perf: optimize DoublePairingCheck i.e. e(a,b)e(c,d) == 1

std/algebra/emulated/sw_bls12381/hints.go

@@ -0,0 +1,142 @@
+package sw_bls12381
+
+import (
+	"math/big"
+
+	"github.com/consensys/gnark-crypto/ecc/bls12-381"
+	"github.com/consensys/gnark/constraint/solver"
+	"github.com/consensys/gnark/std/math/emulated"
+)
+
+func init() {
+	solver.RegisterHint(GetHints()...)
+}
+
+// GetHints returns all hint functions used in the package.
+func GetHints() []solver.Hint {
+	return []solver.Hint{
+		doublePairingCheckHint,
+	}
+}
+
+func doublePairingCheckHint(nativeMod *big.Int, nativeInputs, nativeOutputs []*big.Int) error {
+	// This is inspired from https://eprint.iacr.org/2024/640.pdf
+	// and based on a personal communication with the author Andrija Novakovic.
+	return emulated.UnwrapHint(nativeInputs, nativeOutputs,
+		func(mod *big.Int, inputs, outputs []*big.Int) error {
+			var P0, P1 bls12381.G1Affine
+			var Q0, Q1 bls12381.G2Affine
+
+			P0.X.SetBigInt(inputs[0])
+			P0.Y.SetBigInt(inputs[1])
+			P1.X.SetBigInt(inputs[2])
+			P1.Y.SetBigInt(inputs[3])
+			Q0.X.A0.SetBigInt(inputs[4])
+			Q0.X.A1.SetBigInt(inputs[5])
+			Q0.Y.A0.SetBigInt(inputs[6])
+			Q0.Y.A1.SetBigInt(inputs[7])
+			Q1.X.A0.SetBigInt(inputs[8])
+			Q1.X.A1.SetBigInt(inputs[9])
+			Q1.Y.A0.SetBigInt(inputs[10])
+			Q1.Y.A1.SetBigInt(inputs[11])
+
+			lines0 := bls12381.PrecomputeLines(Q0)
+			lines1 := bls12381.PrecomputeLines(Q1)
+			millerLoop, err := bls12381.MillerLoopFixedQ(
+				[]bls12381.G1Affine{P0, P1},
+				[][2][len(bls12381.LoopCounter) - 1]bls12381.LineEvaluationAff{lines0, lines1},
+			)
+			millerLoop.Conjugate(&millerLoop)
+			if err != nil {
+				return err
+			}
+
+			var root, rootPthInverse, root27thInverse, residueWitness, scalingFactor bls12381.E12
+			var order3rd, order3rdPower, exponent, exponentInv, finalExpFactor, polyFactor big.Int
+			// polyFactor = (1-x)/3
+			polyFactor.SetString("5044125407647214251", 10)
+			// finalExpFactor = ((q^12 - 1) / r) / (27 * polyFactor)
+			finalExpFactor.SetString("2366356426548243601069753987687709088104621721678962410379583120840019275952471579477684846670499039076873213559162845121989217658133790336552276567078487633052653005423051750848782286407340332979263075575489766963251914185767058009683318020965829271737924625612375201545022326908440428522712877494557944965298566001441468676802477524234094954960009227631543471415676620753242466901942121887152806837594306028649150255258504417829961387165043999299071444887652375514277477719817175923289019181393803729926249507024121957184340179467502106891835144220611408665090353102353194448552304429530104218473070114105759487413726485729058069746063140422361472585604626055492939586602274983146215294625774144156395553405525711143696689756441298365274341189385646499074862712688473936093315628166094221735056483459332831845007196600723053356837526749543765815988577005929923802636375670820616189737737304893769679803809426304143627363860243558537831172903494450556755190448279875942974830469855835666815454271389438587399739607656399812689280234103023464545891697941661992848552456326290792224091557256350095392859243101357349751064730561345062266850238821755009430903520645523345000326783803935359711318798844368754833295302563158150573540616830138810935344206231367357992991289265295323280", 10)
+
+			// 1. get pth-root inverse
+			exponent.Mul(&finalExpFactor, big.NewInt(27))
+			root.Exp(millerLoop, &exponent)
+			if root.IsOne() {
+				rootPthInverse.SetOne()
+			} else {
+				exponentInv.ModInverse(&exponent, &polyFactor)
+				exponent.Neg(&exponentInv).Mod(&exponent, &polyFactor)
+				rootPthInverse.Exp(root, &exponent)
+			}
+
+			// 2.1. get order of 3rd primitive root
+			var three big.Int
+			three.SetUint64(3)
+			exponent.Mul(&polyFactor, &finalExpFactor)
+			root.Exp(millerLoop, &exponent)
+			if root.IsOne() {
+				order3rdPower.SetUint64(0)
+			}
+			root.Exp(root, &three)
+			if root.IsOne() {
+				order3rdPower.SetUint64(1)
+			}
+			root.Exp(root, &three)
+			if root.IsOne() {
+				order3rdPower.SetUint64(2)
+			}
+			root.Exp(root, &three)
+			if root.IsOne() {
+				order3rdPower.SetUint64(3)
+			}
+
+			// 2.2. get 27th root inverse
+			if order3rdPower.Uint64() == 0 {
+				root27thInverse.SetOne()
+			} else {
+				order3rd.Exp(&three, &order3rdPower, nil)
+				exponent.Mul(&polyFactor, &finalExpFactor)
+				root.Exp(millerLoop, &exponent)
+				exponentInv.ModInverse(&exponent, &order3rd)
+				exponent.Neg(&exponentInv).Mod(&exponent, &order3rd)
+				root27thInverse.Exp(root, &exponent)
+			}
+
+			// 2.3. shift the Miller loop result so that millerLoop * scalingFactor
+			// is of order finalExpFactor
+			scalingFactor.Mul(&rootPthInverse, &root27thInverse)
+			millerLoop.Mul(&millerLoop, &scalingFactor)
+
+			// 3. get the witness residue
+			//
+			// lambda = q - u, the optimal exponent
+			var lambda big.Int
+			lambda.SetString("4002409555221667393417789825735904156556882819939007885332058136124031650490837864442687629129030796414117214202539", 10)
+			exponent.ModInverse(&lambda, &finalExpFactor)
+			residueWitness.Exp(millerLoop, &exponent)
+
+			// return the witness residue
+			residueWitness.C0.B0.A0.BigInt(outputs[0])
+			residueWitness.C0.B0.A1.BigInt(outputs[1])
+			residueWitness.C0.B1.A0.BigInt(outputs[2])
+			residueWitness.C0.B1.A1.BigInt(outputs[3])
+			residueWitness.C0.B2.A0.BigInt(outputs[4])
+			residueWitness.C0.B2.A1.BigInt(outputs[5])
+			residueWitness.C1.B0.A0.BigInt(outputs[6])
+			residueWitness.C1.B0.A1.BigInt(outputs[7])
+			residueWitness.C1.B1.A0.BigInt(outputs[8])
+			residueWitness.C1.B1.A1.BigInt(outputs[9])
+			residueWitness.C1.B2.A0.BigInt(outputs[10])
+			residueWitness.C1.B2.A1.BigInt(outputs[11])
+
+			// return the scaling factor
+			scalingFactor.C0.B0.A0.BigInt(outputs[12])
+			scalingFactor.C0.B0.A1.BigInt(outputs[13])
+			scalingFactor.C0.B1.A0.BigInt(outputs[14])
+			scalingFactor.C0.B1.A1.BigInt(outputs[15])
+			scalingFactor.C0.B2.A0.BigInt(outputs[16])
+			scalingFactor.C0.B2.A1.BigInt(outputs[17])
+
+			return nil
+		})
+}

std/algebra/emulated/sw_bls12381/pairing.go

@@ -265,6 +265,121 @@ func (pr Pairing) PairingCheck(P []*G1Affine, Q []*G2Affine) error {
 	return nil
 }
 
+// DoublePairingCheck calculates the reduced pairing for a 2 pairs of points and asserts if the result is One
+// e(P0, Q0) * e(P1, Q1) =? 1
+//
+// This function doesn't check that the inputs are in the correct subgroups. See AssertIsOnG1 and AssertIsOnG2.
+func (pr Pairing) DoublePairingCheck(P [2]*G1Affine, Q [2]*G2Affine) error {
+	// hint the non-residue witness
+	hint, err := pr.curveF.NewHint(doublePairingCheckHint, 18, &P[0].X, &P[0].Y, &P[1].X, &P[1].Y, &Q[0].P.X.A0, &Q[0].P.X.A1, &Q[0].P.Y.A0, &Q[0].P.Y.A1, &Q[1].P.X.A0, &Q[1].P.X.A1, &Q[1].P.Y.A0, &Q[1].P.Y.A1)
+	if err != nil {
+		// err is non-nil only for invalid number of inputs
+		panic(err)
+	}
+
+	residueWitness := fields_bls12381.E12{
+		C0: fields_bls12381.E6{
+			B0: fields_bls12381.E2{A0: *hint[0], A1: *hint[1]},
+			B1: fields_bls12381.E2{A0: *hint[2], A1: *hint[3]},
+			B2: fields_bls12381.E2{A0: *hint[4], A1: *hint[5]},
+		},
+		C1: fields_bls12381.E6{
+			B0: fields_bls12381.E2{A0: *hint[6], A1: *hint[7]},
+			B1: fields_bls12381.E2{A0: *hint[8], A1: *hint[9]},
+			B2: fields_bls12381.E2{A0: *hint[10], A1: *hint[11]},
+		},
+	}
+	// constrain scaling factor to be in Fp6
+	scalingFactor := fields_bls12381.E12{
+		C0: fields_bls12381.E6{
+			B0: fields_bls12381.E2{A0: *hint[12], A1: *hint[13]},
+			B1: fields_bls12381.E2{A0: *hint[14], A1: *hint[15]},
+			B2: fields_bls12381.E2{A0: *hint[16], A1: *hint[17]},
+		},
+		C1: (*pr.Ext6.Zero()),
+	}
+
+	// residueWitnessInv = 1 / residueWitness
+	residueWitnessInv := pr.Inverse(&residueWitness)
+
+	if Q[0].Lines == nil {
+		Q0lines := pr.computeLines(&Q[0].P)
+		Q[0].Lines = &Q0lines
+	}
+	lines0 := *Q[0].Lines
+	if Q[1].Lines == nil {
+		Q1lines := pr.computeLines(&Q[1].P)
+		Q[1].Lines = &Q1lines
+	}
+	lines1 := *Q[1].Lines
+
+	// precomputations
+	y0Inv := pr.curveF.Inverse(&P[0].Y)
+	x0NegOverY0 := pr.curveF.Mul(&P[0].X, y0Inv)
+	x0NegOverY0 = pr.curveF.Neg(x0NegOverY0)
+	y1Inv := pr.curveF.Inverse(&P[1].Y)
+	x1NegOverY1 := pr.curveF.Mul(&P[1].X, y1Inv)
+	x1NegOverY1 = pr.curveF.Neg(x1NegOverY1)
+
+	// init Miller loop accumulator to residueWitnessInv to share the squarings
+	// of residueWitnessInv^{-x₀}
+	res := residueWitnessInv
+
+	// Compute f_{x₀,Q}(P)
+	for i := 62; i >= 0; i-- {
+		res = pr.Square(res)
+
+		if loopCounter[i] == 0 {
+			// ℓ × res
+			res = pr.MulBy014(res,
+				pr.MulByElement(&lines0[0][i].R1, y0Inv),
+				pr.MulByElement(&lines0[0][i].R0, x0NegOverY0),
+			)
+
+			// ℓ × res
+			res = pr.MulBy014(res,
+				pr.MulByElement(&lines1[0][i].R1, y1Inv),
+				pr.MulByElement(&lines1[0][i].R0, x1NegOverY1),
+			)
+		} else {
+			// multiply by residueWitnessInv when bit=1
+			res = pr.Mul(res, residueWitnessInv)
+
+			res = pr.MulBy014(res,
+				pr.MulByElement(&lines0[0][i].R1, y0Inv),
+				pr.MulByElement(&lines0[0][i].R0, x0NegOverY0),
+			)
+			res = pr.MulBy014(res,
+				pr.MulByElement(&lines0[1][i].R1, y0Inv),
+				pr.MulByElement(&lines0[1][i].R0, x0NegOverY0),
+			)
+
+			res = pr.MulBy014(res,
+				pr.MulByElement(&lines1[0][i].R1, y1Inv),
+				pr.MulByElement(&lines1[0][i].R0, x1NegOverY1),
+			)
+			res = pr.MulBy014(res,
+				pr.MulByElement(&lines1[1][i].R1, y1Inv),
+				pr.MulByElement(&lines1[1][i].R0, x1NegOverY1),
+			)
+		}
+	}
+
+	// Check that  res * scalingFactor * residueWitnessInv^λ' == 1
+	// where λ' = q, with u the BLS12-381 seed
+	// and residueWitnessInv, scalingFactor from the hint.
+	// Note that res is already MillerLoop(P,Q) * residueWitnessInv^{x₀} since
+	// we initialized the Miller loop accumulator with residueWitnessInv.
+	t0 := pr.Frobenius(residueWitnessInv)
+	t0 = pr.Mul(t0, res)
+	t0 = pr.Mul(t0, &scalingFactor)
+
+	pr.AssertIsEqual(t0, pr.One())
+
+	return nil
+
+}
+
 func (pr Pairing) AssertIsEqual(x, y *GTEl) {
 	pr.Ext12.AssertIsEqual(x, y)
 }

std/algebra/emulated/sw_bls12381/pairing_test.go

@@ -165,26 +165,26 @@ func TestMultiPairTestSolve(t *testing.T) {
 	}
 }
 
-type PairingCheckCircuit struct {
+type DoublePairingCheckCircuit struct {
 	In1G1 G1Affine
 	In2G1 G1Affine
 	In1G2 G2Affine
 	In2G2 G2Affine
 }
 
-func (c *PairingCheckCircuit) Define(api frontend.API) error {
+func (c *DoublePairingCheckCircuit) Define(api frontend.API) error {
 	pairing, err := NewPairing(api)
 	if err != nil {
 		return fmt.Errorf("new pairing: %w", err)
 	}
-	err = pairing.PairingCheck([]*G1Affine{&c.In1G1, &c.In2G1}, []*G2Affine{&c.In1G2, &c.In2G2})
+	err = pairing.DoublePairingCheck([2]*G1Affine{&c.In1G1, &c.In2G1}, [2]*G2Affine{&c.In1G2, &c.In2G2})
 	if err != nil {
 		return fmt.Errorf("pair: %w", err)
 	}
 	return nil
 }
 
-func TestPairingCheckTestSolve(t *testing.T) {
+func TestDoublePairingCheckTestSolve(t *testing.T) {
 	assert := test.NewAssert(t)
 	// e(a,2b) * e(-2a,b) == 1
 	p1, q1 := randomG1G2Affines()
@@ -193,6 +193,41 @@ func TestPairingCheckTestSolve(t *testing.T) {
 	var q2 bls12381.G2Affine
 	q2.Set(&q1)
 	q1.Double(&q1)
+	witness := DoublePairingCheckCircuit{
+		In1G1: NewG1Affine(p1),
+		In1G2: NewG2Affine(q1),
+		In2G1: NewG1Affine(p2),
+		In2G2: NewG2Affine(q2),
+	}
+	err := test.IsSolved(&DoublePairingCheckCircuit{}, &witness, ecc.BN254.ScalarField())
+	assert.NoError(err)
+}
+
+type PairingCheckCircuit struct {
+	In1G1 G1Affine
+	In2G1 G1Affine
+	In1G2 G2Affine
+	In2G2 G2Affine
+}
+
+func (c *PairingCheckCircuit) Define(api frontend.API) error {
+	pairing, err := NewPairing(api)
+	if err != nil {
+		return fmt.Errorf("new pairing: %w", err)
+	}
+	err = pairing.PairingCheck([]*G1Affine{&c.In1G1, &c.In1G1, &c.In2G1, &c.In2G1}, []*G2Affine{&c.In1G2, &c.In2G2, &c.In1G2, &c.In2G2})
+	if err != nil {
+		return fmt.Errorf("pair: %w", err)
+	}
+	return nil
+}
+
+func TestPairingCheckTestSolve(t *testing.T) {
+	assert := test.NewAssert(t)
+	p1, q1 := randomG1G2Affines()
+	_, q2 := randomG1G2Affines()
+	var p2 bls12381.G1Affine
+	p2.Neg(&p1)
 	witness := PairingCheckCircuit{
 		In1G1: NewG1Affine(p1),
 		In1G2: NewG2Affine(q1),