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" ) // 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 G1T [][3]*big.Int // t encrypted in G1 curve, G1T == Pk.H G2T [][3][2]*big.Int // t encrypted in G2 curve Pk struct { // Proving Key pk:=(pkA, pkB, pkC, pkH) 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 } 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 } } // 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.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, setup Setup, 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(setup.Pk.A[i], w[i])) proof.PiAp = Utils.Bn.G1.Add(proof.PiAp, Utils.Bn.G1.MulScalar(setup.Pk.Ap[i], w[i])) } for i := 0; i < circuit.NVars; i++ { proof.PiB = Utils.Bn.G2.Add(proof.PiB, Utils.Bn.G2.MulScalar(setup.Pk.B[i], w[i])) proof.PiBp = Utils.Bn.G1.Add(proof.PiBp, Utils.Bn.G1.MulScalar(setup.Pk.Bp[i], w[i])) proof.PiC = Utils.Bn.G1.Add(proof.PiC, Utils.Bn.G1.MulScalar(setup.Pk.C[i], w[i])) proof.PiCp = Utils.Bn.G1.Add(proof.PiCp, Utils.Bn.G1.MulScalar(setup.Pk.Cp[i], w[i])) proof.PiKp = Utils.Bn.G1.Add(proof.PiKp, Utils.Bn.G1.MulScalar(setup.Pk.Kp[i], w[i])) } hx := Utils.PF.DivisorPolynomial(px, setup.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, setup.G1T[0]) for i := 0; i < len(hx); i++ { proof.PiH = Utils.Bn.G1.Add(proof.PiH, Utils.Bn.G1.MulScalar(setup.G1T[i], hx[i])) } return proof, nil } // VerifyProof verifies over the BN128 the Pairings of the Proof func VerifyProof(circuit circuitcompiler.Circuit, setup Setup, proof Proof, publicSignals []*big.Int, debug bool) bool { // e(piA, Va) == e(piA', g2) pairingPiaVa := Utils.Bn.Pairing(proof.PiA, setup.Vk.Vka) pairingPiapG2 := Utils.Bn.Pairing(proof.PiAp, Utils.Bn.G2.G) if !Utils.Bn.Fq12.Equal(pairingPiaVa, pairingPiapG2) { 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(setup.Vk.Vkb, proof.PiB) pairingPibpG2 := Utils.Bn.Pairing(proof.PiBp, Utils.Bn.G2.G) if !Utils.Bn.Fq12.Equal(pairingVbPib, pairingPibpG2) { 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, setup.Vk.Vkc) pairingPicpG2 := Utils.Bn.Pairing(proof.PiCp, Utils.Bn.G2.G) if !Utils.Bn.Fq12.Equal(pairingPicVc, pairingPicpG2) { 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 := setup.Vk.IC[0] for i := 0; i < len(publicSignals); i++ { vkxpia = Utils.Bn.G1.Add(vkxpia, Utils.Bn.G1.MulScalar(setup.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, setup.Vk.Vkz), Utils.Bn.Pairing(proof.PiC, Utils.Bn.G2.G))) { 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, setup.Vk.G2Kbg) pairingG1KbgPiB := Utils.Bn.Pairing(setup.Vk.G1Kbg, proof.PiB) pairingL := Utils.Bn.Fq12.Mul(pairingPiACG2Kbg, pairingG1KbgPiB) pairingR := Utils.Bn.Pairing(proof.PiKp, setup.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 }