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// implementation of https://eprint.iacr.org/2013/879.pdf
package snark
import (
"fmt"
"math/big"
"os"
"github.com/arnaucube/go-snark/bn128"
"github.com/arnaucube/go-snark/circuitcompiler"
"github.com/arnaucube/go-snark/fields"
"github.com/arnaucube/go-snark/r1csqap"
)
type Pk struct { // Proving Key pk:=(pkA, pkB, pkC, pkH)
G1T [][3]*big.Int // t encrypted in G1 curve, G1T == Pk.H
A [][3]*big.Int
B [][3][2]*big.Int
C [][3]*big.Int
Kp [][3]*big.Int
Ap [][3]*big.Int
Bp [][3]*big.Int
Cp [][3]*big.Int
Z []*big.Int
}
type Vk struct {
Vka [3][2]*big.Int
Vkb [3]*big.Int
Vkc [3][2]*big.Int
IC [][3]*big.Int
G1Kbg [3]*big.Int // g1 * Kbeta * Kgamma
G2Kbg [3][2]*big.Int // g2 * Kbeta * Kgamma
G2Kg [3][2]*big.Int // g2 * Kgamma
Vkz [3][2]*big.Int
}
// Setup is the data structure holding the Trusted Setup data. The Setup.Toxic sub struct must be destroyed after the GenerateTrustedSetup function is completed
type Setup struct {
Toxic struct {
T *big.Int // trusted setup secret
Ka *big.Int // prover
Kb *big.Int // prover
Kc *big.Int // prover
Kbeta *big.Int
Kgamma *big.Int
RhoA *big.Int
RhoB *big.Int
RhoC *big.Int
}
// public
Pk Pk
Vk Vk
}
// Proof contains the parameters to proof the zkSNARK
type Proof struct {
PiA [3]*big.Int
PiAp [3]*big.Int
PiB [3][2]*big.Int
PiBp [3]*big.Int
PiC [3]*big.Int
PiCp [3]*big.Int
PiH [3]*big.Int
PiKp [3]*big.Int
// PublicSignals []*big.Int
}
type utils struct {
Bn bn128.Bn128
FqR fields.Fq
PF r1csqap.PolynomialField
}
// Utils is the data structure holding the BN128, FqR Finite Field over R, PolynomialField, that will be used inside the snarks operations
var Utils = prepareUtils()
func prepareUtils() utils {
bn, err := bn128.NewBn128()
if err != nil {
panic(err)
}
// new Finite Field
fqR := fields.NewFq(bn.R)
// new Polynomial Field
pf := r1csqap.NewPolynomialField(fqR)
return utils{
Bn: bn,
FqR: fqR,
PF: pf,
}
}
// GenerateTrustedSetup generates the Trusted Setup from a compiled Circuit. The Setup.Toxic sub data structure must be destroyed
func GenerateTrustedSetup(witnessLength int, circuit circuitcompiler.Circuit, alphas, betas, gammas [][]*big.Int) (Setup, error) {
var setup Setup
var err error
// input soundness
// for i := 0; i < len(alphas); i++ {
// for j := 0; j < len(alphas[i]); j++ {
// if j <= circuit.NPublic {
// if bytes.Equal(alphas[i][j].Bytes(), Utils.FqR.Zero().Bytes()) {
// alphas[i][j] = Utils.FqR.One()
// }
// }
// }
// }
// generate random t value
setup.Toxic.T, err = Utils.FqR.Rand()
if err != nil {
return Setup{}, err
}
// k for calculating pi' and Vk
setup.Toxic.Ka, err = Utils.FqR.Rand()
if err != nil {
return Setup{}, err
}
setup.Toxic.Kb, err = Utils.FqR.Rand()
if err != nil {
return Setup{}, err
}
setup.Toxic.Kc, err = Utils.FqR.Rand()
if err != nil {
return Setup{}, err
}
// generate Kβ (Kbeta) and Kγ (Kgamma)
setup.Toxic.Kbeta, err = Utils.FqR.Rand()
if err != nil {
return Setup{}, err
}
setup.Toxic.Kgamma, err = Utils.FqR.Rand()
if err != nil {
return Setup{}, err
}
// generate ρ (Rho): ρA, ρB, ρC
setup.Toxic.RhoA, err = Utils.FqR.Rand()
if err != nil {
return Setup{}, err
}
setup.Toxic.RhoB, err = Utils.FqR.Rand()
if err != nil {
return Setup{}, err
}
setup.Toxic.RhoC = Utils.FqR.Mul(setup.Toxic.RhoA, setup.Toxic.RhoB)
// calculated more down
// for i := 0; i < witnessLength; i++ {
// tPow := Utils.FqR.Exp(setup.Toxic.T, big.NewInt(int64(i)))
// tEncr1 := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, tPow)
// gt1 = append(gt1, tEncr1)
// tEncr2 := Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, tPow)
// gt2 = append(gt2, tEncr2)
// }
// gt1: g1, g1*t, g1*t^2, g1*t^3, ...
// gt2: g2, g2*t, g2*t^2, ...
setup.Vk.Vka = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Ka)
setup.Vk.Vkb = Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, setup.Toxic.Kb)
setup.Vk.Vkc = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Kc)
/*
Verification keys:
- Vk_betagamma1: setup.G1Kbg = g1 * Kbeta*Kgamma
- Vk_betagamma2: setup.G2Kbg = g2 * Kbeta*Kgamma
- Vk_gamma: setup.G2Kg = g2 * Kgamma
*/
kbg := Utils.FqR.Mul(setup.Toxic.Kbeta, setup.Toxic.Kgamma)
setup.Vk.G1Kbg = Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, kbg)
setup.Vk.G2Kbg = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, kbg)
setup.Vk.G2Kg = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Kgamma)
// for i := 0; i < circuit.NVars; i++ {
for i := 0; i < len(circuit.Signals); i++ {
at := Utils.PF.Eval(alphas[i], setup.Toxic.T)
// rhoAat := Utils.Bn.Fq1.Mul(setup.Toxic.RhoA, at)
rhoAat := Utils.FqR.Mul(setup.Toxic.RhoA, at)
a := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, rhoAat)
setup.Pk.A = append(setup.Pk.A, a)
if i <= circuit.NPublic {
setup.Vk.IC = append(setup.Vk.IC, a)
}
bt := Utils.PF.Eval(betas[i], setup.Toxic.T)
// rhoBbt := Utils.Bn.Fq1.Mul(setup.Toxic.RhoB, bt)
rhoBbt := Utils.FqR.Mul(setup.Toxic.RhoB, bt)
bg1 := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, rhoBbt)
bg2 := Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, rhoBbt)
setup.Pk.B = append(setup.Pk.B, bg2)
ct := Utils.PF.Eval(gammas[i], setup.Toxic.T)
// rhoCct := Utils.Bn.Fq1.Mul(setup.Toxic.RhoC, ct)
rhoCct := Utils.FqR.Mul(setup.Toxic.RhoC, ct)
c := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, rhoCct)
setup.Pk.C = append(setup.Pk.C, c)
kt := Utils.FqR.Add(Utils.FqR.Add(rhoAat, rhoBbt), rhoCct)
k := Utils.Bn.G1.Affine(Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, kt))
ktest := Utils.Bn.G1.Affine(Utils.Bn.G1.Add(Utils.Bn.G1.Add(a, bg1), c))
if !Utils.Bn.Fq2.Equal(k, ktest) {
os.Exit(1)
return setup, err
}
setup.Pk.Ap = append(setup.Pk.Ap, Utils.Bn.G1.MulScalar(a, setup.Toxic.Ka))
setup.Pk.Bp = append(setup.Pk.Bp, Utils.Bn.G1.MulScalar(bg1, setup.Toxic.Kb))
setup.Pk.Cp = append(setup.Pk.Cp, Utils.Bn.G1.MulScalar(c, setup.Toxic.Kc))
k_ := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, kt)
setup.Pk.Kp = append(setup.Pk.Kp, Utils.Bn.G1.MulScalar(k_, setup.Toxic.Kbeta))
}
// z pol
zpol := []*big.Int{big.NewInt(int64(1))}
// for i := 0; i < len(circuit.Constraints); i++ {
for i := 1; i < len(alphas)-1; i++ {
zpol = Utils.PF.Mul(
zpol,
[]*big.Int{
Utils.FqR.Neg( // neg over R
big.NewInt(int64(i))),
big.NewInt(int64(1)),
})
}
setup.Pk.Z = zpol
zt := Utils.PF.Eval(zpol, setup.Toxic.T)
// rhoCzt := Utils.Bn.Fq1.Mul(setup.Toxic.RhoC, zt)
rhoCzt := Utils.FqR.Mul(setup.Toxic.RhoC, zt)
setup.Vk.Vkz = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, rhoCzt)
// encrypt t values with curve generators
var gt1 [][3]*big.Int
gt1 = append(gt1, Utils.Bn.G1.G) // the first is t**0 * G1 = 1 * G1 = G1
tEncr := setup.Toxic.T
for i := 1; i < len(zpol); i++ { //should be G1T = pkH = (tau**i * G1) from i=0 to d, where d is degree of pol Z(x)
gt1 = append(gt1, Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, tEncr))
// tEncr = Utils.Bn.Fq1.Mul(tEncr, setup.Toxic.T)
tEncr = Utils.FqR.Mul(tEncr, setup.Toxic.T)
}
setup.Pk.G1T = gt1
return setup, nil
}
// GenerateProofs generates all the parameters to proof the zkSNARK from the Circuit, Setup and the Witness
func GenerateProofs(circuit circuitcompiler.Circuit, pk Pk, w []*big.Int, px []*big.Int) (Proof, error) {
var proof Proof
proof.PiA = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
proof.PiAp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
proof.PiB = Utils.Bn.Fq6.Zero()
proof.PiBp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
proof.PiC = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
proof.PiCp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
proof.PiH = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
proof.PiKp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
for i := circuit.NPublic + 1; i < circuit.NVars; i++ {
proof.PiA = Utils.Bn.G1.Add(proof.PiA, Utils.Bn.G1.MulScalar(pk.A[i], w[i]))
proof.PiAp = Utils.Bn.G1.Add(proof.PiAp, Utils.Bn.G1.MulScalar(pk.Ap[i], w[i]))
}
for i := 0; i < circuit.NVars; i++ {
proof.PiB = Utils.Bn.G2.Add(proof.PiB, Utils.Bn.G2.MulScalar(pk.B[i], w[i]))
proof.PiBp = Utils.Bn.G1.Add(proof.PiBp, Utils.Bn.G1.MulScalar(pk.Bp[i], w[i]))
proof.PiC = Utils.Bn.G1.Add(proof.PiC, Utils.Bn.G1.MulScalar(pk.C[i], w[i]))
proof.PiCp = Utils.Bn.G1.Add(proof.PiCp, Utils.Bn.G1.MulScalar(pk.Cp[i], w[i]))
proof.PiKp = Utils.Bn.G1.Add(proof.PiKp, Utils.Bn.G1.MulScalar(pk.Kp[i], w[i]))
}
hx := Utils.PF.DivisorPolynomial(px, pk.Z) // maybe move this calculation to a previous step
// piH = pkH,0 + sum ( hi * pk H,i ), where pkH = G1T, hi=hx
// proof.PiH = Utils.Bn.G1.Add(proof.PiH, pk.G1T[0])
for i := 0; i < len(hx); i++ {
proof.PiH = Utils.Bn.G1.Add(proof.PiH, Utils.Bn.G1.MulScalar(pk.G1T[i], hx[i]))
}
return proof, nil
}
// VerifyProof verifies over the BN128 the Pairings of the Proof
func VerifyProof(vk Vk, proof Proof, publicSignals []*big.Int, debug bool) bool {
// e(piA, Va) == e(piA', g2)
pairingPiaVa := Utils.Bn.Pairing(proof.PiA, vk.Vka)
pairingPiapG2 := Utils.Bn.Pairing(proof.PiAp, Utils.Bn.G2.G)
if !Utils.Bn.Fq12.Equal(pairingPiaVa, pairingPiapG2) {
if debug {
fmt.Println("❌ e(piA, Va) == e(piA', g2), valid knowledge commitment for A")
}
return false
}
if debug {
fmt.Println("✓ e(piA, Va) == e(piA', g2), valid knowledge commitment for A")
}
// e(Vb, piB) == e(piB', g2)
pairingVbPib := Utils.Bn.Pairing(vk.Vkb, proof.PiB)
pairingPibpG2 := Utils.Bn.Pairing(proof.PiBp, Utils.Bn.G2.G)
if !Utils.Bn.Fq12.Equal(pairingVbPib, pairingPibpG2) {
if debug {
fmt.Println("❌ e(Vb, piB) == e(piB', g2), valid knowledge commitment for B")
}
return false
}
if debug {
fmt.Println("✓ e(Vb, piB) == e(piB', g2), valid knowledge commitment for B")
}
// e(piC, Vc) == e(piC', g2)
pairingPicVc := Utils.Bn.Pairing(proof.PiC, vk.Vkc)
pairingPicpG2 := Utils.Bn.Pairing(proof.PiCp, Utils.Bn.G2.G)
if !Utils.Bn.Fq12.Equal(pairingPicVc, pairingPicpG2) {
if debug {
fmt.Println("❌ e(piC, Vc) == e(piC', g2), valid knowledge commitment for C")
}
return false
}
if debug {
fmt.Println("✓ e(piC, Vc) == e(piC', g2), valid knowledge commitment for C")
}
// Vkx, to then calculate Vkx+piA
vkxpia := vk.IC[0]
for i := 0; i < len(publicSignals); i++ {
vkxpia = Utils.Bn.G1.Add(vkxpia, Utils.Bn.G1.MulScalar(vk.IC[i+1], publicSignals[i]))
}
// e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2)
if !Utils.Bn.Fq12.Equal(
Utils.Bn.Pairing(Utils.Bn.G1.Add(vkxpia, proof.PiA), proof.PiB), // TODO Add(vkxpia, proof.PiA) can go outside in order to save computation, as is reused later
Utils.Bn.Fq12.Mul(
Utils.Bn.Pairing(proof.PiH, vk.Vkz),
Utils.Bn.Pairing(proof.PiC, Utils.Bn.G2.G))) {
if debug {
fmt.Println("❌ e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2), QAP disibility checked")
}
return false
}
if debug {
fmt.Println("✓ e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2), QAP disibility checked")
}
// e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB)
// == e(piK, g2Kgamma)
piApiC := Utils.Bn.G1.Add(Utils.Bn.G1.Add(vkxpia, proof.PiA), proof.PiC)
pairingPiACG2Kbg := Utils.Bn.Pairing(piApiC, vk.G2Kbg)
pairingG1KbgPiB := Utils.Bn.Pairing(vk.G1Kbg, proof.PiB)
pairingL := Utils.Bn.Fq12.Mul(pairingPiACG2Kbg, pairingG1KbgPiB)
pairingR := Utils.Bn.Pairing(proof.PiKp, vk.G2Kg)
if !Utils.Bn.Fq12.Equal(pairingL, pairingR) {
fmt.Println("❌ e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB) == e(piK, g2Kgamma)")
return false
}
if debug {
fmt.Println("✓ e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB) == e(piK, g2Kgamma)")
}
return true
}