spread(g) W14e(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) W14a(3)​ spread(W14a(3)​) W14c(3)​ spread(W14c(3)​) W14f(1)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) σ1​v2R0​ σ1​v2R1​ σ1​v2R0even​ σ1​v2R0odd​ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 1 0 0 1 0 0 0 0 {0,1,2,3} W49d(13)​ spread(W49d(13)​) W49lo​ W49hi​ W49​ 0 0 0 0 0 0 0 0 0 {0,1} W49a(10)​ spread(W49a(10)​) W49c(2)​ W49b(7)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) W49b(7)lo​ spread(W49b(7)lo​) W49b(7)mid​ spread(W49b(7)mid​) 0 0 0 0 0 0 0 0 1 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(a) spread(d) W1b(49)​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) W49c(2)​ spread(W49c(2)​) W49b(7)hi​ spread(W49b(7)hi​) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) σ1​v1R0​ σ1​v1R1​ σ1​v1R0even​ σ1​v1R0odd​ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W62lo​ spread(W62lo​) W62lo​ W62hi​ W62​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W62hi​ spread(W62hi​) 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W63lo​ spread(W63lo​) W63lo​ W63hi​ W63​ 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W63hi​ spread(W63hi​) Constraints: sw: construct word using reduce4​ sd0: decomposition gate for W0​,W62​,W63​ Wlo+216Whi−W=0 sd1: decomposition gate for W1..13​ (split into (3,4,11,14)-bit pieces) Wa(3)+23Wb(4)lo+25Wb(4)hi+27Wc(11)+218Wd(14)−W=0 sd2: decomposition gate for W14..48​ (split into (3,4,3,7,1,1,13)-bit pieces) Wa(3)+23Wb(4)lo+25Wb(4)hi+27Wc(11)+210Wd(14)+217We(1)+218Wf(1)+219Wg(13)−W=0 sd3: decomposition gate for W49..61​ (split into (10,7,2,13)-bit pieces) Wa(10)+210Wb(7)lo+212Wb(7)mid+215Wb(7)hi+217Wc(2)+219Wd(13)−W=0","breadcrumbs":"Design » Gadgets » SHA-256 » Message scheduling region","id":"70","title":"Message scheduling region"},"71":{"body":"+----------------------------------------------------------+\n| |\n| decompose E, |\n| Σ_1(E) |\n| |\n| +---------------------------------------+\n| | |\n| | reduce_5() to get H' |\n| | |\n+----------------------------------------------------------+\n| decompose F, decompose G |\n| |\n| Ch(E,F,G) |\n| |\n+----------------------------------------------------------+\n| |\n| decompose A, |\n| Σ_0(A) |\n| |\n| |\n| +---------------------------------------+\n| | |\n| | reduce_7() to get A_new, |\n| | using H' |\n| | |\n+------------------+---------------------------------------+\n| decompose B, decompose C |\n| |\n| Maj(A,B,C) |\n| |\n| +---------------------------------------+\n| | reduce_6() to get E_new, |\n| | using H' |\n+------------------+---------------------------------------+ Initial round: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0​d(7) spread(E0​d(7)) E0​b(5)lo spread(E0​b(5)lo) E0​b(5)hi spread(E0​b(5)hi) E0lo​ spread(E0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} E0​c(14) spread(E0​c(14)) E0​a(6)lo spread(E0​a(6)lo) E0​a(6)hi spread(E0​a(6)hi) E0hi​ spread(E0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(E0​b(5)lo) spread(E0​b(5)hi) 0 0 0 0 1 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(E0​d(7)) spread(E0​c(14)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(E0​a(6)lo) spread(E0​a(6)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0​d(7) spread(F0​d(7)) F0​b(5)lo spread(F0​b(5)lo) F0​b(5)hi spread(F0​b(5)hi) F0lo​ spread(F0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} F0​c(14) spread(F0​c(14)) F0​a(6)lo spread(F0​a(6)lo) F0​a(6)hi spread(F0​a(6)hi) F0hi​ spread(F0hi​) 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} G0​d(7) spread(G0​d(7)) G0​b(5)lo spread(G0​b(5)lo) G0​b(5)hi spread(G0​b(5)hi) G0lo​ spread(G0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} G0​c(14) spread(G0​c(14)) G0​a(6)lo spread(G0​a(6)lo) G0​a(6)hi spread(G0​a(6)hi) G0hi​ spread(G0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P0even​ spread(P0even​) spread(Elo) spread(Ehi) Q0odd​ K0lo​ H0lo​ W0lo​ 0 0 0 0 0 0 0 1 0 0 1 {0,1,2,3,4,5} P0odd​ spread(P0odd​) spread(P1odd​) Σ1​(E0​)lo Σ1​(E0​)hi K0hi​ H0hi​ W0hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1even​ spread(P1even​) spread(Flo) spread(Fhi) Q1odd​ P1odd​ Hprime0lo​ Hprime0hi​ Hprime0​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1odd​ spread(P1odd​) D0lo​ E1lo​ 0 0 0 0 0 0 0 0 0 1 0 {0,1,2,3,4,5} Q0even​ spread(Q0even​) spread(Eneglo​) spread(Eneghi​) spread(Elo) D0hi​ E1hi​ E1​carry 0 0 0 0 0 0 1 0 0 0 0 {0,1,2,3,4,5} Q0odd​ spread(Q0odd​) spread(Q1odd​) spread(Ehi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1even​ spread(Q1even​) spread(Glo) spread(Ghi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1odd​ spread(Q1odd​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} A0​b(11) spread(A0​b(11)) A0​c(9)lo spread(A0​c(9)lo) A0​c(9)mid spread(A0​c(9)mid) A0lo​ spread(A0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} A0​d(10) spread(A0​d(10)) A0​a(2) spread(A0​a(2)) A0​c(9)hi spread(A0​c(9)hi) A0hi​ spread(A0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(c(9)lo) spread(c(9)mid) 0 0 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(10)) spread(b(11)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(a(2)) spread(c(9)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} B0​b(11) spread(B0​b(11)) B0​c(9)lo spread(B0​c(9)lo) B0​c(9)mid spread(B0​c(9)mid) B0lo​ spread(B0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} B0​d(10) spread(B0​d(10)) B0​a(2) spread(B0​a(2)) B0​c(9)hi spread(B0​c(9)hi) B0hi​ spread(B0hi​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} C0​b(11) spread(C0​b(11)) C0​c(9)lo spread(C0​c(9)lo) C0​c(9)mid spread(C0​c(9)mid) C0lo​ spread(C0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} C0​d(10) spread(C0​d(10)) C0​a(2) spread(C0​a(2)) C0​c(9)hi spread(C0​c(9)hi) C0hi​ spread(C0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M0even​ spread(M0even​) M1odd​ spread(A0lo​) spread(A0hi​) Hprime0lo​ Hprime0hi​ 0 0 0 0 0 1 0 0 1 0 0 {0,1,2,3,4,5} M0odd​ spread(M0odd​) spread(M1odd​) spread(B0lo​) spread(B0hi​) Σ0​(A0​)lo A1lo​ A1​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1even​ spread(M1even​) spread(C0lo​) spread(C0hi​) Σ0​(A0​)hi A1hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1odd​ spread(M1odd​) Steady-state: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0​d(7) spread(E0​d(7)) E0​b(5)lo spread(E0​b(5)lo) E0​b(5)hi spread(E0​b(5)hi) E0lo​ spread(E0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} E0​c(14) spread(E0​c(14)) E0​a(6)lo spread(E0​a(6)lo) E0​a(6)hi spread(E0​a(6)hi) E0hi​ spread(E0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(E0​b(5)lo) spread(E0​b(5)hi) 0 0 0 0 1 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(E0​d(7)) spread(E0​c(14)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(E0​a(6)lo) spread(E0​a(6)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P0even​ spread(P0even​) spread(Elo) spread(Ehi) Q0odd​ K0lo​ H0lo​ W0lo​ 0 0 0 0 0 0 0 1 0 0 1 {0,1,2,3,4,5} P0odd​ spread(P0odd​) spread(P1odd​) Σ1​(E0​)lo Σ1​(E0​)hi K0hi​ H0hi​ W0hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1even​ spread(P1even​) spread(Flo) spread(Fhi) Q1odd​ P1odd​ Hprime0lo​ Hprime0hi​ Hprime0​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1odd​ spread(P1odd​) D0lo​ E1lo​ 0 0 0 0 0 0 0 0 0 1 0 {0,1,2,3,4,5} Q0even​ spread(Q0even​) spread(Eneglo​) spread(Eneghi​) spread(Elo) D0hi​ E1hi​ E1​carry 0 0 0 0 0 0 1 0 0 0 0 {0,1,2,3,4,5} Q0odd​ spread(Q0odd​) spread(Q1odd​) spread(Ehi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1even​ spread(Q1even​) spread(Glo) spread(Ghi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1odd​ spread(Q1odd​) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} A0​b(11) spread(A0​b(11)) A0​c(9)lo spread(A0​c(9)lo) A0​c(9)mid spread(A0​c(9)mid) A0lo​ spread(A0lo​) 0 0 0 0 0 0 0 0 0 0 0 {0,1} A0​d(10) spread(A0​d(10)) A0​a(2) spread(A0​a(2)) A0​c(9)hi spread(A0​c(9)hi) A0hi​ spread(A0hi​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even​ spread(R0even​) spread(c(9)lo) spread(c(9)mid) 0 0 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd​ spread(R0odd​) spread(R1odd​) spread(d(10)) spread(b(11)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even​ spread(R1even​) spread(a(2)) spread(c(9)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd​ spread(R1odd​) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M0even​ spread(M0even​) M1odd​ spread(A0lo​) spread(A0hi​) Hprime0lo​ Hprime0hi​ 0 0 0 0 0 1 0 0 1 0 0 {0,1,2,3,4,5} M0odd​ spread(M0odd​) spread(M1odd​) spread(B0lo​) spread(B0hi​) Σ0​(A0​)lo A1lo​ A1​carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1even​ spread(M1even​) spread(C0lo​) spread(C0hi​) Σ0​(A0​)hi A1hi​ 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1odd​ spread(M1odd​) Final digest: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0​ a1​ a2​ a3​ a4​ a5​ a6​ a7​ a8​ a9​ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 A63lo​ A63hi​ A63​ B63lo​ B63hi​ B63​ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C63lo​ C63hi​ C63​ C63lo​ C63hi​ C63​ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 E63lo​ E63hi​ E63​ G63lo​ G63hi​ G63​ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 F63lo​ F63hi​ F63​ H63lo​ H63hi​ H63​","breadcrumbs":"Design » Gadgets » SHA-256 » Compression region","id":"71","title":"Compression region"},"72":{"body":"This section covers the background material required to understand the Halo 2 proving system. It is targeted at an ELI15 (Explain It Like I'm 15) level; if you think anything could do with additional explanation, let us know !","breadcrumbs":"Background Material","id":"72","title":"Background Material"},"73":{"body":"A fundamental component of many cryptographic protocols is the algebraic structure known as a field . Fields are sets of objects (usually numbers) with two associated binary operators + and × such that various field axioms hold. The real numbers R are an example of a field with an uncountably infinite number of elements. Halo makes use of finite fields which have a finite number of elements. Finite fields are fully classified as follows: if F is a finite field, it contains ∣F∣=pk elements for some integer k≥1 and some prime p; any two finite fields with the same number of elements are isomorphic. In particular, all of the arithmetic in a prime field Fp​ is isomorphic to addition and multiplication of integers modulo p, i.e. in Zp​. This is why we often refer to p as the modulus . We'll write a field as Fq​ where q=pk. The prime p is called its characteristic . In the cases where k>1 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":"73","title":"Fields"},"74":{"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∏k​giai​​. 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":"74","title":"Groups"},"75":{"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 0≤i,j<p−1. As a result the multiplication of nonzero field elements can be interpreted as addition modulo p−1 with respect to some fixed generator α. The addition just happens \"in the exponent.\" This is another way to look at where ap−2 comes from for computing inverses in the field: p−2≡−1(modp−1), so ap−2=a−1.","breadcrumbs":"Background Material » The multiplicative group of a finite field","id":"75","title":"The multiplicative group of a finite field"},"76":{"body":"Montgomery's trick, named after Peter Montgomery (RIP) is a way to compute many group inversions at the same time. It is commonly used to compute inversions in Fp×​, which are quite computationally expensive compared to multiplication. Imagine we need to compute the inverses of three nonzero elements a,b,c∈Fp×​. Instead, we'll compute the products x=ab and y=xc=abc, and compute the inversion z=yp−2=abc1​. We can now multiply z by x to obtain c1​ and multiply z by c to obtain ab1​, which we can then multiply by a,b to obtain their respective inverses. This technique generalizes to arbitrary numbers of group elements with just a single inversion necessary.","breadcrumbs":"Background Material » Montgomery's Trick","id":"76","title":"Montgomery's Trick"},"77":{"body":"A subgroup of a group G with operation ⋅, is a subset of elements of G that also form a group under ⋅. In the previous section we said that α is a generator of the (p−1)-order multiplicative group Fp×​. This group has composite order, and so by the Chinese remainder theorem [1] it has strict subgroups. As an example let's imagine that p=11, and so p−1 factors into 5⋅2. Thus, there is a generator β of the 5-order subgroup and a generator γ of the 2-order subgroup. All elements in Fp×​, therefore, can be written uniquely as βi⋅γj for some i (modulo 5) and some j (modulo 2). If we have a=βi⋅γj notice what happens when we compute a5=(βi⋅γj)5=βi⋅5⋅γj⋅5=β0⋅γj⋅5=γj⋅5; we have effectively \"killed\" the 5-order subgroup component, producing a value in the 2-order subgroup. Lagrange's theorem (group theory) states that the order of any subgroup H of a finite group G divides the order of G. Therefore, the order of any subgroup of Fp×​ must divide p−1.","breadcrumbs":"Background Material » Multiplicative subgroups","id":"77","title":"Multiplicative subgroups"},"78":{"body":"In a field Fp​ exactly half of all nonzero elements are squares; the remainder are non-squares or \"quadratic non-residues\". In order to see why, consider an α that generates the 2-order multiplicative subgroup of Fp×​ (this exists because p−1 is divisible by 2 since p is a prime greater than 2) and β that generates the t-order multiplicative subgroup of Fp×​ where p−1=2t. Then every element a∈Fp×​ can be written uniquely as αi⋅βj with i∈Z2​ and j∈Zt​. Half of all elements will have i=0 and the other half will have i=1. Let's consider the simple case where p≡3(mod4) and so t is odd (if t is even, then p−1 would be divisible by 4, which contradicts p being 3(mod4)). If a∈Fp×​ is a square, then there must exist b=αi⋅βj such that b2=a. But this means that a=(αi⋅βj)2=α2i⋅β2j=β2j. In other words, all squares in this particular field do not generate the 2-order multiplicative subgroup, and so since half of the elements generate the 2-order subgroup then at most half of the elements are square. In fact exactly half of the elements are square (since squaring each nonsquare element gives a unique square). This means we can assume all squares can be written as βm for some m, and therefore finding the square root is a matter of exponentiating by 2−1(modt). In the event that p≡1(mod4) then things get more complicated because 2−1(modt) does not exist. Let's write p−1 as 2k⋅t with t odd. The case k=0 is impossible, and the case k=1 is what we already described, so consider k≥2. α generates a 2k-order multiplicative subgroup and β generates the odd t-order multiplicative subgroup. Then every element a∈Fp×​ can be written as αi⋅βj for i∈Z2k​ and j∈Zt​. If the element is a square, then there exists some b=a​ which can be written b=αi′⋅βj′ for i′∈Z2k​ and j′∈Zt​. This means that a=b2=α2i′⋅β2j′, therefore we have i≡2i′(mod2k), and j≡2j′(modt). i would have to be even in this case because otherwise it would be impossible to have i≡2i′(mod2k) for any i′. In the case that a is not a square, then i is odd, and so half of all elements are squares. In order to compute the square root, we can first raise the element a=αi⋅βj to the power t to \"kill\" the t-order component, giving at=αit(mod2)k⋅βjt(modt)=αit(mod2)k and then raise this result to the power t−1(mod2k) to undo the effect of the original exponentiation on the 2k-order component: (αitmod2k)t−1(mod2k)=αi (since t is relatively prime to 2k). This leaves bare the αi value which we can trivially handle. We can similarly kill the 2k-order component to obtain βj⋅2−1(modt), and put the values together to obtain the square root. It turns out that in the cases k=2,3 there are simpler algorithms that merge several of these exponentiations together for efficiency. For other values of k, the only known way is to manually extract i by squaring until you obtain the identity for every single bit of i. This is the essence of the Tonelli-Shanks square root algorithm and describes the general strategy. (There is another square root algorithm that uses quadratic extension fields, but it doesn't pay off in efficiency until the prime becomes quite large.) TODO: describe more recent algorithms, e.g. Bernstein's table-based method and eprint 2020/1407 .","breadcrumbs":"Background Material » Square roots","id":"78","title":"Square roots"},"79":{"body":"In the previous sections we wrote p−1=2k⋅t with t odd, and stated that an element α∈Fp×​ generated the 2k-order subgroup. For convenience, let's denote n:=2k. The elements {1,α,α2,αn−1} are known as the nth roots of unity . The primitive root of unity , ω, is an nth root of unity such that ωi=1 except when i≡0(modn). Important notes: If α is an nth root of unity, α satisfies αn−1=0. If α=1, then 1+α+α2+⋯+αn−1=0. Equivalently, the roots of unity are solutions to the equation Xn−1=(X−1)(X−α)(X−α2)⋯(X−αn−1). ω2n​+i=−ωi​ (\"Negation lemma\") . Proof: ωn=1​⟹ωn−1=0⟹(ωn/2+1)(ωn/2−1)=0.​ Since the order of ω is n, ωn/2=1. Therefore, ωn/2=−1. (ω2n​+i)2=(ωi)2​ (\"Halving lemma\") . Proof: (ω2n​+i)2=ωn+2i=ωn⋅ω2i=ω2i=(ωi)2. In other words, if we square each element in the nth roots of unity, we would get back only half the elements, {(ωni​)2}={ωn/2​} (i.e. the 2n​th roots of unity). There is a two-to-one mapping between the elements and their squares.","breadcrumbs":"Background Material » Roots of unity","id":"79","title":"Roots of unity"},"8":{"body":"When implementing a circuit, we could use the features of the cores we've selected directly via the chip. Typically, though, we will use them via gadgets . This indirection is useful because, for reasons of efficiency and limitations imposed by UPA, the core interfaces will often be dependent on low-level implementation details. The gadget interface can provide a more convenient and stable API that abstracts away from extraneous detail. For example, consider a hash function such as SHA-256. The interface of a core supporting SHA-256 might be dependent on internals of the hash function design such as the separation between message schedule and compression function. The corresponding gadget interface can provide a more convenient and familiar update/finalize API, and can also handle parts of the hash function that do not need core support, such as padding. This is similar to how accelerated instructions for cryptographic primitives on CPUs are typically accessed via software libraries, rather than directly. Gadgets can also provide modular and reusable abstractions for circuit programming at a higher level, similar to their use in libraries such as libsnark and bellman . As well as abstracting functions , they can also abstract types , such as elliptic curve points or integers of specific sizes.","breadcrumbs":"Concepts » Gadgets","id":"8","title":"Gadgets"},"80":{"body":"Friedman, R. (n.d.) \"Cyclic Groups and Elementary Number Theory II\" (p. 5). http://www.math.columbia.edu/~rf/numbertheory2.pdf","breadcrumbs":"Background Material » References","id":"80","title":"References"},"81":{"body":"Let A(X) be a polynomial over Fp​ with formal indeterminate X. As an example, A(X)=a0​+a1​X+a2​X2+a3​X3 defines a degree-3 polynomial. a0​ is referred to as the constant term. Polynomials of degree n−1 have n coefficients. Important notes: Multiplication of polynomials produces a product polynomial that is the sum of the degrees of its factors. Polynomial division subtracts from the degree. deg(A(X)B(X))=deg(A(X))+deg(B(X)), deg(A(X)/B(X))=deg(A(X))−deg(B(X)). Given a polynomial A(X) of degree n−1, if we obtain n evaluations of the polynomial at distinct points then these evaluations perfectly define the polynomial. In other words, given these evaluations we can obtain a unique polynomial A(X) of degree n−1 via polynomial interpolation. [a0​,a1​,⋯,an−1​] is the coefficient representation of the polynomial A(X). Equivalently, we could use its evaluation representation [(x0​,A(x0​)),(x1​,A(x1​)),⋯,(xn−1​,A(xn−1​))] at n distinct points. Either representation uniquely specifies the same polynomial. (aside) Horner's rule Horner's rule allows for efficient evaluation of a polynomial of degree n−1, using only n−1 multiplications and n−1 additions. It is the following identity: a0​​+a1​X+a2​X2+⋯+an−1​Xn−1=a0​+X(a1​+X(a2​+⋯+X(an−2​+Xan−1​))),​","breadcrumbs":"Background Material » Polynomials","id":"81","title":"Polynomials"},"82":{"body":"The FFT is an efficient way of converting between the coefficient and evaluation representations of a polynomial. It evaluates the polynomial at the nth roots of unity {ω0,ω1,⋯,ωn−1}, where ω is a primitive nth root of unity. By exploiting symmetries in the roots of unity, each round of the FFT reduces the evaluation into a problem only half the size. Most commonly we use polynomials of length some power of two, n=2k, and apply the halving reduction recursively.","breadcrumbs":"Background Material » Fast Fourier Transform (FFT)","id":"82","title":"Fast Fourier Transform (FFT)"},"83":{"body":"In the coefficient representation, it takes O(n2) operations to multiply two polynomials A(X)⋅B(X)=C(X): A(X)B(X)C(X)​=a0​+a1​X+a2​X2+⋯+an−1​Xn−1,=b0​+b1​X+b2​X2+⋯+bn−1​Xn−1,=a0​⋅(b0​+b1​X+b2​X2+⋯+bn−1​Xn−1)+a1​X⋅(b0​+b1​X+b2​X2+⋯+bn−1​Xn−1)+⋯+an−1​Xn−1⋅(b0​+b1​X+b2​X2+⋯+bn−1​Xn−1),​ where each of the n terms in the first polynomial has to be multiplied by the n terms of the second polynomial. In the evaluation representation, however, polynomial multiplication only requires O(n) operations: ABC​:{(x0​,A(x0​)),(x1​,A(x1​)),⋯,(xn−1​,A(xn−1​))},:{(x0​,B(x0​)),(x1​,B(x1​)),⋯,(xn−1​,B(xn−1​))},:{(x0​,A(x0​)B(x0​)),(x1​,A(x1​)B(x1​)),⋯,(xn−1​,A(xn−1​)B(xn−1​))},​ where each evaluation is multiplied pointwise. This suggests the following strategy for fast polynomial multiplication: Evaluate polynomials at all n points; Perform fast pointwise multiplication in the evaluation representation (O(n)); Convert back to the coefficient representation. The challenge now is how to evaluate and interpolate the polynomials efficiently. Naively, evaluating a polynomial at n points would require O(n2) operations (we use the O(n) Horner's rule at each point): ⎣⎢⎢⎢⎢⎢⎢⎡​A(1)A(ω)A(ω2)⋮A(ωn−1)​⎦⎥⎥⎥⎥⎥⎥⎤​=⎣⎢⎢⎢⎢⎢⎢⎡​111⋮1​1ωω2⋮ωn−1​1ω2ω2⋅2⋮ω2(n−1)​………⋯​1ωn−1ω2⋅(n−1)⋮ω(n−1)2​⎦⎥⎥⎥⎥⎥⎥⎤​⋅⎣⎢⎢⎢⎢⎢⎢⎡​a0​a1​a2​⋮an−1​​⎦⎥⎥⎥⎥⎥⎥⎤​. For convenience, we will denote the matrices above as: A^=Vω​⋅A. (A^ is known as the Discrete Fourier Transform of A; Vω​ is also called the Vandermonde matrix .)","breadcrumbs":"Background Material » Motivation: Fast polynomial multiplication","id":"83","title":"Motivation: Fast polynomial multiplication"},"84":{"body":"Our strategy is to divide a DFT of size n into two interleaved DFTs of size n/2. Given the polynomial A(X)=a0​+a1​X+a2​X2+⋯+an−1​Xn−1, we split it up into even and odd terms: Aeven​Aodd​​=a0​+a2​X+⋯+an−2​X2n​−1,=a1​+a3​X+⋯+an−1​X2n​−1.​ To recover the original polynomial, we do A(X)=Aeven​(X2)+XAodd​(X2). Trying this out on points ωni​ and ωn2n​+i​, i∈[0..2n​−1], we start to notice some symmetries: A(ωni​)A(ωn2n​+i​)​=Aeven​((ωni​)2)+ωni​Aodd​((ωni​)2),=Aeven​((ωn2n​+i​)2)+ωn2n​+i​Aodd​((ωn2n​+i​)2)=Aeven​((−ωni​)2)−ωni​Aodd​((−ωni​)2)←(negation lemma)=Aeven​((ωni​)2)−ωni​Aodd​((ωni​)2).​ Notice that we are only evaluating Aeven​(X) and Aodd​(X) over half the domain {(ωn0​)2,(ωn​)2,⋯,(ωn2n​−1​)2}={ωn/2i​},i=[0..2n​−1] (halving lemma). This gives us all the terms we need to reconstruct A(X) over the full domain {ω0,ω,⋯,ωn−1}: which means we have transformed a length-n DFT into two length-2n​ DFTs. We choose n=2k to be a power of two (by zero-padding if needed), and apply this divide-and-conquer strategy recursively. By the Master Theorem [1] , this gives us an evaluation algorithm with O(nlog2​n) operations, also known as the Fast Fourier Transform (FFT).","breadcrumbs":"Background Material » The (radix-2) Cooley-Tukey algorithm","id":"84","title":"The (radix-2) Cooley-Tukey algorithm"},"85":{"body":"So we've evaluated our polynomials and multiplied them pointwise. What remains is to convert the product from the evaluation representation back to coefficient representation. To do this, we simply call the FFT on the evaluation representation. However, this time we also: replace ωi by ω−i in the Vandermonde matrix, and multiply our final result by a factor of 1/n. In other words: A=n1​Vω−1​⋅A^. (To understand why the inverse FFT has a similar form to the FFT, refer to Slide 13-1 of [2] . The below image was also taken from [2] .)","breadcrumbs":"Background Material » Inverse FFT","id":"85","title":"Inverse FFT"},"86":{"body":"The Schwartz-Zippel lemma informally states that \"different polynomials are different at most points.\" Formally, it can be written as follows: Let p(x1​,x2​,⋯,xn​) be a nonzero polynomial of n variables with degree d. Let S be a finite set of numbers with at least d elements in it. If we choose random α1​,α1​,⋯,αn​ from S, Pr[p(α1​,α2​,⋯,αn​)=0]≤∣S∣d​. In the familiar univariate case p(X), this reduces to saying that a nonzero polynomial of degree d has at most d roots. The Schwartz-Zippel lemma is used in polynomial equality testing. Given two multi-variate polynomials p1​(x1​,⋯,xn​) and p2​(x1​,⋯,xn​) of degrees d1​,d2​ respectively, we can test if p1​(α1​,⋯,αn​)−p2​(α1​,⋯,αn​)=0 for random α1​,⋯,αn​←S, where the size of S is at least ∣S∣≥(d1​+d2​). If the two polynomials are identical, this will always be true, whereas if the two polynomials are different then the equality holds with probability at most ∣S∣max(d1​,d2​)​.","breadcrumbs":"Background Material » The Schwartz-Zippel lemma","id":"86","title":"The Schwartz-Zippel lemma"},"87":{"body":"Consider the order-n multiplicative subgroup H with primitive root of unity ω. For all ωi∈H,i∈[n−1], we have (ωi)n=(ωn)i=(ω0)i=1. In other words, every element of H fulfils the equation ZH​(X)​=Xn−1=(X−ω0)(X−ω1)(X−ω2)⋯(X−ωn−1),​ meaning every element is a root of ZH​(X). We call ZH​(X) the vanishing polynomial over H because it evaluates to zero on all elements of H. This comes in particularly handy when checking polynomial constraints. For instance, to check that A(X)+B(X)=C(X) over H, we simply have to check that A(X)+B(X)−C(X) is some multiple of ZH​(X). In other words, if dividing our constraint by the vanishing polynomial still yields some polynomial ZH​(X)A(X)+B(X)−C(X)​=H(X), we are satisfied that A(X)+B(X)−C(X)=0 over H.","breadcrumbs":"Background Material » Vanishing polynomial","id":"87","title":"Vanishing polynomial"},"88":{"body":"TODO: explain what a basis is in general (briefly). Polynomials are commonly written in the monomial basis (e.g. X,X2,...Xn). However, when working over a multiplicative subgroup of order n, we find a more natural expression in the Lagrange basis. Consider the order-n multiplicative subgroup H with primitive root of unity ω. The Lagrange basis corresponding to this subgroup is a set of functions {Li​}i=0n−1​, where Li​(ωj)={10​if i=j,otherwise.​ We can write this more compactly as Li​(ωj)=δij​, where δ is the Kronecker delta function. Now, we can write our polynomial as a linear combination of Lagrange basis functions, A(X)=i=0∑n−1​ai​Li​(X),X∈H, which is equivalent to saying that p(X) evaluates to a0​ at ω0, to a1​ at ω1, to a2​ at ω2,⋯, and so on. When working over a multiplicative subgroup, the Lagrange basis function has a convenient sparse representation of the form Li​(X)=X−ωici​⋅(Xn−1)​, where ci​ is the barycentric weight. (To understand how this form was derived, refer to [3] .) For i=0, we have c=1/n⟹L0​(X)=n1​X−1(Xn−1)​. Suppose we are given a set of evaluation points {x0​,x1​,⋯,xn−1​}. Since we cannot assume that the xi​'s form a multiplicative subgroup, we consider also the Lagrange polynomials Li​'s in the general case. Then we can construct: Li​(X)=j=i∏​xi​−xj​X−xj​​,i∈[0..n−1]. Here, every X=xj​=xi​ will produce a zero numerator term (xj​−xj​), causing the whole product to evaluate to zero. On the other hand, X=xi​ will evaluate to xi​−xj​xi​−xj​​ at every term, resulting in an overall product of one. This gives the desired Kronecker delta behaviour Li​(xj​)=δij​ on the set {x0​,x1​,⋯,xn−1​}.","breadcrumbs":"Background Material » Lagrange basis functions","id":"88","title":"Lagrange basis functions"},"89":{"body":"Given a polynomial in its evaluation representation A:{(x0​,A(x0​)),(x1​,A(x1​)),⋯,(xn−1​,A(xn−1​))}, we can reconstruct its coefficient form in the Lagrange basis: A(X)=i=0∑n−1​A(xi​)Li​(X), where X∈{x0​,x1​,⋯,x1−n​}.","breadcrumbs":"Background Material » Lagrange interpolation","id":"89","title":"Lagrange interpolation"},"9":{"body":"You're probably here because you want to write circuits? Excellent! This section will guide you through the process of creating circuits with halo2.","breadcrumbs":"User Documentation","id":"9","title":"User Documentation"},"90":{"body":"Dasgupta, S., Papadimitriou, C. H., & Vazirani, U. V. (2008). \"Algorithms\" (ch. 2). New York: McGraw-Hill Higher Education. https://people.eecs.berkeley.edu/~vazirani/algorithms/chap2.pdf http://www.cs.ust.hk/mjg_lib/Classes/COMP3711H_Fall16/lectures/FFT_Slides.pdf Berrut, J. and Trefethen, L. (2004). \"Barycentric Lagrange Interpolation.\" https://people.maths.ox.ac.uk/trefethen/barycentric.pdf","breadcrumbs":"Background Material » References","id":"90","title":"References"},"91":{"body":"In the section Inverses and groups we introduced the concept of groups . A group has an identity and a group operation. In this section we will write groups additively, i.e. the identity is O and the group operation is +. Some groups can be used as cryptographic groups . At the risk of oversimplifying, this means that the problem of finding a discrete logarithm of a group element P to a given base G, i.e. finding x such that P=[x]G, is hard in general.","breadcrumbs":"Background Material » Cryptographic groups","id":"91","title":"Cryptographic groups"},"92":{"body":"The Pedersen commitment [ P99 ] is a way to commit to a secret message in a verifiable way. It uses two random public generators G,H∈G, where G is a cryptographic group of order p. A random secret r is chosen in Zq​, and the message to commit to m is from any subset of Zq​. The commitment is c=Commit(m,r)=[m]G+[r]H. To open the commitment, the committer reveals m and r, thus allowing anyone to verify that c is indeed a commitment to m. Notice that the Pedersen commitment scheme is homomorphic: Commit(m,r)+Commit(m′,r′)​=[m]G+[r]H+[m′]G+[r′]H=[m+m′]G+[r+r′]H=Commit(m+m′,r+r′).​ Assuming the discrete log assumption holds, Pedersen commitments are also perfectly hiding and computationally binding: hiding : the adversary chooses messages m0​,m1​. The committer commits to one of these messages c=Commit(mb​;r),b∈{0,1}. Given c, the probability of the adversary guessing the correct b is no more than 21​. binding : the adversary cannot pick two different messages m0​=m1​, and randomness r0​,r1​, such that Commit(m0​,r0​)=Commit(m1​,r1​).","breadcrumbs":"Background Material » Pedersen commitment","id":"92","title":"Pedersen commitment"},"93":{"body":"We can use a variant of the Pedersen commitment scheme to commit to multiple messages at once, m=(m1​,⋯,mn​). This time, we'll have to sample a corresponding number of random public generators G=(G0​,⋯,Gn−1​), along with a single random generator H as before (for use in hiding). Then, our commitment scheme is: Commit(m;r)​=Commit((m0​,⋯,mn−1​);r)=[r]H+[m0​]G0​+⋯+[mn−1​]Gn−1​=[r]H+i=0∑n−1​[mi​]Gi​.​ TODO: is this positionally binding?","breadcrumbs":"Background Material » Vector Pedersen commitment","id":"93","title":"Vector Pedersen commitment"},"94":{"body":"An example of a protocol that uses cryptographic groups is Diffie--Hellman key agreement [ DH1976 ]. The Diffie--Hellman protocol is a method for two users, Alice and Bob, to generate a shared private key. It proceeds as follows: Alice and Bob publicly agree on two prime numbers, p and G, where p is large and G is a primitive root (modp). (Note that g is a generator of the group Fp×​.) Alice chooses a large random number a as her private key. She computes her public key A=[a]G(modp), and sends A to Bob. Similarly, Bob chooses a large random number b as his private key. He computes his public key B=[b]G(modp), and sends B to Alice. Now both Alice and Bob compute their shared key K=[ab]G(modp), which Alice computes as K=[a]B(modp)=[a]([b]G)(modp), and Bob computes as K=[b]A(modp)=[b]([a]G)(modp). A potential eavesdropper would need to derive K=[ab]g(modp) knowing only g,p,A=[a]G, and B=[b]G: in other words, they would need to either get the discrete logarithm a from A=[a]G or b from B=[b]G, which we assume to be computationally infeasible in Fp×​. More generally, protocols that use similar ideas to Diffie--Hellman are used throughout cryptography. One way of instantiating a cryptographic group is as an elliptic curve . Before we go into detail on elliptic curves, we'll describe some algorithms that can be used for any group.","breadcrumbs":"Background Material » Diffie--Hellman","id":"94","title":"Diffie--Hellman"},"95":{"body":"","breadcrumbs":"Background Material » Multiscalar multiplication","id":"95","title":"Multiscalar multiplication"},"96":{"body":"Reference: https://jbootle.github.io/Misc/pippenger.pdf","breadcrumbs":"Background Material » TODO: Pippenger's algorithm","id":"96","title":"TODO: Pippenger's algorithm"},"97":{"body":"Elliptic curves constructed over finite fields are another important cryptographic tool. We use elliptic curves because they provide a cryptographic group , i.e. a group in which the discrete logarithm problem (discussed below) is hard. There are several ways to define the curve equation, but for our purposes, let Fp​ be a large (255-bit) field, and then let the set of solutions (x,y) to y2=x3+b for some constant b define the Fp​-rational points on an elliptic curve E(Fp​). These (x,y) coordinates are called \"affine coordinates\". Each of the Fp​-rational points, together with a \"point at infinity\" O that serves as the group identity, can be interpreted as an element of a group. By convention, elliptic curve groups are written additively. \"Three points on a line sum to zero, which is the point at infinity.\" The group addition law is simple: to add two points together, find the line that intersects both points and obtain the third point, and then negate its y-coordinate. The case that a point is being added to itself, called point doubling, requires special handling: we find the line tangent to the point, and then find the single other point that intersects this line and then negate. Otherwise, in the event that a point is being \"added\" to its negation, the result is the point at infinity. The ability to add and double points naturally gives us a way to scale them by integers, called scalars . The number of points on the curve is the group order. If this number is a prime q, then the scalars can be considered as elements of a scalar field , Fq​. Elliptic curves, when properly designed, have an important security property. Given two random elements G,H∈E(Fp​) finding a such that [a]G=H, otherwise known as the discrete log of H with respect to G, is considered computationally infeasible with classical computers. This is called the elliptic curve discrete log assumption. If an elliptic curve group G has prime order q (like the ones used in Halo 2), then it is a finite cyclic group. Recall from the section on groups that this implies it is isomorphic to Z/qZ, or equivalently, to the scalar field Fq​. Each possible generator G fixes the isomorphism; then an element on the scalar side is precisely the discrete log of the corresponding group element with respect to G. In the case of a cryptographically secure elliptic curve, the isomorphism is hard to compute in the G→Fq​ direction because the elliptic curve discrete log problem is hard. It is sometimes helpful to make use of this isomorphism by thinking of group-based cryptographic protocols and algorithms in terms of the scalars instead of in terms of the group elements. This can make proofs and notation simpler. For instance, it has become common in papers on proof systems to use the notation [x] to denote a group element with discrete log x, where the generator is implicit. We also used this idea in the \" distinct-x theorem \", in order to prove correctness of optimizations for elliptic curve scalar multiplication in Sapling, and an endomorphism-based optimization in Appendix C of the original Halo paper .","breadcrumbs":"Background Material » Elliptic curves","id":"97","title":"Elliptic curves"},"98":{"body":"","breadcrumbs":"Background Material » Curve arithmetic","id":"98","title":"Curve arithmetic"},"99":{"body":"The simplest situation is doubling a point (x0​,y0​). Continuing with our example y2=x3+b, this is done first by computing the derivative λ=dxdy​=2y3x2​. 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