perf(bn254): include G2 membership check in ML (#1387)

Author: yelhousni

internal/stats/latest_stats.csv

@@ -181,14 +181,14 @@ pairing_bls24315,bls24_315,plonk,0,0
 pairing_bls24315,bls24_317,plonk,0,0
 pairing_bls24315,bw6_761,plonk,0,0
 pairing_bls24315,bw6_633,plonk,141249,141249
-pairing_bn254,bn254,groth16,604783,990919
+pairing_bn254,bn254,groth16,607378,995098
 pairing_bn254,bls12_377,groth16,0,0
 pairing_bn254,bls12_381,groth16,0,0
 pairing_bn254,bls24_315,groth16,0,0
 pairing_bn254,bls24_317,groth16,0,0
 pairing_bn254,bw6_761,groth16,0,0
 pairing_bn254,bw6_633,groth16,0,0
-pairing_bn254,bn254,plonk,2319665,2030447
+pairing_bn254,bn254,plonk,2329131,2039205
 pairing_bn254,bls12_377,plonk,0,0
 pairing_bn254,bls12_381,plonk,0,0
 pairing_bn254,bls24_315,plonk,0,0

std/algebra/emulated/sw_bn254/g2.go

@@ -101,7 +101,7 @@ func (g2 *G2) phi(q *G2Affine) *G2Affine {
 	return &G2Affine{
 		P: g2AffP{
 			X: *x,
-			Y: q.P.Y,
+			Y: *g2.Ext2.Neg(&q.P.Y),
 		},
 	}
 }

std/algebra/emulated/sw_bn254/g2_test.go

@@ -126,7 +126,6 @@ type endomorphismG2Circuit struct {
 func (c *endomorphismG2Circuit) Define(api frontend.API) error {
 	g2 := NewG2(api)
 	res1 := g2.phi(&c.In1)
-	res1 = g2.neg(res1)
 	res2 := g2.psi(&c.In1)
 	res2 = g2.psi(res2)
 	g2.AssertIsEqual(res1, res2)

std/algebra/emulated/sw_bn254/pairing.go

@@ -81,10 +81,10 @@ func NewPairing(api frontend.API) (*Pairing, error) {
 	}, nil
 }
 
-// Pair calculates the reduced pairing for a set of points
-// ∏ᵢ e(Pᵢ, Qᵢ).
+// Pair calculates the reduced pairing for a set of points ∏ᵢ e(Pᵢ, Qᵢ).
 //
-// This function doesn't check that the inputs are in the correct subgroups. See AssertIsOnG1 and AssertIsOnG2.
+// This function checks that the Qᵢ are in the correct subgroup, but does not
+// check Pᵢ. See AssertIsOnG1.
 func (pr Pairing) Pair(P []*G1Affine, Q []*G2Affine) (*GTEl, error) {
 	res, err := pr.MillerLoop(P, Q)
 	if err != nil {
@@ -206,10 +206,13 @@ func (pr Pairing) AssertFinalExponentiationIsOne(a *GTEl) {
 	pr.AssertIsEqual(t0, t2)
 }
 
-// PairingCheck calculates the reduced pairing for a set of points and asserts if the result is One
-// ∏ᵢ e(Pᵢ, Qᵢ) =? 1
+// PairingCheck calculates the reduced pairing for a set of points and asserts
+// if the result is one:
 //
-// This function doesn't check that the inputs are in the correct subgroups. See AssertIsOnG1 and AssertIsOnG2.
+//	∏ᵢ e(Pᵢ, Qᵢ) =? 1
+//
+// This function checks that the Qᵢ are in the correct subgroup, but does not
+// check Pᵢ. See AssertIsOnG1.
 func (pr Pairing) PairingCheck(P []*G1Affine, Q []*G2Affine) error {
 	f, err := pr.MillerLoop(P, Q)
 	if err != nil {
@@ -266,6 +269,7 @@ func (pr Pairing) AssertIsOnG1(P *G1Affine) {
 	pr.AssertIsOnCurve(P)
 }
 
+// computeG2ShortVector computes ψ³([2x₀]Q) - ψ²([x₀]Q) - ψ([x₀]Q) - [x₀]Q
 func (pr Pairing) computeG2ShortVector(Q *G2Affine) (_Q *G2Affine) {
 	// [x₀]Q
 	xQ := pr.g2.scalarMulBySeed(Q)
@@ -278,7 +282,7 @@ func (pr Pairing) computeG2ShortVector(Q *G2Affine) (_Q *G2Affine) {
 	psi3xxQ = pr.g2.psi(psi3xxQ)
 
 	// _Q = ψ³([2x₀]Q) - ψ²([x₀]Q) - ψ([x₀]Q) - [x₀]Q
-	_Q = pr.g2.sub(psi2xQ, psi3xxQ)
+	_Q = pr.g2.sub(psi3xxQ, psi2xQ)
 	_Q = pr.g2.sub(_Q, psixQ)
 	_Q = pr.g2.sub(_Q, xQ)
 	return _Q
@@ -289,8 +293,10 @@ func (pr Pairing) AssertIsOnG2(Q *G2Affine) {
 	pr.AssertIsOnTwist(Q)
 
 	// 2- Check Q has the right subgroup order
+	// 		[r]Q == 0 <==>  ψ³([2x₀]Q) - ψ²([x₀]Q) - ψ([x₀]Q) - [x₀]Q == Q
+	// This is a valid short vector since x₀ ≠ 5422 mod 2196.
+	// See Sec. 3.1.2 (Example 1) in https://eprint.iacr.org/2022/348.
 	_Q := pr.computeG2ShortVector(Q)
-	// [r]Q == 0 <==>  _Q == Q
 	pr.g2.AssertIsEqual(Q, _Q)
 }
 

std/algebra/emulated/sw_bn254/precomputations.go

@@ -31,6 +31,10 @@ func precomputeLines(Q bn254.G2Affine) lineEvaluations {
 
 func (p *Pairing) computeLines(Q *g2AffP) lineEvaluations {
 
+	// check Q is on curve
+	Qaff := G2Affine{P: *Q, Lines: nil}
+	p.IsOnTwist(&Qaff)
+
 	var cLines lineEvaluations
 	Qacc := Q
 	n := len(bn254.LoopCounter)
@@ -50,6 +54,20 @@ func (p *Pairing) computeLines(Q *g2AffP) lineEvaluations {
 		}
 	}
 
+	// Check that Q is on G2 subgroup:
+	// 		[r]Q == 0 <==> [6x₀+2]Q + ψ(Q) + ψ³(Q) = ψ²(Q).
+	// This is a valid short vector since x₀ ≠ 4 mod 13 and x₀ ≠ 92 mod 97.
+	// See Sec. 3.1.2 (Remark 2) in https://eprint.iacr.org/2022/348.
+	// This test is equivalent to [computeG2ShortVector] in [AssertIsOnG2].
+	//
+	// At this point Qacc = [6x₀+2]Q.
+	psiQ := p.g2.psi(&Qaff)  // ψ(Q)
+	psi2Q := p.g2.phi(&Qaff) // ϕ(Q)=ψ²(Q)
+	psi3Q := p.g2.psi(psi2Q) // ψ³(Q)
+	lhs := p.g2.add(&G2Affine{P: *Qacc, Lines: nil}, psiQ)
+	lhs = p.g2.add(lhs, psi3Q)
+	p.g2.AssertIsEqual(lhs, psi2Q)
+
 	Q1X := p.Ext2.Conjugate(&Q.X)
 	Q1X = p.Ext2.MulByNonResidue1Power2(Q1X)
 	Q1Y := p.Ext2.Conjugate(&Q.Y)

std/evmprecompiles/08-bnpairing.go

@@ -40,11 +40,8 @@ func ECPair(api frontend.API, P []*sw_bn254.G1Affine, Q []*sw_bn254.G2Affine) {
 	if err != nil {
 		panic(err)
 	}
-	// 1- Check that Pᵢ are on G1 (done in the zkEVM ⚠️ )
-	// 2- Check that Qᵢ are on G2
-	for i := 0; i < len(Q); i++ {
-		pair.AssertIsOnG2(Q[i])
-	}
+	// 1- Check that Pᵢ are on G1 (done in the zkEVM ⚠️
+	// 2- Check that Qᵢ are on G2 (done in `computeLines` in `MillerLoopAndMul` and `MillerLoopAndFinalExpCheck)
 
 	// 3- Check that ∏ᵢ e(Pᵢ, Qᵢ) == 1
 	ml := pair.Ext12.One()
@@ -79,7 +76,6 @@ func ECPairMillerLoopAndMul(api frontend.API, accumulator *sw_bn254.GTEl, P *sw_
 	if err != nil {
 		return fmt.Errorf("new pairing: %w", err)
 	}
-	pairing.AssertIsOnG2(Q)
 	ml, err := pairing.MillerLoopAndMul(P, Q, accumulator)
 	if err != nil {
 		return fmt.Errorf("miller loop and mul: %w", err)
@@ -97,7 +93,6 @@ func ECPairMillerLoopAndFinalExpCheck(api frontend.API, accumulator *sw_bn254.GT
 	if err != nil {
 		return fmt.Errorf("new pairing: %w", err)
 	}
-	pairing.AssertIsOnG2(Q)
 
 	isSuccess := pairing.IsMillerLoopAndFinalExpOne(P, Q, accumulator)
 	api.AssertIsEqual(expectedIsSuccess, isSuccess)