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permutation.go
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permutation.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 consensys/gnark-crypto DO NOT EDIT
package permutation
import (
"crypto/sha256"
"errors"
"math/big"
"math/bits"
"github.com/consensys/gnark-crypto/ecc/bls12-377"
"github.com/consensys/gnark-crypto/ecc/bls12-377/fr"
"github.com/consensys/gnark-crypto/ecc/bls12-377/fr/fft"
"github.com/consensys/gnark-crypto/ecc/bls12-377/fr/kzg"
fiatshamir "github.com/consensys/gnark-crypto/fiat-shamir"
)
var (
ErrIncompatibleSize = errors.New("t1 and t2 should be of the same size")
ErrSize = errors.New("t1 and t2 should be of size a power of 2")
ErrPermutationProof = errors.New("permutation proof verification failed")
ErrGenerator = errors.New("wrong generator")
)
// Proof proof that the commitments of t1 and t2 come from
// the same vector but permuted.
type Proof struct {
// size of the polynomials
size int
// generator of the fft domain, used for shifting the evaluation point
g fr.Element
// commitments of t1 & t2, the permuted vectors, and z, the accumulation
// polynomial
t1, t2, z kzg.Digest
// commitment to the quotient polynomial
q kzg.Digest
// opening proofs of t1, t2, z, q (in that order)
batchedProof kzg.BatchOpeningProof
// shifted opening proof of z
shiftedProof kzg.OpeningProof
}
// evaluateAccumulationPolynomialBitReversed returns the accumulation polynomial in Lagrange basis.
func evaluateAccumulationPolynomialBitReversed(lt1, lt2 []fr.Element, epsilon fr.Element) []fr.Element {
s := len(lt1)
z := make([]fr.Element, s)
d := make([]fr.Element, s)
z[0].SetOne()
d[0].SetOne()
nn := uint64(64 - bits.TrailingZeros64(uint64(s)))
var t fr.Element
for i := 0; i < s-1; i++ {
_i := int(bits.Reverse64(uint64(i)) >> nn)
_ii := int(bits.Reverse64(uint64((i+1)%s)) >> nn)
z[_ii].Mul(&z[_i], t.Sub(&epsilon, <1[i]))
d[i+1].Mul(&d[i], t.Sub(&epsilon, <2[i]))
}
d = fr.BatchInvert(d)
for i := 0; i < s-1; i++ {
_ii := int(bits.Reverse64(uint64((i+1)%s)) >> nn)
z[_ii].Mul(&z[_ii], &d[i+1])
}
return z
}
// evaluateFirstPartNumReverse computes lt2*z(gx) - lt1*z
func evaluateFirstPartNumReverse(lt1, lt2, lz []fr.Element, epsilon fr.Element) []fr.Element {
s := len(lt1)
res := make([]fr.Element, s)
var a, b fr.Element
nn := uint64(64 - bits.TrailingZeros64(uint64(s)))
for i := 0; i < s; i++ {
_i := int(bits.Reverse64(uint64(i)) >> nn)
_ii := int(bits.Reverse64(uint64((i+1)%s)) >> nn)
a.Sub(&epsilon, <2[_i])
a.Mul(&lz[_ii], &a)
b.Sub(&epsilon, <1[_i])
b.Mul(&lz[_i], &b)
res[_i].Sub(&a, &b)
}
return res
}
// evaluateSecondPartNumReverse computes L0 * (z-1)
func evaluateSecondPartNumReverse(lz []fr.Element, d *fft.Domain) []fr.Element {
var tn, o, g fr.Element
o.SetOne()
tn.Exp(d.FrMultiplicativeGen, big.NewInt(int64(d.Cardinality))).
Sub(&tn, &o)
s := len(lz)
u := make([]fr.Element, s)
g.Set(&d.FrMultiplicativeGen)
for i := 0; i < s; i++ {
u[i].Sub(&g, &o)
g.Mul(&g, &d.Generator)
}
u = fr.BatchInvert(u)
res := make([]fr.Element, s)
nn := uint64(64 - bits.TrailingZeros64(uint64(s)))
for i := 0; i < s; i++ {
_i := int(bits.Reverse64(uint64(i)) >> nn)
res[_i].Sub(&lz[_i], &o).
Mul(&res[_i], &u[i]).
Mul(&res[_i], &tn)
}
return res
}
// Prove generates a proof that t1 and t2 are the same but permuted.
// The size of t1 and t2 should be the same and a power of 2.
func Prove(pk kzg.ProvingKey, t1, t2 []fr.Element) (Proof, error) {
// res
var proof Proof
var err error
// size checking
if len(t1) != len(t2) {
return proof, ErrIncompatibleSize
}
// create the domains
d := fft.NewDomain(uint64(len(t1)))
if d.Cardinality != uint64(len(t1)) {
return proof, ErrSize
}
s := int(d.Cardinality)
proof.size = s
proof.g.Set(&d.Generator)
// hash function for Fiat Shamir
hFunc := sha256.New()
// transcript to derive the challenge
fs := fiatshamir.NewTranscript(hFunc, "epsilon", "omega", "eta")
// commit t1, t2
ct1 := make([]fr.Element, s)
ct2 := make([]fr.Element, s)
copy(ct1, t1)
copy(ct2, t2)
d.FFTInverse(ct1, fft.DIF)
d.FFTInverse(ct2, fft.DIF)
fft.BitReverse(ct1)
fft.BitReverse(ct2)
proof.t1, err = kzg.Commit(ct1, pk)
if err != nil {
return proof, err
}
proof.t2, err = kzg.Commit(ct2, pk)
if err != nil {
return proof, err
}
// derive challenge for z
epsilon, err := deriveRandomness(&fs, "epsilon", &proof.t1, &proof.t2)
if err != nil {
return proof, err
}
// compute Z and commit it
cz := evaluateAccumulationPolynomialBitReversed(t1, t2, epsilon)
d.FFTInverse(cz, fft.DIT)
proof.z, err = kzg.Commit(cz, pk)
if err != nil {
return proof, err
}
lz := make([]fr.Element, s)
copy(lz, cz)
d.FFT(lz, fft.DIF, fft.OnCoset())
// compute the first part of the numerator
lt1 := make([]fr.Element, s)
lt2 := make([]fr.Element, s)
copy(lt1, ct1)
copy(lt2, ct2)
d.FFT(lt1, fft.DIF, fft.OnCoset())
d.FFT(lt2, fft.DIF, fft.OnCoset())
lsNumFirstPart := evaluateFirstPartNumReverse(lt1, lt2, lz, epsilon)
// compute second part of the numerator
lsNum := evaluateSecondPartNumReverse(lz, d)
// derive challenge used for the folding
omega, err := deriveRandomness(&fs, "omega", &proof.z)
if err != nil {
return proof, err
}
// fold the numerator and divide it by x^n-1
var t, one fr.Element
one.SetOne()
t.Exp(d.FrMultiplicativeGen, big.NewInt(int64(d.Cardinality))).Sub(&t, &one).Inverse(&t)
for i := 0; i < s; i++ {
lsNum[i].Mul(&omega, &lsNum[i]).
Add(&lsNum[i], &lsNumFirstPart[i]).
Mul(&lsNum[i], &t)
}
// get the quotient and commit it
d.FFTInverse(lsNum, fft.DIT, fft.OnCoset())
proof.q, err = kzg.Commit(lsNum, pk)
if err != nil {
return proof, err
}
// derive the evaluation challenge
eta, err := deriveRandomness(&fs, "eta", &proof.q)
if err != nil {
return proof, err
}
// compute the opening proofs
proof.batchedProof, err = kzg.BatchOpenSinglePoint(
[][]fr.Element{
ct1,
ct2,
cz,
lsNum,
},
[]kzg.Digest{
proof.t1,
proof.t2,
proof.z,
proof.q,
},
eta,
hFunc,
pk,
)
if err != nil {
return proof, err
}
var shiftedEta fr.Element
shiftedEta.Mul(&eta, &d.Generator)
proof.shiftedProof, err = kzg.Open(
cz,
shiftedEta,
pk,
)
if err != nil {
return proof, err
}
// done
return proof, nil
}
// Verify verifies a permutation proof.
func Verify(vk kzg.VerifyingKey, proof Proof) error {
// hash function that is used for Fiat Shamir
hFunc := sha256.New()
// transcript to derive the challenge
fs := fiatshamir.NewTranscript(hFunc, "epsilon", "omega", "eta")
// derive the challenges
epsilon, err := deriveRandomness(&fs, "epsilon", &proof.t1, &proof.t2)
if err != nil {
return err
}
omega, err := deriveRandomness(&fs, "omega", &proof.z)
if err != nil {
return err
}
eta, err := deriveRandomness(&fs, "eta", &proof.q)
if err != nil {
return err
}
// check the relation
bs := big.NewInt(int64(proof.size))
var l0, a, b, one, rhs, lhs fr.Element
one.SetOne()
rhs.Exp(eta, bs).
Sub(&rhs, &one)
a.Sub(&eta, &one)
l0.Div(&rhs, &a)
rhs.Mul(&rhs, &proof.batchedProof.ClaimedValues[3])
a.Sub(&epsilon, &proof.batchedProof.ClaimedValues[1]).
Mul(&a, &proof.shiftedProof.ClaimedValue)
b.Sub(&epsilon, &proof.batchedProof.ClaimedValues[0]).
Mul(&b, &proof.batchedProof.ClaimedValues[2])
lhs.Sub(&a, &b)
a.Sub(&proof.batchedProof.ClaimedValues[2], &one).
Mul(&a, &l0).
Mul(&a, &omega)
lhs.Add(&a, &lhs)
if !lhs.Equal(&rhs) {
return ErrPermutationProof
}
// check the opening proofs
err = kzg.BatchVerifySinglePoint(
[]kzg.Digest{
proof.t1,
proof.t2,
proof.z,
proof.q,
},
&proof.batchedProof,
eta,
hFunc,
vk,
)
if err != nil {
return err
}
var shiftedEta fr.Element
shiftedEta.Mul(&eta, &proof.g)
err = kzg.Verify(&proof.z, &proof.shiftedProof, shiftedEta, vk)
if err != nil {
return err
}
// check the generator is correct
var checkOrder fr.Element
checkOrder.Exp(proof.g, big.NewInt(int64(proof.size/2)))
if checkOrder.Equal(&one) {
return ErrGenerator
}
checkOrder.Square(&checkOrder)
if !checkOrder.Equal(&one) {
return ErrGenerator
}
return nil
}
// TODO put that in fiat-shamir package
func deriveRandomness(fs *fiatshamir.Transcript, challenge string, points ...*bls12377.G1Affine) (fr.Element, error) {
var buf [bls12377.SizeOfG1AffineUncompressed]byte
var r fr.Element
for _, p := range points {
buf = p.RawBytes()
if err := fs.Bind(challenge, buf[:]); err != nil {
return r, err
}
}
b, err := fs.ComputeChallenge(challenge)
if err != nil {
return r, err
}
r.SetBytes(b)
return r, nil
}