proofs

README.md

@@ -112,4 +112,4 @@ 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.
+Thanks to [@jbaylina](https://github.com/jbaylina), [@bellesmarta](https://github.com/bellesmarta), [@adriamb](https://github.com/adriamb) for their explanations that helped to understand this a little bit.

zk/zk.go

@@ -8,22 +8,48 @@ import (
 	"github.com/arnaucube/go-snark/bn128"
 )
 
+type Setup struct {
+	T  *big.Int // trusted setup secret
+	Ka *big.Int // trusted setup
+	Kb *big.Int // trusted setup
+	Kc *big.Int // trusted setup
+
+	// public
+	G1T [][3]*big.Int    // t encrypted in G1 curve
+	G2T [][3][2]*big.Int // t encrypted in G2 curve
+}
+type Proof struct {
+	PiA  [3]*big.Int
+	PiAp [3]*big.Int
+	PiB  [3][2]*big.Int
+	PiBp [3][2]*big.Int
+	PiC  [3]*big.Int
+	PiCp [3]*big.Int
+	PiH  [3]*big.Int
+	Va   [3][2]*big.Int
+	Vb   [3][2]*big.Int
+	Vc   [3][2]*big.Int
+	Vz   [3][2]*big.Int
+}
+
 const bits = 512
 
-func GenerateTrustedSetup(bn bn128.Bn128, pollength int) ([][3]*big.Int, [][3][2]*big.Int, error) {
+func GenerateTrustedSetup(bn bn128.Bn128, pollength int) (Setup, error) {
+	var setup Setup
+	var err error
 	// generate random t value
-	t, err := rand.Prime(rand.Reader, bits)
+	setup.T, err = rand.Prime(rand.Reader, bits)
 	if err != nil {
-		return [][3]*big.Int{}, [][3][2]*big.Int{}, err
+		return Setup{}, err
 	}
 	fmt.Print("trusted t: ")
-	fmt.Println(t)
+	fmt.Println(setup.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)))
+		tPow := bn.Fq1.Exp(setup.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)
@@ -31,39 +57,103 @@ func GenerateTrustedSetup(bn bn128.Bn128, pollength int) ([][3]*big.Int, [][3][2
 	}
 	// gt1: g1, g1*t, g1*t^2, g1*t^3, ...
 	// gt2: g2, g2*t, g2*t^2, ...
-	return gt1, gt2, nil
+	setup.G1T = gt1
-}
+	setup.G2T = gt2
-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)
+	// k for pi'
+	setup.Ka, err = rand.Prime(rand.Reader, bits)
+	if err != nil {
+		return Setup{}, err
+	}
+	setup.Kb, err = rand.Prime(rand.Reader, bits)
+	if err != nil {
+		return Setup{}, err
+	}
+	setup.Kc, err = rand.Prime(rand.Reader, bits)
+	if err != nil {
+		return Setup{}, err
+	}
+
+	return setup, nil
+}
+
+func GenerateProofs(bn bn128.Bn128, setup Setup, ax, bx, cx, hx, zx []*big.Int) Proof {
+	var proof Proof
 
 	// g1*A(x)
-	g1At := [3]*big.Int{bn.G1.F.Zero(), bn.G1.F.Zero(), bn.G1.F.Zero()}
+	proof.PiA = [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])
+		m := bn.G1.MulScalar(setup.G1T[i], ax[i])
-		g1At = bn.G1.Add(g1At, m)
+		proof.PiA = bn.G1.Add(proof.PiA, 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)
 	}
+	proof.PiAp = bn.G1.MulScalar(proof.PiA, setup.Ka)
 
-	g1Ct := [3]*big.Int{bn.G1.F.Zero(), bn.G1.F.Zero(), bn.G1.F.Zero()}
+	// g1*B(x)
-	for i := 0; i < len(cx); i++ {
+	proof.PiB = bn.Fq6.Zero()
-		m := bn.G1.MulScalar(gt1[i], cx[i])
+	for i := 0; i < len(bx); i++ {
-		g1Ct = bn.G1.Add(g1Ct, m)
+		m := bn.G2.MulScalar(setup.G2T[i], bx[i])
+		proof.PiB = bn.G2.Add(proof.PiB, m)
 	}
+	proof.PiBp = bn.G2.MulScalar(proof.PiB, setup.Kb)
+
+	// g1*C(x)
+	proof.PiC = [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(setup.G1T[i], cx[i])
+		proof.PiC = bn.G1.Add(proof.PiC, m)
+	}
+	proof.PiCp = bn.G1.MulScalar(proof.PiC, setup.Kc)
+
 	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])
+		m := bn.G1.MulScalar(setup.G1T[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])
+		m := bn.G2.MulScalar(setup.G2T[i], zx[i])
 		g2Zt = bn.G2.Add(g2Zt, m)
 	}
+	proof.PiH = g1Ht
+	proof.Vz = g2Zt
+	proof.Va = bn.G2.MulScalar(bn.G2.G, setup.Ka)
+	proof.Vb = bn.G2.MulScalar(bn.G2.G, setup.Kb)
+	proof.Vc = bn.G2.MulScalar(bn.G2.G, setup.Kc)
 
-	return g1At, g2Bt, g1Ct, g1Ht, g2Zt
+	return proof
+}
+
+func VerifyProof(bn bn128.Bn128, setup Setup, proof Proof) bool {
+
+	// e(piA, Va) == e(piA', g2)
+	pairingPiaVa, err := bn.Pairing(proof.PiA, proof.Va)
+	if err != nil {
+		return false
+	}
+	pairingPiapG2, err := bn.Pairing(proof.PiAp, bn.G2.G)
+	if err != nil {
+		return false
+	}
+	if !bn.Fq12.Equal(pairingPiaVa, pairingPiapG2) {
+		return false
+	}
+
+	// e(piB, Vb) == e(piB', g2)
+
+	// e(piC, Vc) == e(piC', g2)
+	pairingPicVc, err := bn.Pairing(proof.PiC, proof.Vc)
+	if err != nil {
+		return false
+	}
+	pairingPicpG2, err := bn.Pairing(proof.PiCp, bn.G2.G)
+	if err != nil {
+		return false
+	}
+	if !bn.Fq12.Equal(pairingPicVc, pairingPicpG2) {
+		return false
+	}
+
+	//
+
+	return true
 }

zk/zk_test.go

@@ -64,33 +64,20 @@ func TestZk(t *testing.T) {
 	assert.Equal(t, abc, hz)
 
 	// calculate trusted setup
-	gt1, gt2, err := GenerateTrustedSetup(bn, len(ax))
+	setup, err := GenerateTrustedSetup(bn, len(ax))
 	assert.Nil(t, err)
 	fmt.Println("trusted setup:")
-	fmt.Println(gt1)
+	fmt.Println(setup.G1T)
-	fmt.Println(gt2)
+	fmt.Println(setup.G2T)
 
 	// 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)
+	proof := GenerateProofs(bn, setup, ax, bx, cx, hx, zx)
 	fmt.Println("proofs:")
-	fmt.Println(piA)
+	fmt.Println(proof.PiA)
-	fmt.Println(piB)
+	fmt.Println(proof.PiB)
-	fmt.Println(piC)
+	fmt.Println(proof.PiC)
-	fmt.Println(piH)
+	fmt.Println(proof.PiH)
-	fmt.Println(piZ)
+	fmt.Println(proof.Vz)
 
-	// pairing
+	assert.True(t, VerifyProof(bn, setup, proof))
-	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))
 }