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prove.go
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prove.go
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// Copyright 2020 ConsenSys Software Inc.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
// Code generated by gnark DO NOT EDIT
package plonk
import (
"crypto/sha256"
"math/big"
"runtime"
"sync"
"time"
"github.com/consensys/gnark/backend/witness"
"github.com/consensys/gnark-crypto/ecc/bn254/fr"
curve "github.com/consensys/gnark-crypto/ecc/bn254"
"github.com/consensys/gnark-crypto/ecc/bn254/fr/kzg"
"github.com/consensys/gnark-crypto/ecc/bn254/fr/fft"
"github.com/consensys/gnark-crypto/ecc/bn254/fr/iop"
cs "github.com/consensys/gnark/constraint/bn254"
"github.com/consensys/gnark-crypto/fiat-shamir"
"github.com/consensys/gnark/backend"
"github.com/consensys/gnark/constraint"
"github.com/consensys/gnark/constraint/solver"
"github.com/consensys/gnark/internal/utils"
"github.com/consensys/gnark/logger"
)
type Proof struct {
// Commitments to the solution vectors
LRO [3]kzg.Digest
// Commitment to Z, the permutation polynomial
Z kzg.Digest
// Commitments to h1, h2, h3 such that h = h1 + Xh2 + X**2h3 is the quotient polynomial
H [3]kzg.Digest
Bsb22Commitments []kzg.Digest
// Batch opening proof of h1 + zeta*h2 + zeta**2h3, linearizedPolynomial, l, r, o, s1, s2, qCPrime
BatchedProof kzg.BatchOpeningProof
// Opening proof of Z at zeta*mu
ZShiftedOpening kzg.OpeningProof
}
// Computing and verifying Bsb22 multi-commits explained in https://hackmd.io/x8KsadW3RRyX7YTCFJIkHg
func bsb22ComputeCommitmentHint(spr *cs.SparseR1CS, pk *ProvingKey, proof *Proof, cCommitments []*iop.Polynomial, res *fr.Element, commDepth int) solver.Hint {
return func(_ *big.Int, ins, outs []*big.Int) error {
commitmentInfo := spr.CommitmentInfo.(constraint.PlonkCommitments)[commDepth]
committedValues := make([]fr.Element, pk.Domain[0].Cardinality)
offset := spr.GetNbPublicVariables()
for i := range ins {
committedValues[offset+commitmentInfo.Committed[i]].SetBigInt(ins[i])
}
var (
err error
hashRes []fr.Element
)
if _, err = committedValues[offset+commitmentInfo.CommitmentIndex].SetRandom(); err != nil { // Commitment injection constraint has qcp = 0. Safe to use for blinding.
return err
}
if _, err = committedValues[offset+spr.GetNbConstraints()-1].SetRandom(); err != nil { // Last constraint has qcp = 0. Safe to use for blinding
return err
}
pi2iop := iop.NewPolynomial(&committedValues, iop.Form{Basis: iop.Lagrange, Layout: iop.Regular})
cCommitments[commDepth] = pi2iop.ShallowClone()
cCommitments[commDepth].ToCanonical(&pk.Domain[0]).ToRegular()
if proof.Bsb22Commitments[commDepth], err = kzg.Commit(cCommitments[commDepth].Coefficients(), pk.Kzg); err != nil {
return err
}
if hashRes, err = fr.Hash(proof.Bsb22Commitments[commDepth].Marshal(), []byte("BSB22-Plonk"), 1); err != nil {
return err
}
res.Set(&hashRes[0]) // TODO @Tabaie use CommitmentIndex for this; create a new variable CommitmentConstraintIndex for other uses
res.BigInt(outs[0])
return nil
}
}
func Prove(spr *cs.SparseR1CS, pk *ProvingKey, fullWitness witness.Witness, opts ...backend.ProverOption) (*Proof, error) {
log := logger.Logger().With().Str("curve", spr.CurveID().String()).Int("nbConstraints", spr.GetNbConstraints()).Str("backend", "plonk").Logger()
opt, err := backend.NewProverConfig(opts...)
if err != nil {
return nil, err
}
start := time.Now()
// pick a hash function that will be used to derive the challenges
hFunc := sha256.New()
// create a transcript manager to apply Fiat Shamir
fs := fiatshamir.NewTranscript(hFunc, "gamma", "beta", "alpha", "zeta")
// result
proof := &Proof{}
commitmentInfo := spr.CommitmentInfo.(constraint.PlonkCommitments)
commitmentVal := make([]fr.Element, len(commitmentInfo)) // TODO @Tabaie get rid of this
cCommitments := make([]*iop.Polynomial, len(commitmentInfo))
proof.Bsb22Commitments = make([]kzg.Digest, len(commitmentInfo))
for i := range commitmentInfo {
opt.SolverOpts = append(opt.SolverOpts, solver.OverrideHint(commitmentInfo[i].HintID,
bsb22ComputeCommitmentHint(spr, pk, proof, cCommitments, &commitmentVal[i], i)))
}
if spr.GkrInfo.Is() {
var gkrData cs.GkrSolvingData
opt.SolverOpts = append(opt.SolverOpts,
solver.OverrideHint(spr.GkrInfo.SolveHintID, cs.GkrSolveHint(spr.GkrInfo, &gkrData)),
solver.OverrideHint(spr.GkrInfo.ProveHintID, cs.GkrProveHint(spr.GkrInfo.HashName, &gkrData)))
}
// query l, r, o in Lagrange basis, not blinded
_solution, err := spr.Solve(fullWitness, opt.SolverOpts...)
if err != nil {
return nil, err
}
// TODO @gbotrel deal with that conversion lazily
lcCommitments := make([]*iop.Polynomial, len(cCommitments))
for i := range cCommitments {
lcCommitments[i] = cCommitments[i].Clone(int(pk.Domain[1].Cardinality)).ToLagrangeCoset(&pk.Domain[1]) // lagrange coset form
}
solution := _solution.(*cs.SparseR1CSSolution)
evaluationLDomainSmall := []fr.Element(solution.L)
evaluationRDomainSmall := []fr.Element(solution.R)
evaluationODomainSmall := []fr.Element(solution.O)
lagReg := iop.Form{Basis: iop.Lagrange, Layout: iop.Regular}
// l, r, o and blinded versions
var (
wliop,
wriop,
woiop,
bwliop,
bwriop,
bwoiop *iop.Polynomial
)
var wgLRO sync.WaitGroup
wgLRO.Add(3)
go func() {
// we keep in lagrange regular form since iop.BuildRatioCopyConstraint prefers it in this form.
wliop = iop.NewPolynomial(&evaluationLDomainSmall, lagReg)
// we set the underlying slice capacity to domain[1].Cardinality to minimize mem moves.
bwliop = wliop.Clone(int(pk.Domain[1].Cardinality)).ToCanonical(&pk.Domain[0]).ToRegular().Blind(1)
wgLRO.Done()
}()
go func() {
wriop = iop.NewPolynomial(&evaluationRDomainSmall, lagReg)
bwriop = wriop.Clone(int(pk.Domain[1].Cardinality)).ToCanonical(&pk.Domain[0]).ToRegular().Blind(1)
wgLRO.Done()
}()
go func() {
woiop = iop.NewPolynomial(&evaluationODomainSmall, lagReg)
bwoiop = woiop.Clone(int(pk.Domain[1].Cardinality)).ToCanonical(&pk.Domain[0]).ToRegular().Blind(1)
wgLRO.Done()
}()
fw, ok := fullWitness.Vector().(fr.Vector)
if !ok {
return nil, witness.ErrInvalidWitness
}
// start computing lcqk
var lcqk *iop.Polynomial
chLcqk := make(chan struct{}, 1)
go func() {
// compute qk in canonical basis, completed with the public inputs
// We copy the coeffs of qk to pk is not mutated
lqkcoef := pk.lQk.Coefficients()
qkCompletedCanonical := make([]fr.Element, len(lqkcoef))
copy(qkCompletedCanonical, fw[:len(spr.Public)])
copy(qkCompletedCanonical[len(spr.Public):], lqkcoef[len(spr.Public):])
for i := range commitmentInfo {
qkCompletedCanonical[spr.GetNbPublicVariables()+commitmentInfo[i].CommitmentIndex] = commitmentVal[i]
}
pk.Domain[0].FFTInverse(qkCompletedCanonical, fft.DIF)
fft.BitReverse(qkCompletedCanonical)
canReg := iop.Form{Basis: iop.Canonical, Layout: iop.Regular}
lcqk = iop.NewPolynomial(&qkCompletedCanonical, canReg)
lcqk.ToLagrangeCoset(&pk.Domain[1])
close(chLcqk)
}()
// The first challenge is derived using the public data: the commitments to the permutation,
// the coefficients of the circuit, and the public inputs.
// derive gamma from the Comm(blinded cl), Comm(blinded cr), Comm(blinded co)
if err := bindPublicData(&fs, "gamma", *pk.Vk, fw[:len(spr.Public)], proof.Bsb22Commitments); err != nil {
return nil, err
}
// wait for polys to be blinded
wgLRO.Wait()
if err := commitToLRO(bwliop.Coefficients(), bwriop.Coefficients(), bwoiop.Coefficients(), proof, pk.Kzg); err != nil {
return nil, err
}
gamma, err := deriveRandomness(&fs, "gamma", &proof.LRO[0], &proof.LRO[1], &proof.LRO[2]) // TODO @Tabaie @ThomasPiellard add BSB commitment here?
if err != nil {
return nil, err
}
// Fiat Shamir this
bbeta, err := fs.ComputeChallenge("beta")
if err != nil {
return nil, err
}
var beta fr.Element
beta.SetBytes(bbeta)
// l, r, o are already blinded
wgLRO.Add(3)
go func() {
bwliop.ToLagrangeCoset(&pk.Domain[1])
wgLRO.Done()
}()
go func() {
bwriop.ToLagrangeCoset(&pk.Domain[1])
wgLRO.Done()
}()
go func() {
bwoiop.ToLagrangeCoset(&pk.Domain[1])
wgLRO.Done()
}()
// compute the copy constraint's ratio
// note that wliop, wriop and woiop are fft'ed (mutated) in the process.
ziop, err := iop.BuildRatioCopyConstraint(
[]*iop.Polynomial{
wliop,
wriop,
woiop,
},
pk.trace.S,
beta,
gamma,
iop.Form{Basis: iop.Canonical, Layout: iop.Regular},
&pk.Domain[0],
)
if err != nil {
return proof, err
}
// commit to the blinded version of z
chZ := make(chan error, 1)
var bwziop, bwsziop *iop.Polynomial
var alpha fr.Element
go func() {
bwziop = ziop // iop.NewWrappedPolynomial(&ziop)
bwziop.Blind(2)
proof.Z, err = kzg.Commit(bwziop.Coefficients(), pk.Kzg, runtime.NumCPU()*2)
if err != nil {
chZ <- err
}
// derive alpha from the Comm(l), Comm(r), Comm(o), Com(Z)
alpha, err = deriveRandomness(&fs, "alpha", &proof.Z)
if err != nil {
chZ <- err
}
// Store z(g*x), without reallocating a slice
bwsziop = bwziop.ShallowClone().Shift(1)
bwsziop.ToLagrangeCoset(&pk.Domain[1])
chZ <- nil
close(chZ)
}()
// Full capture using latest gnark crypto...
fic := func(fql, fqr, fqm, fqo, fqk, l, r, o fr.Element, pi2QcPrime []fr.Element) fr.Element { // TODO @Tabaie make use of the fact that qCPrime is a selector: sparse and binary
var ic, tmp fr.Element
ic.Mul(&fql, &l)
tmp.Mul(&fqr, &r)
ic.Add(&ic, &tmp)
tmp.Mul(&fqm, &l).Mul(&tmp, &r)
ic.Add(&ic, &tmp)
tmp.Mul(&fqo, &o)
ic.Add(&ic, &tmp).Add(&ic, &fqk)
nbComms := len(commitmentInfo)
for i := range commitmentInfo {
tmp.Mul(&pi2QcPrime[i], &pi2QcPrime[i+nbComms])
ic.Add(&ic, &tmp)
}
return ic
}
fo := func(l, r, o, fid, fs1, fs2, fs3, fz, fzs fr.Element) fr.Element {
u := &pk.Domain[0].FrMultiplicativeGen
var a, b, tmp fr.Element
b.Mul(&beta, &fid)
a.Add(&b, &l).Add(&a, &gamma)
b.Mul(&b, u)
tmp.Add(&b, &r).Add(&tmp, &gamma)
a.Mul(&a, &tmp)
tmp.Mul(&b, u).Add(&tmp, &o).Add(&tmp, &gamma)
a.Mul(&a, &tmp).Mul(&a, &fz)
b.Mul(&beta, &fs1).Add(&b, &l).Add(&b, &gamma)
tmp.Mul(&beta, &fs2).Add(&tmp, &r).Add(&tmp, &gamma)
b.Mul(&b, &tmp)
tmp.Mul(&beta, &fs3).Add(&tmp, &o).Add(&tmp, &gamma)
b.Mul(&b, &tmp).Mul(&b, &fzs)
b.Sub(&b, &a)
return b
}
fone := func(fz, flone fr.Element) fr.Element {
one := fr.One()
one.Sub(&fz, &one).Mul(&one, &flone)
return one
}
// 0 , 1 , 2, 3 , 4 , 5 , 6 , 7, 8 , 9 , 10, 11, 12, 13, 14, 15:15+nbComm , 15+nbComm:15+2×nbComm
// l , r , o, id, s1, s2, s3, z, zs, ql, qr, qm, qo, qk ,lone, Bsb22Commitments, qCPrime
fm := func(x ...fr.Element) fr.Element {
a := fic(x[9], x[10], x[11], x[12], x[13], x[0], x[1], x[2], x[15:])
b := fo(x[0], x[1], x[2], x[3], x[4], x[5], x[6], x[7], x[8])
c := fone(x[7], x[14])
c.Mul(&c, &alpha).Add(&c, &b).Mul(&c, &alpha).Add(&c, &a)
return c
}
// wait for lcqk
<-chLcqk
// wait for Z part
if err := <-chZ; err != nil {
return proof, err
}
// wait for l, r o lagrange coset conversion
wgLRO.Wait()
toEval := []*iop.Polynomial{
bwliop,
bwriop,
bwoiop,
pk.lcIdIOP,
pk.lcS1,
pk.lcS2,
pk.lcS3,
bwziop,
bwsziop,
pk.lcQl,
pk.lcQr,
pk.lcQm,
pk.lcQo,
lcqk,
pk.lLoneIOP,
}
toEval = append(toEval, lcCommitments...) // TODO: Add this at beginning
toEval = append(toEval, pk.lcQcp...)
systemEvaluation, err := iop.Evaluate(fm, iop.Form{Basis: iop.LagrangeCoset, Layout: iop.BitReverse}, toEval...)
if err != nil {
return nil, err
}
// open blinded Z at zeta*z
chbwzIOP := make(chan struct{}, 1)
go func() {
bwziop.ToCanonical(&pk.Domain[1]).ToRegular()
close(chbwzIOP)
}()
h, err := iop.DivideByXMinusOne(systemEvaluation, [2]*fft.Domain{&pk.Domain[0], &pk.Domain[1]}) // TODO Rename to DivideByXNMinusOne or DivideByVanishingPoly etc
if err != nil {
return nil, err
}
// compute kzg commitments of h1, h2 and h3
if err := commitToQuotient(
h.Coefficients()[:pk.Domain[0].Cardinality+2],
h.Coefficients()[pk.Domain[0].Cardinality+2:2*(pk.Domain[0].Cardinality+2)],
h.Coefficients()[2*(pk.Domain[0].Cardinality+2):3*(pk.Domain[0].Cardinality+2)],
proof, pk.Kzg); err != nil {
return nil, err
}
// derive zeta
zeta, err := deriveRandomness(&fs, "zeta", &proof.H[0], &proof.H[1], &proof.H[2])
if err != nil {
return nil, err
}
// compute evaluations of (blinded version of) l, r, o, z, qCPrime at zeta
var blzeta, brzeta, bozeta fr.Element
qcpzeta := make([]fr.Element, len(commitmentInfo))
var wgEvals sync.WaitGroup
wgEvals.Add(3)
evalAtZeta := func(poly *iop.Polynomial, res *fr.Element) {
poly.ToCanonical(&pk.Domain[1]).ToRegular()
*res = poly.Evaluate(zeta)
wgEvals.Done()
}
go evalAtZeta(bwliop, &blzeta)
go evalAtZeta(bwriop, &brzeta)
go evalAtZeta(bwoiop, &bozeta)
evalQcpAtZeta := func(begin, end int) {
for i := begin; i < end; i++ {
qcpzeta[i] = pk.trace.Qcp[i].Evaluate(zeta)
}
}
utils.Parallelize(len(commitmentInfo), evalQcpAtZeta)
var zetaShifted fr.Element
zetaShifted.Mul(&zeta, &pk.Vk.Generator)
<-chbwzIOP
proof.ZShiftedOpening, err = kzg.Open(
bwziop.Coefficients()[:bwziop.BlindedSize()],
zetaShifted,
pk.Kzg,
)
if err != nil {
return nil, err
}
// start to compute foldedH and foldedHDigest while computeLinearizedPolynomial runs.
computeFoldedH := make(chan struct{}, 1)
var foldedH []fr.Element
var foldedHDigest kzg.Digest
go func() {
// foldedHDigest = Comm(h1) + ζᵐ⁺²*Comm(h2) + ζ²⁽ᵐ⁺²⁾*Comm(h3)
var bZetaPowerm, bSize big.Int
bSize.SetUint64(pk.Domain[0].Cardinality + 2) // +2 because of the masking (h of degree 3(n+2)-1)
var zetaPowerm fr.Element
zetaPowerm.Exp(zeta, &bSize)
zetaPowerm.BigInt(&bZetaPowerm)
foldedHDigest = proof.H[2]
foldedHDigest.ScalarMultiplication(&foldedHDigest, &bZetaPowerm)
foldedHDigest.Add(&foldedHDigest, &proof.H[1]) // ζᵐ⁺²*Comm(h3)
foldedHDigest.ScalarMultiplication(&foldedHDigest, &bZetaPowerm) // ζ²⁽ᵐ⁺²⁾*Comm(h3) + ζᵐ⁺²*Comm(h2)
foldedHDigest.Add(&foldedHDigest, &proof.H[0]) // ζ²⁽ᵐ⁺²⁾*Comm(h3) + ζᵐ⁺²*Comm(h2) + Comm(h1)
// foldedH = h1 + ζ*h2 + ζ²*h3
foldedH = h.Coefficients()[2*(pk.Domain[0].Cardinality+2) : 3*(pk.Domain[0].Cardinality+2)]
h2 := h.Coefficients()[pk.Domain[0].Cardinality+2 : 2*(pk.Domain[0].Cardinality+2)]
h1 := h.Coefficients()[:pk.Domain[0].Cardinality+2]
utils.Parallelize(len(foldedH), func(start, end int) {
for i := start; i < end; i++ {
foldedH[i].Mul(&foldedH[i], &zetaPowerm) // ζᵐ⁺²*h3
foldedH[i].Add(&foldedH[i], &h2[i]) // ζ^{m+2)*h3+h2
foldedH[i].Mul(&foldedH[i], &zetaPowerm) // ζ²⁽ᵐ⁺²⁾*h3+h2*ζᵐ⁺²
foldedH[i].Add(&foldedH[i], &h1[i]) // ζ^{2(m+2)*h3+ζᵐ⁺²*h2 + h1
}
})
close(computeFoldedH)
}()
wgEvals.Wait() // wait for the evaluations
var (
linearizedPolynomialCanonical []fr.Element
linearizedPolynomialDigest curve.G1Affine
errLPoly error
)
// blinded z evaluated at u*zeta
bzuzeta := proof.ZShiftedOpening.ClaimedValue
// compute the linearization polynomial r at zeta
// (goal: save committing separately to z, ql, qr, qm, qo, k
// note: we linearizedPolynomialCanonical reuses bwziop memory
linearizedPolynomialCanonical = computeLinearizedPolynomial(
blzeta,
brzeta,
bozeta,
alpha,
beta,
gamma,
zeta,
bzuzeta,
qcpzeta,
bwziop.Coefficients()[:bwziop.BlindedSize()],
coefficients(cCommitments),
pk,
)
// TODO this commitment is only necessary to derive the challenge, we should
// be able to avoid doing it and get the challenge in another way
linearizedPolynomialDigest, errLPoly = kzg.Commit(linearizedPolynomialCanonical, pk.Kzg, runtime.NumCPU()*2)
if errLPoly != nil {
return nil, errLPoly
}
// wait for foldedH and foldedHDigest
<-computeFoldedH
// Batch open the first list of polynomials
polysQcp := coefficients(pk.trace.Qcp)
polysToOpen := make([][]fr.Element, 7+len(polysQcp))
copy(polysToOpen[7:], polysQcp)
// offset := len(polysQcp)
polysToOpen[0] = foldedH
polysToOpen[1] = linearizedPolynomialCanonical
polysToOpen[2] = bwliop.Coefficients()[:bwliop.BlindedSize()]
polysToOpen[3] = bwriop.Coefficients()[:bwriop.BlindedSize()]
polysToOpen[4] = bwoiop.Coefficients()[:bwoiop.BlindedSize()]
polysToOpen[5] = pk.trace.S1.Coefficients()
polysToOpen[6] = pk.trace.S2.Coefficients()
digestsToOpen := make([]curve.G1Affine, len(pk.Vk.Qcp)+7)
copy(digestsToOpen[7:], pk.Vk.Qcp)
// offset = len(pk.Vk.Qcp)
digestsToOpen[0] = foldedHDigest
digestsToOpen[1] = linearizedPolynomialDigest
digestsToOpen[2] = proof.LRO[0]
digestsToOpen[3] = proof.LRO[1]
digestsToOpen[4] = proof.LRO[2]
digestsToOpen[5] = pk.Vk.S[0]
digestsToOpen[6] = pk.Vk.S[1]
proof.BatchedProof, err = kzg.BatchOpenSinglePoint(
polysToOpen,
digestsToOpen,
zeta,
hFunc,
pk.Kzg,
)
log.Debug().Dur("took", time.Since(start)).Msg("prover done")
if err != nil {
return nil, err
}
return proof, nil
}
func coefficients(p []*iop.Polynomial) [][]fr.Element {
res := make([][]fr.Element, len(p))
for i, pI := range p {
res[i] = pI.Coefficients()
}
return res
}
// fills proof.LRO with kzg commits of bcl, bcr and bco
func commitToLRO(bcl, bcr, bco []fr.Element, proof *Proof, kzgPk kzg.ProvingKey) error {
n := runtime.NumCPU()
var err0, err1, err2 error
chCommit0 := make(chan struct{}, 1)
chCommit1 := make(chan struct{}, 1)
go func() {
proof.LRO[0], err0 = kzg.Commit(bcl, kzgPk, n)
close(chCommit0)
}()
go func() {
proof.LRO[1], err1 = kzg.Commit(bcr, kzgPk, n)
close(chCommit1)
}()
if proof.LRO[2], err2 = kzg.Commit(bco, kzgPk, n); err2 != nil {
return err2
}
<-chCommit0
<-chCommit1
if err0 != nil {
return err0
}
return err1
}
func commitToQuotient(h1, h2, h3 []fr.Element, proof *Proof, kzgPk kzg.ProvingKey) error {
n := runtime.NumCPU()
var err0, err1, err2 error
chCommit0 := make(chan struct{}, 1)
chCommit1 := make(chan struct{}, 1)
go func() {
proof.H[0], err0 = kzg.Commit(h1, kzgPk, n)
close(chCommit0)
}()
go func() {
proof.H[1], err1 = kzg.Commit(h2, kzgPk, n)
close(chCommit1)
}()
if proof.H[2], err2 = kzg.Commit(h3, kzgPk, n); err2 != nil {
return err2
}
<-chCommit0
<-chCommit1
if err0 != nil {
return err0
}
return err1
}
// computeLinearizedPolynomial computes the linearized polynomial in canonical basis.
// The purpose is to commit and open all in one ql, qr, qm, qo, qk.
// * lZeta, rZeta, oZeta are the evaluation of l, r, o at zeta
// * z is the permutation polynomial, zu is Z(μX), the shifted version of Z
// * pk is the proving key: the linearized polynomial is a linear combination of ql, qr, qm, qo, qk.
//
// The Linearized polynomial is:
//
// α²*L₁(ζ)*Z(X)
// + α*( (l(ζ)+β*s1(ζ)+γ)*(r(ζ)+β*s2(ζ)+γ)*Z(μζ)*s3(X) - Z(X)*(l(ζ)+β*id1(ζ)+γ)*(r(ζ)+β*id2(ζ)+γ)*(o(ζ)+β*id3(ζ)+γ))
// + l(ζ)*Ql(X) + l(ζ)r(ζ)*Qm(X) + r(ζ)*Qr(X) + o(ζ)*Qo(X) + Qk(X)
func computeLinearizedPolynomial(lZeta, rZeta, oZeta, alpha, beta, gamma, zeta, zu fr.Element, qcpZeta, blindedZCanonical []fr.Element, pi2Canonical [][]fr.Element, pk *ProvingKey) []fr.Element {
// first part: individual constraints
var rl fr.Element
rl.Mul(&rZeta, &lZeta)
// second part:
// Z(μζ)(l(ζ)+β*s1(ζ)+γ)*(r(ζ)+β*s2(ζ)+γ)*β*s3(X)-Z(X)(l(ζ)+β*id1(ζ)+γ)*(r(ζ)+β*id2(ζ)+γ)*(o(ζ)+β*id3(ζ)+γ)
var s1, s2 fr.Element
chS1 := make(chan struct{}, 1)
go func() {
s1 = pk.trace.S1.Evaluate(zeta) // s1(ζ)
s1.Mul(&s1, &beta).Add(&s1, &lZeta).Add(&s1, &gamma) // (l(ζ)+β*s1(ζ)+γ)
close(chS1)
}()
// ps2 := iop.NewPolynomial(&pk.S2Canonical, iop.Form{Basis: iop.Canonical, Layout: iop.Regular})
tmp := pk.trace.S2.Evaluate(zeta) // s2(ζ)
tmp.Mul(&tmp, &beta).Add(&tmp, &rZeta).Add(&tmp, &gamma) // (r(ζ)+β*s2(ζ)+γ)
<-chS1
s1.Mul(&s1, &tmp).Mul(&s1, &zu).Mul(&s1, &beta) // (l(ζ)+β*s1(β)+γ)*(r(ζ)+β*s2(β)+γ)*β*Z(μζ)
var uzeta, uuzeta fr.Element
uzeta.Mul(&zeta, &pk.Vk.CosetShift)
uuzeta.Mul(&uzeta, &pk.Vk.CosetShift)
s2.Mul(&beta, &zeta).Add(&s2, &lZeta).Add(&s2, &gamma) // (l(ζ)+β*ζ+γ)
tmp.Mul(&beta, &uzeta).Add(&tmp, &rZeta).Add(&tmp, &gamma) // (r(ζ)+β*u*ζ+γ)
s2.Mul(&s2, &tmp) // (l(ζ)+β*ζ+γ)*(r(ζ)+β*u*ζ+γ)
tmp.Mul(&beta, &uuzeta).Add(&tmp, &oZeta).Add(&tmp, &gamma) // (o(ζ)+β*u²*ζ+γ)
s2.Mul(&s2, &tmp) // (l(ζ)+β*ζ+γ)*(r(ζ)+β*u*ζ+γ)*(o(ζ)+β*u²*ζ+γ)
s2.Neg(&s2) // -(l(ζ)+β*ζ+γ)*(r(ζ)+β*u*ζ+γ)*(o(ζ)+β*u²*ζ+γ)
// third part L₁(ζ)*α²*Z
var lagrangeZeta, one, den, frNbElmt fr.Element
one.SetOne()
nbElmt := int64(pk.Domain[0].Cardinality)
lagrangeZeta.Set(&zeta).
Exp(lagrangeZeta, big.NewInt(nbElmt)).
Sub(&lagrangeZeta, &one)
frNbElmt.SetUint64(uint64(nbElmt))
den.Sub(&zeta, &one).
Inverse(&den)
lagrangeZeta.Mul(&lagrangeZeta, &den). // L₁ = (ζⁿ⁻¹)/(ζ-1)
Mul(&lagrangeZeta, &alpha).
Mul(&lagrangeZeta, &alpha).
Mul(&lagrangeZeta, &pk.Domain[0].CardinalityInv) // (1/n)*α²*L₁(ζ)
s3canonical := pk.trace.S3.Coefficients()
utils.Parallelize(len(blindedZCanonical), func(start, end int) {
var t, t0, t1 fr.Element
for i := start; i < end; i++ {
t.Mul(&blindedZCanonical[i], &s2) // -Z(X)*(l(ζ)+β*ζ+γ)*(r(ζ)+β*u*ζ+γ)*(o(ζ)+β*u²*ζ+γ)
if i < len(s3canonical) {
t0.Mul(&s3canonical[i], &s1) // (l(ζ)+β*s1(ζ)+γ)*(r(ζ)+β*s2(ζ)+γ)*Z(μζ)*β*s3(X)
t.Add(&t, &t0)
}
t.Mul(&t, &alpha) // α*( (l(ζ)+β*s1(ζ)+γ)*(r(ζ)+β*s2(ζ)+γ)*Z(μζ)*s3(X) - Z(X)*(l(ζ)+β*ζ+γ)*(r(ζ)+β*u*ζ+γ)*(o(ζ)+β*u²*ζ+γ))
cql := pk.trace.Ql.Coefficients()
cqr := pk.trace.Qr.Coefficients()
cqm := pk.trace.Qm.Coefficients()
cqo := pk.trace.Qo.Coefficients()
cqk := pk.trace.Qk.Coefficients()
if i < len(cqm) {
t1.Mul(&cqm[i], &rl) // linPol = linPol + l(ζ)r(ζ)*Qm(X)
t0.Mul(&cql[i], &lZeta)
t0.Add(&t0, &t1)
t.Add(&t, &t0) // linPol = linPol + l(ζ)*Ql(X)
t0.Mul(&cqr[i], &rZeta)
t.Add(&t, &t0) // linPol = linPol + r(ζ)*Qr(X)
t0.Mul(&cqo[i], &oZeta).Add(&t0, &cqk[i])
t.Add(&t, &t0) // linPol = linPol + o(ζ)*Qo(X) + Qk(X)
for j := range qcpZeta {
t0.Mul(&pi2Canonical[j][i], &qcpZeta[j])
t.Add(&t, &t0)
}
}
t0.Mul(&blindedZCanonical[i], &lagrangeZeta)
blindedZCanonical[i].Add(&t, &t0) // finish the computation
}
})
return blindedZCanonical
}