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refactor(bls12-381, bls12-377): MillerLoopFixed corresponds to gnark #524

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Sep 3, 2024
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51 changes: 48 additions & 3 deletions ecc/bls12-377/pairing.go
Original file line number Diff line number Diff line change
Expand Up @@ -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
Expand Down Expand Up @@ -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)
}
57 changes: 52 additions & 5 deletions ecc/bls12-381/pairing.go
Original file line number Diff line number Diff line change
Expand Up @@ -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
Expand Down Expand Up @@ -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)
}
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