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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 plonkfri
import (
"math/big"
"math/bits"
"runtime"
"github.com/consensys/gnark/backend/witness"
"github.com/consensys/gnark-crypto/ecc/bls12-377/fr"
"github.com/consensys/gnark-crypto/ecc/bls12-377/fr/fft"
cs "github.com/consensys/gnark/constraint/bls12-377"
"github.com/consensys/gnark-crypto/ecc/bls12-377/fr/fri"
fiatshamir "github.com/consensys/gnark-crypto/fiat-shamir"
"github.com/consensys/gnark/backend"
"github.com/consensys/gnark/internal/utils"
)
type Proof struct {
// commitments to the solution vectors
LROpp [3]fri.ProofOfProximity
// commitment to Z (permutation polynomial)
// Z Commitment
Zpp fri.ProofOfProximity
// commitment to h1,h2,h3 such that h = h1 + X**n*h2 + X**2nh3 the quotient polynomial
Hpp [3]fri.ProofOfProximity
// opening proofs for L, R, O
OpeningsLROmp [3]fri.OpeningProof
// opening proofs for Z, Zu
OpeningsZmp [2]fri.OpeningProof
// opening proof for H
OpeningsHmp [3]fri.OpeningProof
// opening proofs for ql, qr, qm, qo, qk
OpeningsQlQrQmQoQkincompletemp [5]fri.OpeningProof
// openings of S1, S2, S3
// OpeningsS1S2S3 [3]OpeningProof
OpeningsS1S2S3mp [3]fri.OpeningProof
// openings of Id1, Id2, Id3
OpeningsId1Id2Id3mp [3]fri.OpeningProof
}
func Prove(spr *cs.SparseR1CS, pk *ProvingKey, fullWitness witness.Witness, opts ...backend.ProverOption) (*Proof, error) {
opt, err := backend.NewProverConfig(opts...)
if err != nil {
return nil, err
}
var proof Proof
// 0 - Fiat Shamir
fs := fiatshamir.NewTranscript(opt.ChallengeHash, "gamma", "beta", "alpha", "zeta")
// 1 - solve the system
_solution, err := spr.Solve(fullWitness, opt.SolverOpts...)
if err != nil {
return nil, err
}
solution := _solution.(*cs.SparseR1CSSolution)
evaluationLDomainSmall := []fr.Element(solution.L)
evaluationRDomainSmall := []fr.Element(solution.R)
evaluationODomainSmall := []fr.Element(solution.O)
// 2 - commit to lro
blindedLCanonical, blindedRCanonical, blindedOCanonical, err := computeBlindedLROCanonical(
evaluationLDomainSmall,
evaluationRDomainSmall,
evaluationODomainSmall,
&pk.Domain[0])
if err != nil {
return nil, err
}
proof.LROpp[0], err = pk.Vk.Iopp.BuildProofOfProximity(blindedLCanonical)
if err != nil {
return nil, err
}
proof.LROpp[1], err = pk.Vk.Iopp.BuildProofOfProximity(blindedRCanonical)
if err != nil {
return nil, err
}
proof.LROpp[2], err = pk.Vk.Iopp.BuildProofOfProximity(blindedOCanonical)
if err != nil {
return nil, err
}
// 3 - compute Z, challenges are derived using L, R, O + public inputs
fw, ok := fullWitness.Vector().(fr.Vector)
if !ok {
return nil, witness.ErrInvalidWitness
}
dataFiatShamir := make([][fr.Bytes]byte, len(spr.Public)+3)
for i := 0; i < len(spr.Public); i++ {
copy(dataFiatShamir[i][:], fw[i].Marshal())
}
copy(dataFiatShamir[len(spr.Public)][:], proof.LROpp[0].ID)
copy(dataFiatShamir[len(spr.Public)+1][:], proof.LROpp[1].ID)
copy(dataFiatShamir[len(spr.Public)+2][:], proof.LROpp[2].ID)
beta, err := deriveRandomnessFixedSize(fs, "gamma", dataFiatShamir...)
if err != nil {
return nil, err
}
gamma, err := deriveRandomness(fs, "beta", nil)
if err != nil {
return nil, err
}
//var beta, gamma fr.Element
//beta.SetUint64(9)
// gamma.SetString("10")
blindedZCanonical, err := computeBlindedZCanonical(
evaluationLDomainSmall,
evaluationRDomainSmall,
evaluationODomainSmall,
pk, beta, gamma)
if err != nil {
return nil, err
}
// 4 - commit Z
proof.Zpp, err = pk.Vk.Iopp.BuildProofOfProximity(blindedZCanonical)
if err != nil {
return nil, err
}
// 5 - compute H
// var alpha fr.Element
alpha, err := deriveRandomness(fs, "alpha", proof.Zpp.ID)
if err != nil {
return nil, err
}
// alpha.SetUint64(11)
evaluationQkCompleteDomainBigBitReversed := make([]fr.Element, pk.Domain[1].Cardinality)
copy(evaluationQkCompleteDomainBigBitReversed, fw[:len(spr.Public)])
copy(evaluationQkCompleteDomainBigBitReversed[len(spr.Public):], pk.LQkIncompleteDomainSmall[len(spr.Public):])
pk.Domain[0].FFTInverse(evaluationQkCompleteDomainBigBitReversed[:pk.Domain[0].Cardinality], fft.DIF)
fft.BitReverse(evaluationQkCompleteDomainBigBitReversed[:pk.Domain[0].Cardinality])
evaluationQkCompleteDomainBigBitReversed = fftBigCosetWOBitReverse(evaluationQkCompleteDomainBigBitReversed, &pk.Domain[1])
evaluationBlindedLDomainBigBitReversed := fftBigCosetWOBitReverse(blindedLCanonical, &pk.Domain[1])
evaluationBlindedRDomainBigBitReversed := fftBigCosetWOBitReverse(blindedRCanonical, &pk.Domain[1])
evaluationBlindedODomainBigBitReversed := fftBigCosetWOBitReverse(blindedOCanonical, &pk.Domain[1])
evaluationConstraintsDomainBigBitReversed := evalConstraintsInd(
pk,
evaluationBlindedLDomainBigBitReversed,
evaluationBlindedRDomainBigBitReversed,
evaluationBlindedODomainBigBitReversed,
evaluationQkCompleteDomainBigBitReversed)
evaluationBlindedZDomainBigBitReversed := fftBigCosetWOBitReverse(blindedZCanonical, &pk.Domain[1])
evaluationOrderingDomainBigBitReversed := evaluateOrderingDomainBigBitReversed(
pk,
evaluationBlindedZDomainBigBitReversed,
evaluationBlindedLDomainBigBitReversed,
evaluationBlindedRDomainBigBitReversed,
evaluationBlindedODomainBigBitReversed,
beta, gamma)
h1Canonical, h2Canonical, h3Canonical := computeQuotientCanonical(
pk,
evaluationConstraintsDomainBigBitReversed,
evaluationOrderingDomainBigBitReversed,
evaluationBlindedZDomainBigBitReversed,
alpha)
// 6 - commit to H
proof.Hpp[0], err = pk.Vk.Iopp.BuildProofOfProximity(h1Canonical)
if err != nil {
return nil, err
}
proof.Hpp[1], err = pk.Vk.Iopp.BuildProofOfProximity(h2Canonical)
if err != nil {
return nil, err
}
proof.Hpp[2], err = pk.Vk.Iopp.BuildProofOfProximity(h3Canonical)
if err != nil {
return nil, err
}
// 7 - build the opening proofs
// compute the size of the domain of evaluation of the committed polynomial,
// the opening position. The challenge zeta will be g^{i} where i is the opening
// position, and g is the generator of the fri domain.
rho := uint64(fri.GetRho())
friSize := 2 * rho * pk.Vk.Size
var bFriSize big.Int
bFriSize.SetInt64(int64(friSize))
frOpeningPosition, err := deriveRandomness(fs, "zeta", proof.Hpp[0].ID, proof.Hpp[1].ID, proof.Hpp[2].ID)
if err != nil {
return nil, err
}
var bOpeningPosition big.Int
bOpeningPosition.SetBytes(frOpeningPosition.Marshal()).Mod(&bOpeningPosition, &bFriSize)
openingPosition := bOpeningPosition.Uint64()
// ql, qr, qm, qo, qkIncomplete
proof.OpeningsQlQrQmQoQkincompletemp[0], err = pk.Vk.Iopp.Open(pk.CQl, openingPosition)
if err != nil {
return &proof, err
}
proof.OpeningsQlQrQmQoQkincompletemp[1], err = pk.Vk.Iopp.Open(pk.CQr, openingPosition)
if err != nil {
return &proof, err
}
proof.OpeningsQlQrQmQoQkincompletemp[2], err = pk.Vk.Iopp.Open(pk.CQm, openingPosition)
if err != nil {
return &proof, err
}
proof.OpeningsQlQrQmQoQkincompletemp[3], err = pk.Vk.Iopp.Open(pk.CQo, openingPosition)
if err != nil {
return &proof, err
}
proof.OpeningsQlQrQmQoQkincompletemp[4], err = pk.Vk.Iopp.Open(pk.CQkIncomplete, openingPosition)
if err != nil {
return &proof, err
}
// l, r, o
proof.OpeningsLROmp[0], err = pk.Vk.Iopp.Open(blindedLCanonical, openingPosition)
if err != nil {
return &proof, err
}
proof.OpeningsLROmp[1], err = pk.Vk.Iopp.Open(blindedRCanonical, openingPosition)
if err != nil {
return &proof, err
}
proof.OpeningsLROmp[2], err = pk.Vk.Iopp.Open(blindedOCanonical, openingPosition)
if err != nil {
return &proof, err
}
// h0, h1, h2
proof.OpeningsHmp[0], err = pk.Vk.Iopp.Open(h1Canonical, openingPosition)
if err != nil {
return &proof, err
}
proof.OpeningsHmp[1], err = pk.Vk.Iopp.Open(h2Canonical, openingPosition)
if err != nil {
return &proof, err
}
proof.OpeningsHmp[2], err = pk.Vk.Iopp.Open(h3Canonical, openingPosition)
if err != nil {
return &proof, err
}
// s0, s1, s2
proof.OpeningsS1S2S3mp[0], err = pk.Vk.Iopp.Open(pk.Vk.SCanonical[0], openingPosition)
if err != nil {
return &proof, err
}
proof.OpeningsS1S2S3mp[1], err = pk.Vk.Iopp.Open(pk.Vk.SCanonical[1], openingPosition)
if err != nil {
return &proof, err
}
proof.OpeningsS1S2S3mp[2], err = pk.Vk.Iopp.Open(pk.Vk.SCanonical[2], openingPosition)
if err != nil {
return &proof, err
}
// id0, id1, id2
proof.OpeningsId1Id2Id3mp[0], err = pk.Vk.Iopp.Open(pk.Vk.IdCanonical[0], openingPosition)
if err != nil {
return &proof, err
}
proof.OpeningsId1Id2Id3mp[1], err = pk.Vk.Iopp.Open(pk.Vk.IdCanonical[1], openingPosition)
if err != nil {
return &proof, err
}
proof.OpeningsId1Id2Id3mp[2], err = pk.Vk.Iopp.Open(pk.Vk.IdCanonical[2], openingPosition)
if err != nil {
return &proof, err
}
// zeta is shifted by g, the generator of Z/nZ where n is the number of constraints. We need
// to query the "rho" factor from FRI to know by what should be shifted the opening position.
// We multiply by 2 because FRI is instantiated with pk.Domain[0].Cardinality+2, which makes
// the iop's domain of size rho*(2*pk.Domain[0].Cardinality).
shiftedOpeningPosition := (openingPosition + uint64(2*rho)) % friSize
proof.OpeningsZmp[0], err = pk.Vk.Iopp.Open(blindedZCanonical, openingPosition)
if err != nil {
return &proof, err
}
proof.OpeningsZmp[1], err = pk.Vk.Iopp.Open(blindedZCanonical, shiftedOpeningPosition)
if err != nil {
return &proof, err
}
return &proof, nil
}
// evaluateOrderingDomainBigBitReversed computes the evaluation of Z(uX)g1g2g3-Z(X)f1f2f3 on the odd
// cosets of the big domain.
//
// * z evaluation of the blinded permutation accumulator polynomial on odd cosets
// * l, r, o evaluation of the blinded solution vectors on odd cosets
// * gamma randomization
func evaluateOrderingDomainBigBitReversed(pk *ProvingKey, z, l, r, o []fr.Element, beta, gamma fr.Element) []fr.Element {
nbElmts := int(pk.Domain[1].Cardinality)
// computes z_(uX)*(l(X)+s₁(X)*β+γ)*(r(X))+s₂(gⁱ)*β+γ)*(o(X))+s₃(X)*β+γ) - z(X)*(l(X)+X*β+γ)*(r(X)+u*X*β+γ)*(o(X)+u²*X*β+γ)
// on the big domain (coset).
res := make([]fr.Element, pk.Domain[1].Cardinality) // re use allocated memory for EvaluationS1BigDomain
// utils variables useful for using bit reversed indices
nn := uint64(64 - bits.TrailingZeros64(uint64(nbElmts)))
// needed to shift LsZ
toShift := int(pk.Domain[1].Cardinality / pk.Domain[0].Cardinality)
var cosetShift, cosetShiftSquare fr.Element
cosetShift.Set(&pk.Vk.CosetShift)
cosetShiftSquare.Square(&pk.Vk.CosetShift)
utils.Parallelize(int(pk.Domain[1].Cardinality), func(start, end int) {
var evaluationIDBigDomain fr.Element
evaluationIDBigDomain.Exp(pk.Domain[1].Generator, big.NewInt(int64(start))).
Mul(&evaluationIDBigDomain, &pk.Domain[1].FrMultiplicativeGen)
var f [3]fr.Element
var g [3]fr.Element
for i := start; i < end; i++ {
_i := bits.Reverse64(uint64(i)) >> nn
_is := bits.Reverse64(uint64((i+toShift)%nbElmts)) >> nn
// in what follows gⁱ is understood as the generator of the chosen coset of domainBig
f[0].Mul(&evaluationIDBigDomain, &beta).Add(&f[0], &l[_i]).Add(&f[0], &gamma) //l(gⁱ)+gⁱ*β+γ
f[1].Mul(&evaluationIDBigDomain, &cosetShift).Mul(&f[1], &beta).Add(&f[1], &r[_i]).Add(&f[1], &gamma) //r(gⁱ)+u*gⁱ*β+γ
f[2].Mul(&evaluationIDBigDomain, &cosetShiftSquare).Mul(&f[2], &beta).Add(&f[2], &o[_i]).Add(&f[2], &gamma) //o(gⁱ)+u²*gⁱ*β+γ
g[0].Mul(&pk.EvaluationS1BigDomain[_i], &beta).Add(&g[0], &l[_i]).Add(&g[0], &gamma) //l(gⁱ))+s1(gⁱ)*β+γ
g[1].Mul(&pk.EvaluationS2BigDomain[_i], &beta).Add(&g[1], &r[_i]).Add(&g[1], &gamma) //r(gⁱ))+s2(gⁱ)*β+γ
g[2].Mul(&pk.EvaluationS3BigDomain[_i], &beta).Add(&g[2], &o[_i]).Add(&g[2], &gamma) //o(gⁱ))+s3(gⁱ)*β+γ
f[0].Mul(&f[0], &f[1]).Mul(&f[0], &f[2]).Mul(&f[0], &z[_i]) // z(gⁱ)*(l(gⁱ)+g^i*β+γ)*(r(g^i)+u*g^i*β+γ)*(o(g^i)+u²*g^i*β+γ)
g[0].Mul(&g[0], &g[1]).Mul(&g[0], &g[2]).Mul(&g[0], &z[_is]) // z_(ugⁱ)*(l(gⁱ))+s₁(gⁱ)*β+γ)*(r(gⁱ))+s₂(gⁱ)*β+γ)*(o(gⁱ))+s₃(gⁱ)*β+γ)
res[_i].Sub(&g[0], &f[0]) // z_(ugⁱ)*(l(gⁱ))+s₁(gⁱ)*β+γ)*(r(gⁱ))+s₂(gⁱ)*β+γ)*(o(gⁱ))+s₃(gⁱ)*β+γ) - z(gⁱ)*(l(gⁱ)+g^i*β+γ)*(r(g^i)+u*g^i*β+γ)*(o(g^i)+u²*g^i*β+γ)
evaluationIDBigDomain.Mul(&evaluationIDBigDomain, &pk.Domain[1].Generator) // gⁱ*g
}
})
return res
}
// evalConstraintsInd computes the evaluation of lL+qrR+qqmL.R+qoO+k on
// the odd coset of (Z/8mZ)/(Z/4mZ), where m=nbConstraints+nbAssertions.
//
// * lsL, lsR, lsO are the evaluation of the blinded solution vectors on odd cosets
// * lsQk is the completed version of qk, in canonical version
//
// lsL, lsR, lsO are in bit reversed order, lsQk is in the correct order.
func evalConstraintsInd(pk *ProvingKey, lsL, lsR, lsO, lsQk []fr.Element) []fr.Element {
res := make([]fr.Element, pk.Domain[1].Cardinality)
// nn := uint64(64 - bits.TrailingZeros64(pk.Domain[1].Cardinality))
utils.Parallelize(len(res), func(start, end int) {
var t0, t1 fr.Element
for i := start; i < end; i++ {
// irev := bits.Reverse64(uint64(i)) >> nn
t1.Mul(&pk.EvaluationQmDomainBigBitReversed[i], &lsR[i]) // qm.r
t1.Add(&t1, &pk.EvaluationQlDomainBigBitReversed[i]) // qm.r + ql
t1.Mul(&t1, &lsL[i]) // qm.l.r + ql.l
t0.Mul(&pk.EvaluationQrDomainBigBitReversed[i], &lsR[i])
t0.Add(&t0, &t1) // qm.l.r + ql.l + qr.r
t1.Mul(&pk.EvaluationQoDomainBigBitReversed[i], &lsO[i])
t0.Add(&t0, &t1) // ql.l + qr.r + qm.l.r + qo.o
res[i].Add(&t0, &lsQk[i]) // ql.l + qr.r + qm.l.r + qo.o + k
}
})
return res
}
// fftBigCosetWOBitReverse evaluates poly (canonical form) of degree m<n where n=domainBig.Cardinality
// on the odd coset of (Z/2nZ)/(Z/nZ).
//
// Puts the result in res of size n.
// Warning: result is in bit reversed order, we do a bit reverse operation only once in computeQuotientCanonical
func fftBigCosetWOBitReverse(poly []fr.Element, domainBig *fft.Domain) []fr.Element {
res := make([]fr.Element, domainBig.Cardinality)
// we copy poly in res and scale by coset here
// to avoid FFT scaling on domainBig.Cardinality (res is very sparse)
cosetTable, err := domainBig.CosetTable()
if err != nil {
panic(err)
}
utils.Parallelize(len(poly), func(start, end int) {
for i := start; i < end; i++ {
res[i].Mul(&poly[i], &cosetTable[i])
}
}, runtime.NumCPU()/2)
domainBig.FFT(res, fft.DIF)
return res
}
// evaluateXnMinusOneDomainBigCoset evalutes Xᵐ-1 on DomainBig coset
func evaluateXnMinusOneDomainBigCoset(domainBig, domainSmall *fft.Domain) []fr.Element {
ratio := domainBig.Cardinality / domainSmall.Cardinality
res := make([]fr.Element, ratio)
expo := big.NewInt(int64(domainSmall.Cardinality))
res[0].Exp(domainBig.FrMultiplicativeGen, expo)
var t fr.Element
t.Exp(domainBig.Generator, big.NewInt(int64(domainSmall.Cardinality)))
for i := 1; i < int(ratio); i++ {
res[i].Mul(&res[i-1], &t)
}
var one fr.Element
one.SetOne()
for i := 0; i < int(ratio); i++ {
res[i].Sub(&res[i], &one)
}
return res
}
// computeQuotientCanonical computes h in canonical form, split as h1+X^mh2+X²mh3 such that
//
// qlL+qrR+qmL.R+qoO+k + alpha.(zu*g1*g2*g3-z*f1*f2*f3) + alpha**2*L1*(z-1)= h.Z
// \------------------/ \------------------------/ \-----/
//
// constraintsInd constraintOrdering startsAtOne
//
// constraintInd, constraintOrdering are evaluated on the odd cosets of (Z/8mZ)/(Z/mZ)
func computeQuotientCanonical(pk *ProvingKey, evaluationConstraintsIndBitReversed, evaluationConstraintOrderingBitReversed, evaluationBlindedZDomainBigBitReversed []fr.Element, alpha fr.Element) ([]fr.Element, []fr.Element, []fr.Element) {
h := make([]fr.Element, pk.Domain[1].Cardinality)
// evaluate Z = Xᵐ-1 on a coset of the big domain
evaluationXnMinusOneInverse := evaluateXnMinusOneDomainBigCoset(&pk.Domain[1], &pk.Domain[0])
evaluationXnMinusOneInverse = fr.BatchInvert(evaluationXnMinusOneInverse)
// computes L₁ (canonical form)
startsAtOne := make([]fr.Element, pk.Domain[1].Cardinality)
for i := 0; i < int(pk.Domain[0].Cardinality); i++ {
startsAtOne[i].Set(&pk.Domain[0].CardinalityInv)
}
pk.Domain[1].FFT(startsAtOne, fft.DIF, fft.OnCoset())
// ql(X)L(X)+qr(X)R(X)+qm(X)L(X)R(X)+qo(X)O(X)+k(X) + α.(z(μX)*g₁(X)*g₂(X)*g₃(X)-z(X)*f₁(X)*f₂(X)*f₃(X)) + α**2*L₁(X)(Z(X)-1)
// on a coset of the big domain
nn := uint64(64 - bits.TrailingZeros64(pk.Domain[1].Cardinality))
var one fr.Element
one.SetOne()
ratio := pk.Domain[1].Cardinality / pk.Domain[0].Cardinality
utils.Parallelize(int(pk.Domain[1].Cardinality), func(start, end int) {
var t fr.Element
for i := uint64(start); i < uint64(end); i++ {
_i := bits.Reverse64(i) >> nn
t.Sub(&evaluationBlindedZDomainBigBitReversed[_i], &one) // evaluates L₁(X)*(Z(X)-1) on a coset of the big domain
h[_i].Mul(&startsAtOne[_i], &t).Mul(&h[_i], &alpha).
Add(&h[_i], &evaluationConstraintOrderingBitReversed[_i]).
Mul(&h[_i], &alpha).
Add(&h[_i], &evaluationConstraintsIndBitReversed[_i]).
Mul(&h[_i], &evaluationXnMinusOneInverse[i%ratio])
}
})
// put h in canonical form. h is of degree 3*(n+1)+2.
// using fft.DIT put h revert bit reverse
pk.Domain[1].FFTInverse(h, fft.DIT, fft.OnCoset())
// degree of hi is n+2 because of the blinding
h1 := h[:pk.Domain[0].Cardinality+2]
h2 := h[pk.Domain[0].Cardinality+2 : 2*(pk.Domain[0].Cardinality+2)]
h3 := h[2*(pk.Domain[0].Cardinality+2) : 3*(pk.Domain[0].Cardinality+2)]
return h1, h2, h3
}
// computeZ computes Z, in canonical basis, where:
//
// - Z of degree n (domainNum.Cardinality)
//
// - Z(1)=1
// (l_i+z**i+gamma)*(r_i+u*z**i+gamma)*(o_i+u**2z**i+gamma)
//
// - for i>0: Z(u**i) = Pi_{k<i} -------------------------------------------------------
// (l_i+s1+gamma)*(r_i+s2+gamma)*(o_i+s3+gamma)
//
// - l, r, o are the solution in Lagrange basis
func computeBlindedZCanonical(l, r, o []fr.Element, pk *ProvingKey, beta, gamma fr.Element) ([]fr.Element, error) {
// note that z has more capacity has its memory is reused for blinded z later on
z := make([]fr.Element, pk.Domain[0].Cardinality, pk.Domain[0].Cardinality+3)
nbElmts := int(pk.Domain[0].Cardinality)
gInv := make([]fr.Element, pk.Domain[0].Cardinality)
z[0].SetOne()
gInv[0].SetOne()
evaluationIDSmallDomain := getIDSmallDomain(&pk.Domain[0])
utils.Parallelize(nbElmts-1, func(start, end int) {
var f [3]fr.Element
var g [3]fr.Element
for i := start; i < end; i++ {
f[0].Mul(&evaluationIDSmallDomain[i], &beta).Add(&f[0], &l[i]).Add(&f[0], &gamma) //lᵢ+g^i*β+γ
f[1].Mul(&evaluationIDSmallDomain[i+nbElmts], &beta).Add(&f[1], &r[i]).Add(&f[1], &gamma) //rᵢ+u*g^i*β+γ
f[2].Mul(&evaluationIDSmallDomain[i+2*nbElmts], &beta).Add(&f[2], &o[i]).Add(&f[2], &gamma) //oᵢ+u²*g^i*β+γ
g[0].Mul(&evaluationIDSmallDomain[pk.Permutation[i]], &beta).Add(&g[0], &l[i]).Add(&g[0], &gamma) //lᵢ+s₁(g^i)*β+γ
g[1].Mul(&evaluationIDSmallDomain[pk.Permutation[i+nbElmts]], &beta).Add(&g[1], &r[i]).Add(&g[1], &gamma) //rᵢ+s₂(g^i)*β+γ
g[2].Mul(&evaluationIDSmallDomain[pk.Permutation[i+2*nbElmts]], &beta).Add(&g[2], &o[i]).Add(&g[2], &gamma) //oᵢ+s₃(g^i)*β+γ
f[0].Mul(&f[0], &f[1]).Mul(&f[0], &f[2]) // (lᵢ+g^i*β+γ)*(rᵢ+u*g^i*β+γ)*(oᵢ+u²*g^i*β+γ)
g[0].Mul(&g[0], &g[1]).Mul(&g[0], &g[2]) // (lᵢ+s₁(g^i)*β+γ)*(rᵢ+s₂(g^i)*β+γ)*(oᵢ+s₃(g^i)*β+γ)
gInv[i+1] = g[0]
z[i+1] = f[0]
}
})
gInv = fr.BatchInvert(gInv)
for i := 1; i < nbElmts; i++ {
z[i].Mul(&z[i], &z[i-1]).
Mul(&z[i], &gInv[i])
}
pk.Domain[0].FFTInverse(z, fft.DIF)
fft.BitReverse(z)
return blindPoly(z, pk.Domain[0].Cardinality, 2)
}
// computeBlindedLROCanonical
// l, r, o in canonical basis with blinding
func computeBlindedLROCanonical(
ll, lr, lo []fr.Element, domain *fft.Domain) (bcl, bcr, bco []fr.Element, err error) {
// note that bcl, bcr and bco reuses cl, cr and co memory
cl := make([]fr.Element, domain.Cardinality, domain.Cardinality+2)
cr := make([]fr.Element, domain.Cardinality, domain.Cardinality+2)
co := make([]fr.Element, domain.Cardinality, domain.Cardinality+2)
chDone := make(chan error, 2)
go func() {
var err error
copy(cl, ll)
domain.FFTInverse(cl, fft.DIF)
fft.BitReverse(cl)
bcl, err = blindPoly(cl, domain.Cardinality, 1)
chDone <- err
}()
go func() {
var err error
copy(cr, lr)
domain.FFTInverse(cr, fft.DIF)
fft.BitReverse(cr)
bcr, err = blindPoly(cr, domain.Cardinality, 1)
chDone <- err
}()
copy(co, lo)
domain.FFTInverse(co, fft.DIF)
fft.BitReverse(co)
if bco, err = blindPoly(co, domain.Cardinality, 1); err != nil {
return
}
err = <-chDone
if err != nil {
return
}
err = <-chDone
return
}
// blindPoly blinds a polynomial by adding a Q(X)*(X**degree-1), where deg Q = order.
//
// * cp polynomial in canonical form
// * rou root of unity, meaning the blinding factor is multiple of X**rou-1
// * bo blinding order, it's the degree of Q, where the blinding is Q(X)*(X**degree-1)
//
// WARNING:
// pre condition degree(cp) ⩽ rou + bo
// pre condition cap(cp) ⩾ int(totalDegree + 1)
func blindPoly(cp []fr.Element, rou, bo uint64) ([]fr.Element, error) {
// degree of the blinded polynomial is max(rou+order, cp.Degree)
totalDegree := rou + bo
// re-use cp
res := cp[:totalDegree+1]
// random polynomial
blindingPoly := make([]fr.Element, bo+1)
for i := uint64(0); i < bo+1; i++ {
if _, err := blindingPoly[i].SetRandom(); err != nil {
return nil, err
}
}
// blinding
for i := uint64(0); i < bo+1; i++ {
res[i].Sub(&res[i], &blindingPoly[i])
res[rou+i].Add(&res[rou+i], &blindingPoly[i])
}
return res, nil
}
func deriveRandomnessFixedSize(fs *fiatshamir.Transcript, challenge string, data ...[fr.Bytes]byte) (fr.Element, error) {
var r fr.Element
for _, d := range data {
if err := fs.Bind(challenge, d[:]); err != nil {
return r, err
}
}
b, err := fs.ComputeChallenge(challenge)
if err != nil {
return r, err
}
r.SetBytes(b)
return r, nil
}
func deriveRandomness(fs *fiatshamir.Transcript, challenge string, data ...[]byte) (fr.Element, error) {
var r fr.Element
for _, d := range data {
if err := fs.Bind(challenge, d); err != nil {
return r, err
}
}
b, err := fs.ComputeChallenge(challenge)
if err != nil {
return r, err
}
r.SetBytes(b)
return r, nil
}