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@ -62,59 +62,85 @@ type Proof struct { |
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PublicSignals []*big.Int |
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PublicSignals []*big.Int |
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} |
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} |
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type utils struct { |
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Bn bn128.Bn128 |
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FqR fields.Fq |
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PF r1csqap.PolynomialField |
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} |
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// Utils is the data structure holding the BN128, FqR Finite Field over R, PolynomialField, that will be used inside the snarks operations
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var Utils = prepareUtils() |
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func prepareUtils() utils { |
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bn, err := bn128.NewBn128() |
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if err != nil { |
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panic(err) |
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} |
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// new Finite Field
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fqR := fields.NewFq(bn.R) |
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// new Polynomial Field
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pf := r1csqap.NewPolynomialField(fqR) |
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return utils{ |
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Bn: bn, |
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FqR: fqR, |
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PF: pf, |
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} |
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} |
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// GenerateTrustedSetup generates the Trusted Setup from a compiled Circuit. The Setup.Toxic sub data structure must be destroyed
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// GenerateTrustedSetup generates the Trusted Setup from a compiled Circuit. The Setup.Toxic sub data structure must be destroyed
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func GenerateTrustedSetup(bn bn128.Bn128, fqR fields.Fq, pf r1csqap.PolynomialField, witnessLength int, circuit circuitcompiler.Circuit, alphas, betas, gammas [][]*big.Int, zx []*big.Int) (Setup, error) { |
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func GenerateTrustedSetup(witnessLength int, circuit circuitcompiler.Circuit, alphas, betas, gammas [][]*big.Int, zx []*big.Int) (Setup, error) { |
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var setup Setup |
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var setup Setup |
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var err error |
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var err error |
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// generate random t value
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// generate random t value
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setup.Toxic.T, err = fqR.Rand() |
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setup.Toxic.T, err = Utils.FqR.Rand() |
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if err != nil { |
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if err != nil { |
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return Setup{}, err |
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return Setup{}, err |
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} |
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} |
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// k for calculating pi' and Vk
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// k for calculating pi' and Vk
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setup.Toxic.Ka, err = fqR.Rand() |
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setup.Toxic.Ka, err = Utils.FqR.Rand() |
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if err != nil { |
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if err != nil { |
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return Setup{}, err |
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return Setup{}, err |
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} |
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} |
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setup.Toxic.Kb, err = fqR.Rand() |
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setup.Toxic.Kb, err = Utils.FqR.Rand() |
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if err != nil { |
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if err != nil { |
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return Setup{}, err |
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return Setup{}, err |
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} |
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} |
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setup.Toxic.Kc, err = fqR.Rand() |
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setup.Toxic.Kc, err = Utils.FqR.Rand() |
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if err != nil { |
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if err != nil { |
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return Setup{}, err |
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return Setup{}, err |
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} |
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} |
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// generate Kβ (Kbeta) and Kγ (Kgamma)
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// generate Kβ (Kbeta) and Kγ (Kgamma)
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setup.Toxic.Kbeta, err = fqR.Rand() |
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setup.Toxic.Kbeta, err = Utils.FqR.Rand() |
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if err != nil { |
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if err != nil { |
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return Setup{}, err |
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return Setup{}, err |
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} |
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} |
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setup.Toxic.Kgamma, err = fqR.Rand() |
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setup.Toxic.Kgamma, err = Utils.FqR.Rand() |
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if err != nil { |
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if err != nil { |
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return Setup{}, err |
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return Setup{}, err |
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} |
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} |
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// generate ρ (Rho): ρA, ρB, ρC
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// generate ρ (Rho): ρA, ρB, ρC
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setup.Toxic.RhoA, err = fqR.Rand() |
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setup.Toxic.RhoA, err = Utils.FqR.Rand() |
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if err != nil { |
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if err != nil { |
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return Setup{}, err |
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return Setup{}, err |
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} |
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} |
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setup.Toxic.RhoB, err = fqR.Rand() |
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setup.Toxic.RhoB, err = Utils.FqR.Rand() |
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if err != nil { |
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if err != nil { |
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return Setup{}, err |
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return Setup{}, err |
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} |
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} |
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setup.Toxic.RhoC = fqR.Mul(setup.Toxic.RhoA, setup.Toxic.RhoB) |
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setup.Toxic.RhoC = Utils.FqR.Mul(setup.Toxic.RhoA, setup.Toxic.RhoB) |
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// encrypt t values with curve generators
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// encrypt t values with curve generators
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var gt1 [][3]*big.Int |
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var gt1 [][3]*big.Int |
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var gt2 [][3][2]*big.Int |
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var gt2 [][3][2]*big.Int |
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for i := 0; i < witnessLength; i++ { |
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for i := 0; i < witnessLength; i++ { |
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tPow := fqR.Exp(setup.Toxic.T, big.NewInt(int64(i))) |
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tEncr1 := bn.G1.MulScalar(bn.G1.G, tPow) |
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tPow := Utils.FqR.Exp(setup.Toxic.T, big.NewInt(int64(i))) |
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tEncr1 := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, tPow) |
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gt1 = append(gt1, tEncr1) |
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gt1 = append(gt1, tEncr1) |
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tEncr2 := bn.G2.MulScalar(bn.G2.G, tPow) |
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tEncr2 := Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, tPow) |
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gt2 = append(gt2, tEncr2) |
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gt2 = append(gt2, tEncr2) |
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} |
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} |
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// gt1: g1, g1*t, g1*t^2, g1*t^3, ...
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// gt1: g1, g1*t, g1*t^2, g1*t^3, ...
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@ -122,9 +148,9 @@ func GenerateTrustedSetup(bn bn128.Bn128, fqR fields.Fq, pf r1csqap.PolynomialFi |
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setup.G1T = gt1 |
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setup.G1T = gt1 |
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setup.G2T = gt2 |
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setup.G2T = gt2 |
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setup.Vk.Vka = bn.G2.MulScalar(bn.G2.G, setup.Toxic.Ka) |
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setup.Vk.Vkb = bn.G1.MulScalar(bn.G1.G, setup.Toxic.Kb) |
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setup.Vk.Vkc = bn.G2.MulScalar(bn.G2.G, setup.Toxic.Kc) |
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setup.Vk.Vka = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Ka) |
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setup.Vk.Vkb = Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, setup.Toxic.Kb) |
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setup.Vk.Vkc = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Kc) |
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/* |
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/* |
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Verification keys: |
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Verification keys: |
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@ -132,78 +158,78 @@ func GenerateTrustedSetup(bn bn128.Bn128, fqR fields.Fq, pf r1csqap.PolynomialFi |
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- Vk_betagamma2: setup.G2Kbg = g2 * Kbeta*Kgamma |
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- Vk_betagamma2: setup.G2Kbg = g2 * Kbeta*Kgamma |
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- Vk_gamma: setup.G2Kg = g2 * Kgamma |
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- Vk_gamma: setup.G2Kg = g2 * Kgamma |
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*/ |
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*/ |
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kbg := fqR.Mul(setup.Toxic.Kbeta, setup.Toxic.Kgamma) |
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setup.Vk.G1Kbg = bn.G1.MulScalar(bn.G1.G, kbg) |
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setup.Vk.G2Kbg = bn.G2.MulScalar(bn.G2.G, kbg) |
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setup.Vk.G2Kg = bn.G2.MulScalar(bn.G2.G, setup.Toxic.Kgamma) |
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kbg := Utils.FqR.Mul(setup.Toxic.Kbeta, setup.Toxic.Kgamma) |
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setup.Vk.G1Kbg = Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, kbg) |
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setup.Vk.G2Kbg = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, kbg) |
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setup.Vk.G2Kg = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Kgamma) |
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// for i := 0; i < circuit.NSignals; i++ {
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// for i := 0; i < circuit.NSignals; i++ {
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for i := 0; i < circuit.NVars; i++ { |
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for i := 0; i < circuit.NVars; i++ { |
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at := pf.Eval(alphas[i], setup.Toxic.T) |
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a := bn.G1.MulScalar(bn.G1.G, at) |
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at := Utils.PF.Eval(alphas[i], setup.Toxic.T) |
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a := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, at) |
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setup.Pk.A = append(setup.Pk.A, a) |
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setup.Pk.A = append(setup.Pk.A, a) |
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if i <= circuit.NPublic { |
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if i <= circuit.NPublic { |
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setup.Vk.A = append(setup.Vk.A, a) |
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setup.Vk.A = append(setup.Vk.A, a) |
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} |
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} |
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bt := pf.Eval(betas[i], setup.Toxic.T) |
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bg1 := bn.G1.MulScalar(bn.G1.G, bt) |
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bg2 := bn.G2.MulScalar(bn.G2.G, bt) |
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bt := Utils.PF.Eval(betas[i], setup.Toxic.T) |
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bg1 := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, bt) |
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bg2 := Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, bt) |
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setup.Pk.B = append(setup.Pk.B, bg2) |
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setup.Pk.B = append(setup.Pk.B, bg2) |
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ct := pf.Eval(gammas[i], setup.Toxic.T) |
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c := bn.G1.MulScalar(bn.G1.G, ct) |
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ct := Utils.PF.Eval(gammas[i], setup.Toxic.T) |
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c := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, ct) |
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setup.Pk.C = append(setup.Pk.C, c) |
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setup.Pk.C = append(setup.Pk.C, c) |
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kt := fqR.Add(fqR.Add(at, bt), ct) |
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k := bn.G1.Affine(bn.G1.MulScalar(bn.G1.G, kt)) |
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kt := Utils.FqR.Add(Utils.FqR.Add(at, bt), ct) |
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k := Utils.Bn.G1.Affine(Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, kt)) |
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ktest := bn.G1.Affine(bn.G1.Add(bn.G1.Add(a, bg1), c)) |
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if !bn.Fq2.Equal(k, ktest) { |
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ktest := Utils.Bn.G1.Affine(Utils.Bn.G1.Add(Utils.Bn.G1.Add(a, bg1), c)) |
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if !Utils.Bn.Fq2.Equal(k, ktest) { |
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os.Exit(1) |
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os.Exit(1) |
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return setup, err |
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return setup, err |
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} |
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} |
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setup.Pk.Ap = append(setup.Pk.Ap, bn.G1.MulScalar(a, setup.Toxic.Ka)) |
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setup.Pk.Bp = append(setup.Pk.Bp, bn.G1.MulScalar(bg1, setup.Toxic.Kb)) |
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setup.Pk.Cp = append(setup.Pk.Cp, bn.G1.MulScalar(c, setup.Toxic.Kc)) |
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k_ := bn.G1.MulScalar(bn.G1.G, kt) |
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setup.Pk.Kp = append(setup.Pk.Kp, bn.G1.MulScalar(k_, setup.Toxic.Kbeta)) |
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setup.Pk.Ap = append(setup.Pk.Ap, Utils.Bn.G1.MulScalar(a, setup.Toxic.Ka)) |
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setup.Pk.Bp = append(setup.Pk.Bp, Utils.Bn.G1.MulScalar(bg1, setup.Toxic.Kb)) |
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setup.Pk.Cp = append(setup.Pk.Cp, Utils.Bn.G1.MulScalar(c, setup.Toxic.Kc)) |
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k_ := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, kt) |
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setup.Pk.Kp = append(setup.Pk.Kp, Utils.Bn.G1.MulScalar(k_, setup.Toxic.Kbeta)) |
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} |
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} |
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setup.Vk.Vkz = bn.G2.MulScalar(bn.G2.G, pf.Eval(zx, setup.Toxic.T)) |
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setup.Vk.Vkz = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, Utils.PF.Eval(zx, setup.Toxic.T)) |
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return setup, nil |
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return setup, nil |
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} |
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} |
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// GenerateProofs generates all the parameters to proof the zkSNARK from the Circuit, Setup and the Witness
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// GenerateProofs generates all the parameters to proof the zkSNARK from the Circuit, Setup and the Witness
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func GenerateProofs(bn bn128.Bn128, f fields.Fq, circuit circuitcompiler.Circuit, setup Setup, hx []*big.Int, w []*big.Int) (Proof, error) { |
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func GenerateProofs(circuit circuitcompiler.Circuit, setup Setup, hx []*big.Int, w []*big.Int) (Proof, error) { |
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var proof Proof |
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var proof Proof |
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proof.PiA = [3]*big.Int{bn.G1.F.Zero(), bn.G1.F.Zero(), bn.G1.F.Zero()} |
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proof.PiAp = [3]*big.Int{bn.G1.F.Zero(), bn.G1.F.Zero(), bn.G1.F.Zero()} |
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proof.PiB = bn.Fq6.Zero() |
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proof.PiBp = [3]*big.Int{bn.G1.F.Zero(), bn.G1.F.Zero(), bn.G1.F.Zero()} |
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proof.PiC = [3]*big.Int{bn.G1.F.Zero(), bn.G1.F.Zero(), bn.G1.F.Zero()} |
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proof.PiCp = [3]*big.Int{bn.G1.F.Zero(), bn.G1.F.Zero(), bn.G1.F.Zero()} |
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proof.PiH = [3]*big.Int{bn.G1.F.Zero(), bn.G1.F.Zero(), bn.G1.F.Zero()} |
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proof.PiKp = [3]*big.Int{bn.G1.F.Zero(), bn.G1.F.Zero(), bn.G1.F.Zero()} |
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proof.PiA = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()} |
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proof.PiAp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()} |
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proof.PiB = Utils.Bn.Fq6.Zero() |
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proof.PiBp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()} |
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proof.PiC = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()} |
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proof.PiCp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()} |
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proof.PiH = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()} |
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proof.PiKp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()} |
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for i := circuit.NPublic + 1; i < circuit.NVars; i++ { |
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for i := circuit.NPublic + 1; i < circuit.NVars; i++ { |
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proof.PiA = bn.G1.Add(proof.PiA, bn.G1.MulScalar(setup.Pk.A[i], w[i])) |
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proof.PiAp = bn.G1.Add(proof.PiAp, bn.G1.MulScalar(setup.Pk.Ap[i], w[i])) |
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proof.PiA = Utils.Bn.G1.Add(proof.PiA, Utils.Bn.G1.MulScalar(setup.Pk.A[i], w[i])) |
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proof.PiAp = Utils.Bn.G1.Add(proof.PiAp, Utils.Bn.G1.MulScalar(setup.Pk.Ap[i], w[i])) |
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} |
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} |
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for i := 0; i < circuit.NVars; i++ { |
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for i := 0; i < circuit.NVars; i++ { |
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proof.PiB = bn.G2.Add(proof.PiB, bn.G2.MulScalar(setup.Pk.B[i], w[i])) |
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proof.PiBp = bn.G1.Add(proof.PiBp, bn.G1.MulScalar(setup.Pk.Bp[i], w[i])) |
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proof.PiB = Utils.Bn.G2.Add(proof.PiB, Utils.Bn.G2.MulScalar(setup.Pk.B[i], w[i])) |
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proof.PiBp = Utils.Bn.G1.Add(proof.PiBp, Utils.Bn.G1.MulScalar(setup.Pk.Bp[i], w[i])) |
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proof.PiC = bn.G1.Add(proof.PiC, bn.G1.MulScalar(setup.Pk.C[i], w[i])) |
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proof.PiCp = bn.G1.Add(proof.PiCp, bn.G1.MulScalar(setup.Pk.Cp[i], w[i])) |
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proof.PiC = Utils.Bn.G1.Add(proof.PiC, Utils.Bn.G1.MulScalar(setup.Pk.C[i], w[i])) |
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proof.PiCp = Utils.Bn.G1.Add(proof.PiCp, Utils.Bn.G1.MulScalar(setup.Pk.Cp[i], w[i])) |
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proof.PiKp = bn.G1.Add(proof.PiKp, bn.G1.MulScalar(setup.Pk.Kp[i], w[i])) |
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proof.PiKp = Utils.Bn.G1.Add(proof.PiKp, Utils.Bn.G1.MulScalar(setup.Pk.Kp[i], w[i])) |
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} |
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} |
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for i := 0; i < len(hx); i++ { |
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for i := 0; i < len(hx); i++ { |
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proof.PiH = bn.G1.Add(proof.PiH, bn.G1.MulScalar(setup.G1T[i], hx[i])) |
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proof.PiH = Utils.Bn.G1.Add(proof.PiH, Utils.Bn.G1.MulScalar(setup.G1T[i], hx[i])) |
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} |
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} |
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proof.PublicSignals = w[1 : circuit.NPublic+1] |
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proof.PublicSignals = w[1 : circuit.NPublic+1] |
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@ -211,64 +237,58 @@ func GenerateProofs(bn bn128.Bn128, f fields.Fq, circuit circuitcompiler.Circuit |
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} |
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} |
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// VerifyProof verifies over the BN128 the Pairings of the Proof
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// VerifyProof verifies over the BN128 the Pairings of the Proof
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func VerifyProof(bn bn128.Bn128, circuit circuitcompiler.Circuit, setup Setup, proof Proof) bool { |
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func VerifyProof(circuit circuitcompiler.Circuit, setup Setup, proof Proof) bool { |
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// e(piA, Va) == e(piA', g2)
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// e(piA, Va) == e(piA', g2)
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pairingPiaVa := bn.Pairing(proof.PiA, setup.Vk.Vka) |
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pairingPiapG2 := bn.Pairing(proof.PiAp, bn.G2.G) |
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if !bn.Fq12.Equal(pairingPiaVa, pairingPiapG2) { |
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pairingPiaVa := Utils.Bn.Pairing(proof.PiA, setup.Vk.Vka) |
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pairingPiapG2 := Utils.Bn.Pairing(proof.PiAp, Utils.Bn.G2.G) |
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if !Utils.Bn.Fq12.Equal(pairingPiaVa, pairingPiapG2) { |
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return false |
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return false |
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} else { |
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fmt.Println("✓ e(piA, Va) == e(piA', g2), valid knowledge commitment for A") |
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} |
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} |
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fmt.Println("✓ e(piA, Va) == e(piA', g2), valid knowledge commitment for A") |
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// e(Vb, piB) == e(piB', g2)
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// e(Vb, piB) == e(piB', g2)
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pairingVbPib := bn.Pairing(setup.Vk.Vkb, proof.PiB) |
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pairingPibpG2 := bn.Pairing(proof.PiBp, bn.G2.G) |
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if !bn.Fq12.Equal(pairingVbPib, pairingPibpG2) { |
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pairingVbPib := Utils.Bn.Pairing(setup.Vk.Vkb, proof.PiB) |
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pairingPibpG2 := Utils.Bn.Pairing(proof.PiBp, Utils.Bn.G2.G) |
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if !Utils.Bn.Fq12.Equal(pairingVbPib, pairingPibpG2) { |
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return false |
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return false |
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} else { |
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fmt.Println("✓ e(Vb, piB) == e(piB', g2), valid knowledge commitment for B") |
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} |
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} |
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fmt.Println("✓ e(Vb, piB) == e(piB', g2), valid knowledge commitment for B") |
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// e(piC, Vc) == e(piC', g2)
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// e(piC, Vc) == e(piC', g2)
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pairingPicVc := bn.Pairing(proof.PiC, setup.Vk.Vkc) |
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pairingPicpG2 := bn.Pairing(proof.PiCp, bn.G2.G) |
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if !bn.Fq12.Equal(pairingPicVc, pairingPicpG2) { |
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pairingPicVc := Utils.Bn.Pairing(proof.PiC, setup.Vk.Vkc) |
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pairingPicpG2 := Utils.Bn.Pairing(proof.PiCp, Utils.Bn.G2.G) |
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if !Utils.Bn.Fq12.Equal(pairingPicVc, pairingPicpG2) { |
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return false |
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return false |
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} else { |
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fmt.Println("✓ e(piC, Vc) == e(piC', g2), valid knowledge commitment for C") |
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} |
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} |
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fmt.Println("✓ e(piC, Vc) == e(piC', g2), valid knowledge commitment for C") |
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// Vkx, to then calculate Vkx+piA
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// Vkx, to then calculate Vkx+piA
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vkxpia := setup.Vk.A[0] |
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|
vkxpia := setup.Vk.A[0] |
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|
for i := 0; i < circuit.NPublic; i++ { |
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|
for i := 0; i < circuit.NPublic; i++ { |
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vkxpia = bn.G1.Add(vkxpia, bn.G1.MulScalar(setup.Vk.A[i+1], proof.PublicSignals[i])) |
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vkxpia = Utils.Bn.G1.Add(vkxpia, Utils.Bn.G1.MulScalar(setup.Vk.A[i+1], proof.PublicSignals[i])) |
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|
} |
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|
} |
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// e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2)
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// e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2)
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|
if !bn.Fq12.Equal( |
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|
|
bn.Pairing(bn.G1.Add(vkxpia, proof.PiA), proof.PiB), |
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|
|
bn.Fq12.Mul( |
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|
|
bn.Pairing(proof.PiH, setup.Vk.Vkz), |
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|
|
bn.Pairing(proof.PiC, bn.G2.G))) { |
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|
|
if !Utils.Bn.Fq12.Equal( |
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|
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|
|
Utils.Bn.Pairing(Utils.Bn.G1.Add(vkxpia, proof.PiA), proof.PiB), |
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|
|
|
|
Utils.Bn.Fq12.Mul( |
|
|
|
|
|
Utils.Bn.Pairing(proof.PiH, setup.Vk.Vkz), |
|
|
|
|
|
Utils.Bn.Pairing(proof.PiC, Utils.Bn.G2.G))) { |
|
|
return false |
|
|
return false |
|
|
} else { |
|
|
|
|
|
fmt.Println("✓ e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2), QAP disibility checked") |
|
|
|
|
|
} |
|
|
} |
|
|
|
|
|
fmt.Println("✓ e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2), QAP disibility checked") |
|
|
|
|
|
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|
|
// e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB)
|
|
|
// e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB)
|
|
|
// == e(piK, g2Kgamma)
|
|
|
// == e(piK, g2Kgamma)
|
|
|
piApiC := bn.G1.Add(bn.G1.Add(vkxpia, proof.PiA), proof.PiC) |
|
|
|
|
|
pairingPiACG2Kbg := bn.Pairing(piApiC, setup.Vk.G2Kbg) |
|
|
|
|
|
pairingG1KbgPiB := bn.Pairing(setup.Vk.G1Kbg, proof.PiB) |
|
|
|
|
|
pairingL := bn.Fq12.Mul(pairingPiACG2Kbg, pairingG1KbgPiB) |
|
|
|
|
|
pairingR := bn.Pairing(proof.PiKp, setup.Vk.G2Kg) |
|
|
|
|
|
if !bn.Fq12.Equal(pairingL, pairingR) { |
|
|
|
|
|
|
|
|
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) { |
|
|
return false |
|
|
return false |
|
|
} else { |
|
|
|
|
|
fmt.Println("✓ e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB) == e(piK, g2Kgamma)") |
|
|
|
|
|
} |
|
|
} |
|
|
|
|
|
fmt.Println("✓ e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB) == e(piK, g2Kgamma)") |
|
|
|
|
|
|
|
|
return true |
|
|
return true |
|
|
} |
|
|
} |