From ffaaea73e1b67358e625343c0d32fc785792b3f6 Mon Sep 17 00:00:00 2001
From: Youssef El Housni <youssef.housni21@gmail.com>
Date: Tue, 30 Jul 2024 15:12:09 -0400
Subject: [PATCH] refactor(bls12-381, bls12-377): MillerLoopFixed corresponds
 to gnark

---
 ecc/bls12-377/pairing.go | 51 ++++++++++++++++++++++++++++++++---
 ecc/bls12-381/pairing.go | 57 ++++++++++++++++++++++++++++++++++++----
 2 files changed, 100 insertions(+), 8 deletions(-)

diff --git a/ecc/bls12-377/pairing.go b/ecc/bls12-377/pairing.go
index dcb142823..1b95a173f 100644
--- a/ecc/bls12-377/pairing.go
+++ b/ecc/bls12-377/pairing.go
@@ -384,11 +384,10 @@ func PrecomputeLines(Q G2Affine) (PrecomputedLines [2][len(LoopCounter) - 1]Line
 	accQ.Set(&Q)
 
 	for i := len(LoopCounter) - 2; i >= 0; i-- {
-		accQ.doubleStep(&PrecomputedLines[0][i])
 		if LoopCounter[i] == 0 {
-			continue
+			accQ.doubleStep(&PrecomputedLines[0][i])
 		} else {
-			accQ.addStep(&PrecomputedLines[1][i], &Q)
+			accQ.doubleAndAddStep(&PrecomputedLines[0][i], &PrecomputedLines[1][i], &Q)
 		}
 	}
 	return PrecomputedLines
@@ -573,3 +572,49 @@ func (p *G2Affine) addStep(evaluations *LineEvaluationAff, a *G2Affine) {
 	p.X.Set(&xr)
 	p.Y.Set(&yr)
 }
+
+func (p *G2Affine) doubleAndAddStep(evaluations1, evaluations2 *LineEvaluationAff, a *G2Affine) {
+	var n, d, l1, x3, l2, x4, y4 fptower.E2
+
+	// compute λ1 = (y2-y1)/(x2-x1)
+	n.Sub(&p.Y, &a.Y)
+	d.Sub(&p.X, &a.X)
+	l1.Div(&n, &d)
+
+	// compute x3 =λ1²-x1-x2
+	x3.Square(&l1)
+	x3.Sub(&x3, &p.X)
+	x3.Sub(&x3, &a.X)
+
+	// omit y3 computation
+
+	// compute line1
+	evaluations1.R0.Set(&l1)
+	evaluations1.R1.Mul(&l1, &p.X)
+	evaluations1.R1.Sub(&evaluations1.R1, &p.Y)
+
+	// compute λ2 = -λ1-2y1/(x3-x1)
+	n.Double(&p.Y)
+	d.Sub(&x3, &p.X)
+	l2.Div(&n, &d)
+	l2.Add(&l2, &l1)
+	l2.Neg(&l2)
+
+	// compute x4 = λ2²-x1-x3
+	x4.Square(&l2)
+	x4.Sub(&x4, &p.X)
+	x4.Sub(&x4, &x3)
+
+	// compute y4 = λ2(x1 - x4)-y1
+	y4.Sub(&p.X, &x4)
+	y4.Mul(&l2, &y4)
+	y4.Sub(&y4, &p.Y)
+
+	// compute line2
+	evaluations2.R0.Set(&l2)
+	evaluations2.R1.Mul(&l2, &p.X)
+	evaluations2.R1.Sub(&evaluations2.R1, &p.Y)
+
+	p.X.Set(&x4)
+	p.Y.Set(&y4)
+}
diff --git a/ecc/bls12-381/pairing.go b/ecc/bls12-381/pairing.go
index b7c1bf66b..be2e8bc74 100644
--- a/ecc/bls12-381/pairing.go
+++ b/ecc/bls12-381/pairing.go
@@ -386,14 +386,15 @@ func PairingCheckFixedQ(P []G1Affine, lines [][2][len(LoopCounter) - 1]LineEvalu
 func PrecomputeLines(Q G2Affine) (PrecomputedLines [2][len(LoopCounter) - 1]LineEvaluationAff) {
 	var accQ G2Affine
 	accQ.Set(&Q)
-
 	n := len(LoopCounter)
-	for i := n - 2; i >= 0; i-- {
-		accQ.doubleStep(&PrecomputedLines[0][i])
+	// i = n - 2
+	accQ.doubleStep(&PrecomputedLines[0][n-2])
+	accQ.addStep(&PrecomputedLines[1][n-2], &Q)
+	for i := n - 3; i >= 0; i-- {
 		if LoopCounter[i] == 0 {
-			continue
+			accQ.doubleStep(&PrecomputedLines[0][i])
 		} else {
-			accQ.addStep(&PrecomputedLines[1][i], &Q)
+			accQ.doubleAndAddStep(&PrecomputedLines[0][i], &PrecomputedLines[1][i], &Q)
 		}
 	}
 	return PrecomputedLines
@@ -541,3 +542,49 @@ func (p *G2Affine) addStep(evaluations *LineEvaluationAff, a *G2Affine) {
 	p.X.Set(&xr)
 	p.Y.Set(&yr)
 }
+
+func (p *G2Affine) doubleAndAddStep(evaluations1, evaluations2 *LineEvaluationAff, a *G2Affine) {
+	var n, d, l1, x3, l2, x4, y4 fptower.E2
+
+	// compute λ1 = (y2-y1)/(x2-x1)
+	n.Sub(&p.Y, &a.Y)
+	d.Sub(&p.X, &a.X)
+	l1.Div(&n, &d)
+
+	// compute x3 =λ1²-x1-x2
+	x3.Square(&l1)
+	x3.Sub(&x3, &p.X)
+	x3.Sub(&x3, &a.X)
+
+	// omit y3 computation
+
+	// compute line1
+	evaluations1.R0.Set(&l1)
+	evaluations1.R1.Mul(&l1, &p.X)
+	evaluations1.R1.Sub(&evaluations1.R1, &p.Y)
+
+	// compute λ2 = -λ1-2y1/(x3-x1)
+	n.Double(&p.Y)
+	d.Sub(&x3, &p.X)
+	l2.Div(&n, &d)
+	l2.Add(&l2, &l1)
+	l2.Neg(&l2)
+
+	// compute x4 = λ2²-x1-x3
+	x4.Square(&l2)
+	x4.Sub(&x4, &p.X)
+	x4.Sub(&x4, &x3)
+
+	// compute y4 = λ2(x1 - x4)-y1
+	y4.Sub(&p.X, &x4)
+	y4.Mul(&l2, &y4)
+	y4.Sub(&y4, &p.Y)
+
+	// compute line2
+	evaluations2.R0.Set(&l2)
+	evaluations2.R1.Mul(&l2, &p.X)
+	evaluations2.R1.Sub(&evaluations2.R1, &p.Y)
+
+	p.X.Set(&x4)
+	p.Y.Set(&y4)
+}