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doing trusted setup

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arnaucube
2018-12-04 23:50:43 +01:00
parent 88c3e98cae
commit 6cd494f36f
7 changed files with 269 additions and 76 deletions
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80
README.md
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@@ -2,15 +2,13 @@
zk-SNARK library implementation in Go zk-SNARK library implementation in Go
Not finished, work in progress (doing this in my free time, so I don't have much time).
#### Test #### Test
``` ```
go test ./... -v go test ./... -v
``` ```
## R1CS to Quadratic Arithmetic Program ## R1CS to Quadratic Arithmetic Program
- `Succinct Non-Interactive Zero Knowledge for a von Neumann Architecture`, Eli Ben-Sasson, Alessandro Chiesa, Eran Tromer, Madars Virza https://eprint.iacr.org/2013/879.pdf - `Succinct Non-Interactive Zero Knowledge for a von Neumann Architecture`, Eli Ben-Sasson, Alessandro Chiesa, Eran Tromer, Madars Virza https://eprint.iacr.org/2013/879.pdf
- Vitalik Buterin blog post about QAP https://medium.com/@VitalikButerin/quadratic-arithmetic-programs-from-zero-to-hero-f6d558cea649 - Vitalik Buterin blog post about QAP https://medium.com/@VitalikButerin/quadratic-arithmetic-programs-from-zero-to-hero-f6d558cea649
@@ -20,39 +18,49 @@ go test ./... -v
#### Usage #### Usage
- R1CS to QAP - R1CS to QAP
```go ```go
b0 := big.NewFloat(float64(0)) pf := NewPolynomialField(f)
b1 := big.NewFloat(float64(1))
b5 := big.NewFloat(float64(5)) b0 := big.NewInt(int64(0))
a := [][]*big.Float{ b1 := big.NewInt(int64(1))
[]*big.Float{b0, b1, b0, b0, b0, b0}, b3 := big.NewInt(int64(3))
[]*big.Float{b0, b0, b0, b1, b0, b0}, b5 := big.NewInt(int64(5))
[]*big.Float{b0, b1, b0, b0, b1, b0}, b9 := big.NewInt(int64(9))
[]*big.Float{b5, b0, b0, b0, b0, b1}, b27 := big.NewInt(int64(27))
b30 := big.NewInt(int64(30))
b35 := big.NewInt(int64(35))
a := [][]*big.Int{
[]*big.Int{b0, b1, b0, b0, b0, b0},
[]*big.Int{b0, b0, b0, b1, b0, b0},
[]*big.Int{b0, b1, b0, b0, b1, b0},
[]*big.Int{b5, b0, b0, b0, b0, b1},
} }
b := [][]*big.Float{ b := [][]*big.Int{
[]*big.Float{b0, b1, b0, b0, b0, b0}, []*big.Int{b0, b1, b0, b0, b0, b0},
[]*big.Float{b0, b1, b0, b0, b0, b0}, []*big.Int{b0, b1, b0, b0, b0, b0},
[]*big.Float{b1, b0, b0, b0, b0, b0}, []*big.Int{b1, b0, b0, b0, b0, b0},
[]*big.Float{b1, b0, b0, b0, b0, b0}, []*big.Int{b1, b0, b0, b0, b0, b0},
} }
c := [][]*big.Float{ c := [][]*big.Int{
[]*big.Float{b0, b0, b0, b1, b0, b0}, []*big.Int{b0, b0, b0, b1, b0, b0},
[]*big.Float{b0, b0, b0, b0, b1, b0}, []*big.Int{b0, b0, b0, b0, b1, b0},
[]*big.Float{b0, b0, b0, b0, b0, b1}, []*big.Int{b0, b0, b0, b0, b0, b1},
[]*big.Float{b0, b0, b1, b0, b0, b0}, []*big.Int{b0, b0, b1, b0, b0, b0},
} }
alpha, beta, gamma, z := R1CSToQAP(a, b, c) alphas, betas, gammas, zx := pf.R1CSToQAP(a, b, c)
fmt.Println(alpha) fmt.Println(alphas)
fmt.Println(beta) fmt.Println(betas)
fmt.Println(gamma) fmt.Println(gammas)
fmt.Println(z) fmt.Println(z)
/*
out: w := []*big.Int{b1, b3, b35, b9, b27, b30}
alpha: [[-5 9.166666666666666 -5 0.8333333333333334] [8 -11.333333333333332 5 -0.6666666666666666] [0 0 0 0] [-6 9.5 -4 0.5] [4 -7 3.5 -0.5] [-1 1.8333333333333333 -1 0.16666666666666666]] ax, bx, cx, px := pf.CombinePolynomials(w, alphas, betas, gammas)
beta: [[3 -5.166666666666667 2.5 -0.33333333333333337] [-2 5.166666666666667 -2.5 0.33333333333333337] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0]] fmt.Println(ax)
gamma: [[0 0 0 0] [0 0 0 0] [-1 1.8333333333333333 -1 0.16666666666666666] [4 -4.333333333333333 1.5 -0.16666666666666666] [-6 9.5 -4 0.5] [4 -7 3.5 -0.5]] fmt.Println(bx)
z: [24 -50 35 -10 1] fmt.Println(cx)
*/ fmt.Println(px)
hx := pf.DivisorPolinomial(px, zx)
fmt.Println(hx)
``` ```
## Bn128 ## Bn128
@@ -97,3 +105,11 @@ pB, err := bn128.Pairing(g1b, g2b)
assert.Nil(t, err) assert.Nil(t, err)
assert.True(t, bn128.Fq12.Equal(pA, pB)) assert.True(t, bn128.Fq12.Equal(pA, pB))
``` ```
---
## Caution
Not finished, work in progress (implementing this in my free time to understand it better, so I don't have much time).
Thanks to @jbaylina, @bellesmarta, @adriamb for their explanations that helped to understand a little bit this.

16
r1csqap/r1csqap.go
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@@ -137,17 +137,17 @@ func (pf PolynomialField) R1CSToQAP(a, b, c [][]*big.Int) ([][]*big.Int, [][]*bi
aT := Transpose(a) aT := Transpose(a)
bT := Transpose(b) bT := Transpose(b)
cT := Transpose(c) cT := Transpose(c)
var alpha [][]*big.Int var alphas [][]*big.Int
for i := 0; i < len(aT); i++ { for i := 0; i < len(aT); i++ {
alpha = append(alpha, pf.LagrangeInterpolation(aT[i])) alphas = append(alphas, pf.LagrangeInterpolation(aT[i]))
} }
var beta [][]*big.Int var betas [][]*big.Int
for i := 0; i < len(bT); i++ { for i := 0; i < len(bT); i++ {
beta = append(beta, pf.LagrangeInterpolation(bT[i])) betas = append(betas, pf.LagrangeInterpolation(bT[i]))
} }
var gamma [][]*big.Int var gammas [][]*big.Int
for i := 0; i < len(cT); i++ { for i := 0; i < len(cT); i++ {
gamma = append(gamma, pf.LagrangeInterpolation(cT[i])) gammas = append(gammas, pf.LagrangeInterpolation(cT[i]))
} }
z := []*big.Int{big.NewInt(int64(1))} z := []*big.Int{big.NewInt(int64(1))}
for i := 1; i < len(aT[0])+1; i++ { for i := 1; i < len(aT[0])+1; i++ {
@@ -155,10 +155,10 @@ func (pf PolynomialField) R1CSToQAP(a, b, c [][]*big.Int) ([][]*big.Int, [][]*bi
b1 := big.NewInt(int64(1)) b1 := big.NewInt(int64(1))
z = pf.Mul(z, []*big.Int{ineg, b1}) z = pf.Mul(z, []*big.Int{ineg, b1})
} }
return alpha, beta, gamma, z return alphas, betas, gammas, z
} }
func (pf PolynomialField) SolPolynomials(r []*big.Int, ap, bp, cp [][]*big.Int) ([]*big.Int, []*big.Int, []*big.Int, []*big.Int) { func (pf PolynomialField) CombinePolynomials(r []*big.Int, ap, bp, cp [][]*big.Int) ([]*big.Int, []*big.Int, []*big.Int, []*big.Int) {
var alpha []*big.Int var alpha []*big.Int
for i := 0; i < len(r); i++ { for i := 0; i < len(r); i++ {
m := pf.Mul([]*big.Int{r[i]}, ap[i]) m := pf.Mul([]*big.Int{r[i]}, ap[i])

35
r1csqap/r1csqap_test.go
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@@ -132,27 +132,30 @@ func TestR1CSToQAP(t *testing.T) {
[]*big.Int{b0, b0, b0, b0, b0, b1}, []*big.Int{b0, b0, b0, b0, b0, b1},
[]*big.Int{b0, b0, b1, b0, b0, b0}, []*big.Int{b0, b0, b1, b0, b0, b0},
} }
ap, bp, cp, z := pf.R1CSToQAP(a, b, c) alphas, betas, gammas, zx := pf.R1CSToQAP(a, b, c)
fmt.Println(ap) fmt.Println(alphas)
fmt.Println(bp) fmt.Println(betas)
fmt.Println(cp) fmt.Println(gammas)
fmt.Println(z) fmt.Print("Z(x): ")
fmt.Println(zx)
w := []*big.Int{b1, b3, b35, b9, b27, b30} w := []*big.Int{b1, b3, b35, b9, b27, b30}
alpha, beta, gamma, px := pf.SolPolynomials(w, ap, bp, cp) ax, bx, cx, px := pf.CombinePolynomials(w, alphas, betas, gammas)
fmt.Println(alpha) fmt.Println(ax)
fmt.Println(beta) fmt.Println(bx)
fmt.Println(gamma) fmt.Println(cx)
fmt.Println(px) fmt.Println(px)
h := pf.DivisorPolinomial(px, z) hx := pf.DivisorPolinomial(px, zx)
fmt.Println(h) fmt.Println(hx)
// h==px/z so px==h*z // hx==px/zx so px==hx*zx
assert.Equal(t, px, pf.Mul(h, z)) assert.Equal(t, px, pf.Mul(hx, zx))
// a(x) * b(x) - c(x) == h * z(x) // p(x) = a(x) * b(x) - c(x) == h(x) * z(x)
abc := pf.Sub(pf.Mul(alpha, beta), gamma) abc := pf.Sub(pf.Mul(ax, bx), cx)
hz := pf.Mul(h, z) assert.Equal(t, abc, px)
hz := pf.Mul(hx, zx)
assert.Equal(t, abc, hz) assert.Equal(t, abc, hz)
} }

16
r1csqapFloat/r1csqapFloat.go
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@@ -137,17 +137,17 @@ func R1CSToQAP(a, b, c [][]*big.Float) ([][]*big.Float, [][]*big.Float, [][]*big
aT := Transpose(a) aT := Transpose(a)
bT := Transpose(b) bT := Transpose(b)
cT := Transpose(c) cT := Transpose(c)
var alpha [][]*big.Float var alphas [][]*big.Float
for i := 0; i < len(aT); i++ { for i := 0; i < len(aT); i++ {
alpha = append(alpha, LagrangeInterpolation(aT[i])) alphas = append(alphas, LagrangeInterpolation(aT[i]))
} }
var beta [][]*big.Float var betas [][]*big.Float
for i := 0; i < len(bT); i++ { for i := 0; i < len(bT); i++ {
beta = append(beta, LagrangeInterpolation(bT[i])) betas = append(betas, LagrangeInterpolation(bT[i]))
} }
var gamma [][]*big.Float var gammas [][]*big.Float
for i := 0; i < len(cT); i++ { for i := 0; i < len(cT); i++ {
gamma = append(gamma, LagrangeInterpolation(cT[i])) gammas = append(gammas, LagrangeInterpolation(cT[i]))
} }
z := []*big.Float{big.NewFloat(float64(1))} z := []*big.Float{big.NewFloat(float64(1))}
for i := 1; i < len(aT[0])+1; i++ { for i := 1; i < len(aT[0])+1; i++ {
@@ -155,10 +155,10 @@ func R1CSToQAP(a, b, c [][]*big.Float) ([][]*big.Float, [][]*big.Float, [][]*big
b1 := big.NewFloat(float64(1)) b1 := big.NewFloat(float64(1))
z = PolMul(z, []*big.Float{ineg, b1}) z = PolMul(z, []*big.Float{ineg, b1})
} }
return alpha, beta, gamma, z return alphas, betas, gammas, z
} }
func SolPolynomials(r []*big.Float, ap, bp, cp [][]*big.Float) ([]*big.Float, []*big.Float, []*big.Float, []*big.Float) { func CombinePolynomials(r []*big.Float, ap, bp, cp [][]*big.Float) ([]*big.Float, []*big.Float, []*big.Float, []*big.Float) {
var alpha []*big.Float var alpha []*big.Float
for i := 0; i < len(r); i++ { for i := 0; i < len(r); i++ {
m := PolMul([]*big.Float{r[i]}, ap[i]) m := PolMul([]*big.Float{r[i]}, ap[i])

33
r1csqapFloat/r1csqapFloat_test.go
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@@ -108,21 +108,30 @@ func TestR1CSToQAP(t *testing.T) {
[]*big.Float{b0, b0, b0, b0, b0, b1}, []*big.Float{b0, b0, b0, b0, b0, b1},
[]*big.Float{b0, b0, b1, b0, b0, b0}, []*big.Float{b0, b0, b1, b0, b0, b0},
} }
ap, bp, cp, z := R1CSToQAP(a, b, c) // alphas, betas, gammas
fmt.Println(ap) alphas, betas, gammas, zx := R1CSToQAP(a, b, c)
fmt.Println(bp) fmt.Println(alphas)
fmt.Println(cp) fmt.Println(betas)
fmt.Println(z) fmt.Println(gammas)
fmt.Print("Z(x): ")
fmt.Println(zx)
zexpected := []*big.Float{big.NewFloat(float64(24)), big.NewFloat(float64(-50)), big.NewFloat(float64(35)), big.NewFloat(float64(-10)), big.NewFloat(float64(1))} zexpected := []*big.Float{big.NewFloat(float64(24)), big.NewFloat(float64(-50)), big.NewFloat(float64(35)), big.NewFloat(float64(-10)), big.NewFloat(float64(1))}
assert.Equal(t, z, zexpected) assert.Equal(t, zx, zexpected)
// witness
w := []*big.Float{b1, b3, b35, b9, b27, b30} w := []*big.Float{b1, b3, b35, b9, b27, b30}
alpha, beta, gamma, px := SolPolynomials(w, ap, bp, cp) fmt.Print("w: ")
fmt.Println(alpha) fmt.Println(w)
fmt.Println(beta) // QAP A(x), B(x), C(x)
fmt.Println(gamma) ax, bx, cx, px := CombinePolynomials(w, alphas, betas, gammas)
fmt.Print("A(x), B(x), C(x), P(x): ")
fmt.Println(ax)
fmt.Println(bx)
fmt.Println(cx)
fmt.Println(px) fmt.Println(px)
h := DivisorPolinomial(px, z) hx := DivisorPolinomial(px, zx)
fmt.Println(h) fmt.Print("H(x): ")
fmt.Println(hx)
} }

69
zk/zk.go Normal file
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@@ -0,0 +1,69 @@
package zk
import (
"crypto/rand"
"fmt"
"math/big"
"github.com/arnaucube/go-snark/bn128"
)
const bits = 512
func GenerateTrustedSetup(bn bn128.Bn128, pollength int) ([][3]*big.Int, [][3][2]*big.Int, error) {
// generate random t value
t, err := rand.Prime(rand.Reader, bits)
if err != nil {
return [][3]*big.Int{}, [][3][2]*big.Int{}, err
}
fmt.Print("trusted t: ")
fmt.Println(t)
// encrypt t values with curve generators
var gt1 [][3]*big.Int
var gt2 [][3][2]*big.Int
for i := 0; i < pollength; i++ {
tPow := bn.Fq1.Exp(t, big.NewInt(int64(i)))
tEncr1 := bn.G1.MulScalar(bn.G1.G, tPow)
gt1 = append(gt1, tEncr1)
tEncr2 := bn.G2.MulScalar(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, ...
return gt1, gt2, nil
}
func GenerateProofs(bn bn128.Bn128, gt1 [][3]*big.Int, gt2 [][3][2]*big.Int, ax, bx, cx, hx, zx []*big.Int) ([3]*big.Int, [3][2]*big.Int, [3]*big.Int, [3]*big.Int, [3][2]*big.Int) {
// multiply g1*A(x), g2*B(x), g1*C(x), g1*H(x)
// g1*A(x)
g1At := [3]*big.Int{bn.G1.F.Zero(), bn.G1.F.Zero(), bn.G1.F.Zero()}
for i := 0; i < len(ax); i++ {
m := bn.G1.MulScalar(gt1[i], ax[i])
g1At = bn.G1.Add(g1At, m)
}
g2Bt := bn.Fq6.Zero()
for i := 0; i < len(bx); i++ {
m := bn.G2.MulScalar(gt2[i], bx[i])
g2Bt = bn.G2.Add(g2Bt, m)
}
g1Ct := [3]*big.Int{bn.G1.F.Zero(), bn.G1.F.Zero(), bn.G1.F.Zero()}
for i := 0; i < len(cx); i++ {
m := bn.G1.MulScalar(gt1[i], cx[i])
g1Ct = bn.G1.Add(g1Ct, m)
}
g1Ht := [3]*big.Int{bn.G1.F.Zero(), bn.G1.F.Zero(), bn.G1.F.Zero()}
for i := 0; i < len(hx); i++ {
m := bn.G1.MulScalar(gt1[i], hx[i])
g1Ht = bn.G1.Add(g1Ht, m)
}
g2Zt := bn.Fq6.Zero()
for i := 0; i < len(bx); i++ {
m := bn.G2.MulScalar(gt2[i], zx[i])
g2Zt = bn.G2.Add(g2Zt, m)
}
return g1At, g2Bt, g1Ct, g1Ht, g2Zt
}

96
zk/zk_test.go Normal file
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@@ -0,0 +1,96 @@
package zk
import (
"fmt"
"math/big"
"testing"
"github.com/arnaucube/go-snark/bn128"
"github.com/arnaucube/go-snark/fields"
"github.com/arnaucube/go-snark/r1csqap"
"github.com/stretchr/testify/assert"
)
func TestZk(t *testing.T) {
bn, err := bn128.NewBn128()
assert.Nil(t, err)
// new Finite Field
f := fields.NewFq(bn.R)
// new Polynomial Field
pf := r1csqap.NewPolynomialField(f)
b0 := big.NewInt(int64(0))
b1 := big.NewInt(int64(1))
b3 := big.NewInt(int64(3))
b5 := big.NewInt(int64(5))
b9 := big.NewInt(int64(9))
b27 := big.NewInt(int64(27))
b30 := big.NewInt(int64(30))
b35 := big.NewInt(int64(35))
a := [][]*big.Int{
[]*big.Int{b0, b1, b0, b0, b0, b0},
[]*big.Int{b0, b0, b0, b1, b0, b0},
[]*big.Int{b0, b1, b0, b0, b1, b0},
[]*big.Int{b5, b0, b0, b0, b0, b1},
}
b := [][]*big.Int{
[]*big.Int{b0, b1, b0, b0, b0, b0},
[]*big.Int{b0, b1, b0, b0, b0, b0},
[]*big.Int{b1, b0, b0, b0, b0, b0},
[]*big.Int{b1, b0, b0, b0, b0, b0},
}
c := [][]*big.Int{
[]*big.Int{b0, b0, b0, b1, b0, b0},
[]*big.Int{b0, b0, b0, b0, b1, b0},
[]*big.Int{b0, b0, b0, b0, b0, b1},
[]*big.Int{b0, b0, b1, b0, b0, b0},
}
alphas, betas, gammas, zx := pf.R1CSToQAP(a, b, c)
w := []*big.Int{b1, b3, b35, b9, b27, b30}
ax, bx, cx, px := pf.CombinePolynomials(w, alphas, betas, gammas)
hx := pf.DivisorPolinomial(px, zx)
// hx==px/zx so px==hx*zx
assert.Equal(t, px, pf.Mul(hx, zx))
// p(x) = a(x) * b(x) - c(x) == h(x) * z(x)
abc := pf.Sub(pf.Mul(ax, bx), cx)
assert.Equal(t, abc, px)
hz := pf.Mul(hx, zx)
assert.Equal(t, abc, hz)
// calculate trusted setup
gt1, gt2, err := GenerateTrustedSetup(bn, len(ax))
assert.Nil(t, err)
fmt.Println("trusted setup:")
fmt.Println(gt1)
fmt.Println(gt2)
// piA = g1 * A(t), piB = g2 * B(t), piC = g1 * C(t), piH = g1 * H(t)
piA, piB, piC, piH, piZ := GenerateProofs(bn, gt1, gt2, ax, bx, cx, hx, zx)
fmt.Println("proofs:")
fmt.Println(piA)
fmt.Println(piB)
fmt.Println(piC)
fmt.Println(piH)
fmt.Println(piZ)
// pairing
fmt.Println("pairing")
pairingAB, err := bn.Pairing(piA, piB)
assert.Nil(t, err)
pairingCg2, err := bn.Pairing(piC, bn.G2.G)
assert.Nil(t, err)
pairingLeft := bn.Fq12.Div(pairingAB, pairingCg2)
pairingHg2Z, err := bn.Pairing(piH, piZ)
assert.Nil(t, err)
fmt.Println(bn.Fq12.Affine(pairingLeft))
fmt.Println(bn.Fq12.Affine(pairingHg2Z))
assert.True(t, bn.Fq12.Equal(pairingLeft, pairingHg2Z))
}