doing trusted setup

README.md

@@ -2,15 +2,13 @@
 
 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
 ```
 go test ./... -v
 ```
 
+
 ## 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
 - 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
 - R1CS to QAP
 ```go
-b0 := big.NewFloat(float64(0))
-b1 := big.NewFloat(float64(1))
-b5 := big.NewFloat(float64(5))
-a := [][]*big.Float{
-  []*big.Float{b0, b1, b0, b0, b0, b0},
-  []*big.Float{b0, b0, b0, b1, b0, b0},
-  []*big.Float{b0, b1, b0, b0, b1, b0},
-  []*big.Float{b5, b0, b0, b0, b0, b1},
+pf := 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.Float{
-  []*big.Float{b0, b1, b0, b0, b0, b0},
-  []*big.Float{b0, b1, b0, b0, b0, b0},
-  []*big.Float{b1, b0, b0, b0, b0, b0},
-  []*big.Float{b1, b0, b0, b0, b0, b0},
+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.Float{
-  []*big.Float{b0, b0, b0, b1, b0, b0},
-  []*big.Float{b0, b0, b0, b0, b1, b0},
-  []*big.Float{b0, b0, b0, b0, b0, b1},
-  []*big.Float{b0, b0, b1, 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},
 }
-alpha, beta, gamma, z := R1CSToQAP(a, b, c)
-fmt.Println(alpha)
-fmt.Println(beta)
-fmt.Println(gamma)
+alphas, betas, gammas, zx := pf.R1CSToQAP(a, b, c)
+fmt.Println(alphas)
+fmt.Println(betas)
+fmt.Println(gammas)
 fmt.Println(z)
-/*
-out:
-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]]
-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]]
-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]]
-z: [24 -50 35 -10 1]
-*/
+
+w := []*big.Int{b1, b3, b35, b9, b27, b30}
+ax, bx, cx, px := pf.CombinePolynomials(w, alphas, betas, gammas)
+fmt.Println(ax)
+fmt.Println(bx)
+fmt.Println(cx)
+fmt.Println(px)
+
+hx := pf.DivisorPolinomial(px, zx)
+fmt.Println(hx)
 ```
 
 ## Bn128
@@ -97,3 +105,11 @@ pB, err := bn128.Pairing(g1b, g2b)
 assert.Nil(t, err)
 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.

r1csqap/r1csqap.go

@@ -137,17 +137,17 @@ func (pf PolynomialField) R1CSToQAP(a, b, c [][]*big.Int) ([][]*big.Int, [][]*bi
 	aT := Transpose(a)
 	bT := Transpose(b)
 	cT := Transpose(c)
-	var alpha [][]*big.Int
+	var alphas [][]*big.Int
 	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++ {
-		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++ {
-		gamma = append(gamma, pf.LagrangeInterpolation(cT[i]))
+		gammas = append(gammas, pf.LagrangeInterpolation(cT[i]))
 	}
 	z := []*big.Int{big.NewInt(int64(1))}
 	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))
 		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
 	for i := 0; i < len(r); i++ {
 		m := pf.Mul([]*big.Int{r[i]}, ap[i])

r1csqap/r1csqap_test.go

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

r1csqapFloat/r1csqapFloat.go

@@ -137,17 +137,17 @@ func R1CSToQAP(a, b, c [][]*big.Float) ([][]*big.Float, [][]*big.Float, [][]*big
 	aT := Transpose(a)
 	bT := Transpose(b)
 	cT := Transpose(c)
-	var alpha [][]*big.Float
+	var alphas [][]*big.Float
 	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++ {
-		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++ {
-		gamma = append(gamma, LagrangeInterpolation(cT[i]))
+		gammas = append(gammas, LagrangeInterpolation(cT[i]))
 	}
 	z := []*big.Float{big.NewFloat(float64(1))}
 	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))
 		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
 	for i := 0; i < len(r); i++ {
 		m := PolMul([]*big.Float{r[i]}, ap[i])

r1csqapFloat/r1csqapFloat_test.go

@@ -108,21 +108,30 @@ func TestR1CSToQAP(t *testing.T) {
 		[]*big.Float{b0, b0, b0, b0, b0, b1},
 		[]*big.Float{b0, b0, b1, b0, b0, b0},
 	}
-	ap, bp, cp, z := R1CSToQAP(a, b, c)
-	fmt.Println(ap)
-	fmt.Println(bp)
-	fmt.Println(cp)
-	fmt.Println(z)
+	// alphas, betas, gammas
+	alphas, betas, gammas, zx := R1CSToQAP(a, b, c)
+	fmt.Println(alphas)
+	fmt.Println(betas)
+	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))}
-	assert.Equal(t, z, zexpected)
+	assert.Equal(t, zx, zexpected)
 
+	// witness
 	w := []*big.Float{b1, b3, b35, b9, b27, b30}
-	alpha, beta, gamma, px := SolPolynomials(w, ap, bp, cp)
-	fmt.Println(alpha)
-	fmt.Println(beta)
-	fmt.Println(gamma)
+	fmt.Print("w: ")
+	fmt.Println(w)
+	// QAP A(x), B(x), C(x)
+	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)
 
-	h := DivisorPolinomial(px, z)
-	fmt.Println(h)
+	hx := DivisorPolinomial(px, zx)
+	fmt.Print("H(x): ")
+	fmt.Println(hx)
+
 }

zk/zk.go

@@ -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
+}

zk/zk_test.go

@@ -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))
+}