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// implementation of https://eprint.iacr.org/2016/260.pdf
package groth16
import (
"fmt"
"math/big"
"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
Kalpha *big.Int
Kbeta *big.Int
Kgamma *big.Int
Kdelta *big.Int
}
// public
Pk struct { // Proving Key
BACDelta [][3]*big.Int // {( βui(x)+αvi(x)+wi(x) ) / δ } from l+1 to m
Z []*big.Int
G1 struct {
Alpha [3]*big.Int
Beta [3]*big.Int
Delta [3]*big.Int
At [][3]*big.Int // {a(τ)} from 0 to m
BACGamma [][3]*big.Int // {( βui(x)+αvi(x)+wi(x) ) / γ } from 0 to m
}
G2 struct {
Beta [3][2]*big.Int
Gamma [3][2]*big.Int
Delta [3][2]*big.Int
BACGamma [][3][2]*big.Int // {( βui(x)+αvi(x)+wi(x) ) / γ } from 0 to m
}
PowersTauDelta [][3]*big.Int // powers of τ encrypted in G1 curve, divided by δ
}
Vk struct {
IC [][3]*big.Int
G1 struct {
Alpha [3]*big.Int
}
G2 struct {
Beta [3][2]*big.Int
Gamma [3][2]*big.Int
Delta [3][2]*big.Int
}
}
}
// Proof contains the parameters to proof the zkSNARK
type Proof struct {
PiA [3]*big.Int
PiB [3][2]*big.Int
PiC [3]*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
// generate random t value
setup.Toxic.T, err = Utils.FqR.Rand()
if err != nil {
return Setup{}, err
}
setup.Toxic.Kalpha, err = Utils.FqR.Rand()
if err != nil {
return Setup{}, err
}
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
}
setup.Toxic.Kdelta, err = Utils.FqR.Rand()
if err != nil {
return Setup{}, err
}
// z pol
zpol := []*big.Int{big.NewInt(int64(1))}
for i := 1; i < len(alphas)-1; i++ {
zpol = Utils.PF.Mul(
zpol,
[]*big.Int{
Utils.FqR.Neg(
big.NewInt(int64(i))),
big.NewInt(int64(1)),
})
}
setup.Pk.Z = zpol
zt := Utils.PF.Eval(zpol, setup.Toxic.T)
invDelta := Utils.FqR.Inverse(setup.Toxic.Kdelta)
ztinvDelta := Utils.FqR.Mul(invDelta, zt)
// encrypt t values with curve generators
// powers of tau divided by delta
var ptd [][3]*big.Int
ini := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, ztinvDelta)
ptd = append(ptd, ini)
tEncr := setup.Toxic.T
for i := 1; i < len(zpol); i++ {
ptd = append(ptd, Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, Utils.FqR.Mul(tEncr, ztinvDelta)))
tEncr = Utils.FqR.Mul(tEncr, setup.Toxic.T)
}
// powers of τ encrypted in G1 curve, divided by δ
// (G1 * τ) / δ
setup.Pk.PowersTauDelta = ptd
setup.Pk.G1.Alpha = Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, setup.Toxic.Kalpha)
setup.Pk.G1.Beta = Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, setup.Toxic.Kbeta)
setup.Pk.G1.Delta = Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, setup.Toxic.Kdelta)
setup.Pk.G2.Beta = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Kbeta)
setup.Pk.G2.Delta = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Kdelta)
setup.Vk.G1.Alpha = Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, setup.Toxic.Kalpha)
setup.Vk.G2.Beta = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Kbeta)
setup.Vk.G2.Gamma = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Kgamma)
setup.Vk.G2.Delta = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Kdelta)
for i := 0; i < len(circuit.Signals); i++ {
// Pk.G1.At: {a(τ)} from 0 to m
at := Utils.PF.Eval(alphas[i], setup.Toxic.T)
a := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, at)
setup.Pk.G1.At = append(setup.Pk.G1.At, a)
bt := Utils.PF.Eval(betas[i], setup.Toxic.T)
g1bt := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, bt)
g2bt := Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, bt)
// G1.BACGamma: {( βui(x)+αvi(x)+wi(x) ) / γ } from 0 to m in G1
setup.Pk.G1.BACGamma = append(setup.Pk.G1.BACGamma, g1bt)
// G2.BACGamma: {( βui(x)+αvi(x)+wi(x) ) / γ } from 0 to m in G2
setup.Pk.G2.BACGamma = append(setup.Pk.G2.BACGamma, g2bt)
}
zero3 := [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
for i := 0; i < circuit.NPublic+1; i++ {
setup.Pk.BACDelta = append(setup.Pk.BACDelta, zero3)
}
for i := circuit.NPublic + 1; i < circuit.NVars; i++ {
// TODO calculate all at, bt, ct outside, to avoid repeating calculations
at := Utils.PF.Eval(alphas[i], setup.Toxic.T)
bt := Utils.PF.Eval(betas[i], setup.Toxic.T)
ct := Utils.PF.Eval(gammas[i], setup.Toxic.T)
c := Utils.FqR.Mul(
invDelta,
Utils.FqR.Add(
Utils.FqR.Add(
Utils.FqR.Mul(at, setup.Toxic.Kbeta),
Utils.FqR.Mul(bt, setup.Toxic.Kalpha),
),
ct,
),
)
g1c := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, c)
// Pk.BACDelta: {( βui(x)+αvi(x)+wi(x) ) / δ } from l+1 to m
setup.Pk.BACDelta = append(setup.Pk.BACDelta, g1c)
}
for i := 0; i <= circuit.NPublic; i++ {
at := Utils.PF.Eval(alphas[i], setup.Toxic.T)
bt := Utils.PF.Eval(betas[i], setup.Toxic.T)
ct := Utils.PF.Eval(gammas[i], setup.Toxic.T)
ic := Utils.FqR.Mul(
Utils.FqR.Inverse(setup.Toxic.Kgamma),
Utils.FqR.Add(
Utils.FqR.Add(
Utils.FqR.Mul(at, setup.Toxic.Kbeta),
Utils.FqR.Mul(bt, setup.Toxic.Kalpha),
),
ct,
),
)
g1ic := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, ic)
// used in verifier
setup.Vk.IC = append(setup.Vk.IC, g1ic)
}
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.PiB = Utils.Bn.Fq6.Zero()
proof.PiC = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
r, err := Utils.FqR.Rand()
if err != nil {
return Proof{}, err
}
s, err := Utils.FqR.Rand()
if err != nil {
return Proof{}, err
}
// piBG1 will hold all the same than proof.PiB but in G1 curve
piBG1 := [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
for i := 0; i < circuit.NVars; i++ {
proof.PiA = Utils.Bn.G1.Add(proof.PiA, Utils.Bn.G1.MulScalar(setup.Pk.G1.At[i], w[i]))
piBG1 = Utils.Bn.G1.Add(piBG1, Utils.Bn.G1.MulScalar(setup.Pk.G1.BACGamma[i], w[i]))
proof.PiB = Utils.Bn.G2.Add(proof.PiB, Utils.Bn.G2.MulScalar(setup.Pk.G2.BACGamma[i], w[i]))
}
for i := circuit.NPublic + 1; i < circuit.NVars; i++ {
proof.PiC = Utils.Bn.G1.Add(proof.PiC, Utils.Bn.G1.MulScalar(setup.Pk.BACDelta[i], w[i]))
}
// piA = (Σ from 0 to m (pk.A * w[i])) + pk.Alpha1 + r * δ
proof.PiA = Utils.Bn.G1.Add(proof.PiA, setup.Pk.G1.Alpha)
deltaR := Utils.Bn.G1.MulScalar(setup.Pk.G1.Delta, r)
proof.PiA = Utils.Bn.G1.Add(proof.PiA, deltaR)
// piBG1 = (Σ from 0 to m (pk.B1 * w[i])) + pk.g1.Beta + s * δ
// piB = piB2 = (Σ from 0 to m (pk.B2 * w[i])) + pk.g2.Beta + s * δ
piBG1 = Utils.Bn.G1.Add(piBG1, setup.Pk.G1.Beta)
proof.PiB = Utils.Bn.G2.Add(proof.PiB, setup.Pk.G2.Beta)
deltaSG1 := Utils.Bn.G1.MulScalar(setup.Pk.G1.Delta, s)
piBG1 = Utils.Bn.G1.Add(piBG1, deltaSG1)
deltaSG2 := Utils.Bn.G2.MulScalar(setup.Pk.G2.Delta, s)
proof.PiB = Utils.Bn.G2.Add(proof.PiB, deltaSG2)
hx := Utils.PF.DivisorPolynomial(px, setup.Pk.Z) // maybe move this calculation to a previous step
// piC = (Σ from l+1 to m (w[i] * (pk.g1.Beta + pk.g1.Alpha + pk.C)) + h(tau)) / δ) + piA*s + r*piB - r*s*δ
for i := 0; i < len(hx); i++ {
proof.PiC = Utils.Bn.G1.Add(proof.PiC, Utils.Bn.G1.MulScalar(setup.Pk.PowersTauDelta[i], hx[i]))
}
proof.PiC = Utils.Bn.G1.Add(proof.PiC, Utils.Bn.G1.MulScalar(proof.PiA, s))
proof.PiC = Utils.Bn.G1.Add(proof.PiC, Utils.Bn.G1.MulScalar(piBG1, r))
negRS := Utils.FqR.Neg(Utils.FqR.Mul(r, s))
proof.PiC = Utils.Bn.G1.Add(proof.PiC, Utils.Bn.G1.MulScalar(setup.Pk.G1.Delta, negRS))
return proof, nil
}
// VerifyProof verifies over the BN128 the Pairings of the Proof
func VerifyProof(setup Setup, proof Proof, publicSignals []*big.Int, debug bool) bool {
icPubl := setup.Vk.IC[0]
for i := 0; i < len(publicSignals); i++ {
icPubl = Utils.Bn.G1.Add(icPubl, Utils.Bn.G1.MulScalar(setup.Vk.IC[i+1], publicSignals[i]))
}
if !Utils.Bn.Fq12.Equal(
Utils.Bn.Pairing(proof.PiA, proof.PiB),
Utils.Bn.Fq12.Mul(
Utils.Bn.Pairing(setup.Vk.G1.Alpha, setup.Vk.G2.Beta),
Utils.Bn.Fq12.Mul(
Utils.Bn.Pairing(icPubl, setup.Vk.G2.Gamma),
Utils.Bn.Pairing(proof.PiC, setup.Vk.G2.Delta)))) {
if debug {
fmt.Println("❌ groth16 verification not passed")
}
return false
}
if debug {
fmt.Println("✓ groth16 verification passed")
}
return true
}