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Perf: faster final exponentiation (hard part) for BW6 curves #459

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merged 6 commits into from
Oct 17, 2023
Merged

Perf: faster final exponentiation (hard part) for BW6 curves #459

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7d13fd0
perf(bw6-761): implement faster final exp hard part
yelhousni Oct 12, 2023
5b7609c
perf(bw6-761): remove unnecessary computations in MillerLoop
yelhousni Oct 12, 2023
19209bb
perf(bw6-756): remove unnecessary computations in MillerLoop
yelhousni Oct 12, 2023
fb97a48
perf(bw6-633): remove unnecessary computations in MillerLoop
yelhousni Oct 12, 2023
2230a7e
perf(bw6-756): implement faster final exp hard part
yelhousni Oct 13, 2023
6cfca29
perf(bw6-633): implement faster final exp hard part
yelhousni Oct 13, 2023
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73 changes: 1 addition & 72 deletions ecc/bw6-633/pairing.go
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Original file line number Diff line number Diff line change
Expand Up @@ -208,30 +208,12 @@ func MillerLoop(P []G1Affine, Q []G2Affine) (GT, error) {
// precomputations
pProj0 := make([]g1Proj, n)
p1 := make([]G1Affine, n)
pProj01 := make([]g1Proj, n) // P0+P1
pProj10 := make([]g1Proj, n) // P0-P1
l01 := make([]lineEvaluation, n)
l10 := make([]lineEvaluation, n)
for k := 0; k < n; k++ {
p1[k].Y.Set(&p0[k].Y)
p1[k].X.Mul(&p0[k].X, &thirdRootOneG1)
p0[k].Neg(&p0[k])
pProj0[k].FromAffine(&p0[k])

// l_{p0,p1}(q)
pProj01[k].Set(&pProj0[k])
pProj01[k].addMixedStep(&l01[k], &p1[k])
l01[k].r1.Mul(&l01[k].r1, &q[k].X)
l01[k].r0.Mul(&l01[k].r0, &q[k].Y)

// l_{-p0,p1}(q)
pProj10[k].Neg(&pProj0[k])
pProj10[k].addMixedStep(&l10[k], &p1[k])
l10[k].r1.Mul(&l10[k].r1, &q[k].X)
l10[k].r0.Mul(&l10[k].r0, &q[k].Y)
}
p01 := BatchProjectiveToAffineG1(pProj01)
p10 := BatchProjectiveToAffineG1(pProj10)

// f_{a0+λ*a1,P}(Q)
var result GT
Expand Down Expand Up @@ -299,20 +281,7 @@ func MillerLoop(P []G1Affine, Q []G2Affine) (GT, error) {
l0.r0.Mul(&l0.r0, &q[k].Y)

switch j {
case -4:
tmp.Neg(&p01[k])
// pProj0[k] ← pProj0[k]-p01[k] and
// l the line ℓ passing pProj0[k] and -p01[k]
pProj0[k].addMixedStep(&l, &tmp)
// line evaluation at Q[k]
l.r1.Mul(&l.r1, &q[k].X)
l.r0.Mul(&l.r0, &q[k].Y)
// ℓ × ℓ
prodLines = fptower.Mul034By034(&l.r0, &l.r1, &l.r2, &l01[k].r0, &l01[k].r1, &l01[k].r2)
// ℓ × res
result.MulBy034(&l0.r0, &l0.r1, &l0.r2)
// (ℓ × ℓ) × res
result.MulBy01234(&prodLines)
// cases -4, -2, 2, 4 do not occur given the static loopCounters
case -3:
tmp.Neg(&p1[k])
// pProj0[k] ← pProj0[k]-p1[k] and
Expand All @@ -325,19 +294,6 @@ func MillerLoop(P []G1Affine, Q []G2Affine) (GT, error) {
prodLines = fptower.Mul034By034(&l.r0, &l.r1, &l.r2, &l0.r0, &l0.r1, &l0.r2)
// (ℓ × ℓ) × res
result.MulBy01234(&prodLines)
case -2:
// pProj0[k] ← pProj0[k]+p10[k] and
// l the line ℓ passing pProj0[k] and p10[k]
pProj0[k].addMixedStep(&l, &p10[k])
// line evaluation at Q[k]
l.r1.Mul(&l.r1, &q[k].X)
l.r0.Mul(&l.r0, &q[k].Y)
// ℓ × ℓ
prodLines = fptower.Mul034By034(&l.r0, &l.r1, &l.r2, &l01[k].r0, &l01[k].r1, &l01[k].r2)
// ℓ × res
result.MulBy034(&l0.r0, &l0.r1, &l0.r2)
// (ℓ × ℓ) × res
result.MulBy01234(&prodLines)
case -1:
tmp.Neg(&p0[k])
// pProj0[k] ← pProj0[k]-p0[k] and
Expand All @@ -364,20 +320,6 @@ func MillerLoop(P []G1Affine, Q []G2Affine) (GT, error) {
prodLines = fptower.Mul034By034(&l.r0, &l.r1, &l.r2, &l0.r0, &l0.r1, &l0.r2)
// (ℓ × ℓ) × res
result.MulBy01234(&prodLines)
case 2:
tmp.Neg(&p10[k])
// pProj0[k] ← pProj0[k]-p10[k] and
// l the line ℓ passing pProj0[k] and -p10[k]
pProj0[k].addMixedStep(&l, &tmp)
// line evaluation at Q[k]
l.r1.Mul(&l.r1, &q[k].X)
l.r0.Mul(&l.r0, &q[k].Y)
// ℓ × ℓ
prodLines = fptower.Mul034By034(&l.r0, &l.r1, &l.r2, &l01[k].r0, &l01[k].r1, &l01[k].r2)
// ℓ × res
result.MulBy034(&l0.r0, &l0.r1, &l0.r2)
// (ℓ × ℓ) × res
result.MulBy01234(&prodLines)
case 3:
// pProj0[k] ← pProj0[k]+p1[k] and
// l the line ℓ passing pProj0[k] and p1[k]
Expand All @@ -389,19 +331,6 @@ func MillerLoop(P []G1Affine, Q []G2Affine) (GT, error) {
prodLines = fptower.Mul034By034(&l.r0, &l.r1, &l.r2, &l0.r0, &l0.r1, &l0.r2)
// (ℓ × ℓ) × res
result.MulBy01234(&prodLines)
case 4:
// pProj0[k] ← pProj0[k]+p01[k] and
// l the line ℓ passing pProj0[k] and p01[k]
pProj0[k].addMixedStep(&l, &p01[k])
// line evaluation at Q[k]
l.r1.Mul(&l.r1, &q[k].X)
l.r0.Mul(&l.r0, &q[k].Y)
// ℓ × ℓ
prodLines = fptower.Mul034By034(&l.r0, &l.r1, &l.r2, &l01[k].r0, &l01[k].r1, &l01[k].r2)
// ℓ × res
result.MulBy034(&l0.r0, &l0.r1, &l0.r2)
// (ℓ × ℓ) × res
result.MulBy01234(&prodLines)
default:
return GT{}, errors.New("invalid loopCounter")
}
Expand Down
72 changes: 0 additions & 72 deletions ecc/bw6-756/pairing.go
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Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -189,29 +189,11 @@ func MillerLoop(P []G1Affine, Q []G2Affine) (GT, error) {
// precomputations
pProj1 := make([]g1Proj, n)
p1 := make([]G1Affine, n)
pProj01 := make([]g1Proj, n) // P0+P1
pProj10 := make([]g1Proj, n) // P0-P1
l01 := make([]lineEvaluation, n)
l10 := make([]lineEvaluation, n)
for k := 0; k < n; k++ {
p1[k].Y.Neg(&p0[k].Y)
p1[k].X.Mul(&p0[k].X, &thirdRootOneG2)
pProj1[k].FromAffine(&p1[k])

// l_{p0,p1}(q)
pProj01[k].Set(&pProj1[k])
pProj01[k].addMixedStep(&l01[k], &p0[k])
l01[k].r1.Mul(&l01[k].r1, &q[k].X)
l01[k].r0.Mul(&l01[k].r0, &q[k].Y)

// l_{p0,-p1}(q)
pProj10[k].Neg(&pProj1[k])
pProj10[k].addMixedStep(&l10[k], &p0[k])
l10[k].r1.Mul(&l10[k].r1, &q[k].X)
l10[k].r0.Mul(&l10[k].r0, &q[k].Y)
}
p01 := BatchProjectiveToAffineG1(pProj01)
p10 := BatchProjectiveToAffineG1(pProj10)

// f_{a0+λ*a1,P}(Q)
var result GT
Expand Down Expand Up @@ -279,20 +261,6 @@ func MillerLoop(P []G1Affine, Q []G2Affine) (GT, error) {
l0.r0.Mul(&l0.r0, &q[k].Y)

switch j {
case -4:
tmp.Neg(&p01[k])
// pProj1[k] ← pProj1[k]-p01[k] and
// l the line ℓ passing pProj1[k] and -p01[k]
pProj1[k].addMixedStep(&l, &tmp)
// line evaluation at Q[k]
l.r1.Mul(&l.r1, &q[k].X)
l.r0.Mul(&l.r0, &q[k].Y)
// ℓ × ℓ
prodLines = fptower.Mul034By034(&l.r0, &l.r1, &l.r2, &l01[k].r0, &l01[k].r1, &l01[k].r2)
// ℓ × res
result.MulBy034(&l0.r0, &l0.r1, &l0.r2)
// (ℓ × ℓ) × res
result.MulBy01234(&prodLines)
case -3:
tmp.Neg(&p1[k])
// pProj1[k] ← pProj1[k]-p1[k] and
Expand All @@ -305,19 +273,6 @@ func MillerLoop(P []G1Affine, Q []G2Affine) (GT, error) {
prodLines = fptower.Mul034By034(&l.r0, &l.r1, &l.r2, &l0.r0, &l0.r1, &l0.r2)
// (ℓ × ℓ) × res
result.MulBy01234(&prodLines)
case -2:
// pProj1[k] ← pProj1[k]+p10[k] and
// l the line ℓ passing pProj1[k] and p10[k]
pProj1[k].addMixedStep(&l, &p10[k])
// line evaluation at Q[k]
l.r1.Mul(&l.r1, &q[k].X)
l.r0.Mul(&l.r0, &q[k].Y)
// ℓ × ℓ
prodLines = fptower.Mul034By034(&l.r0, &l.r1, &l.r2, &l01[k].r0, &l01[k].r1, &l01[k].r2)
// ℓ × res
result.MulBy034(&l0.r0, &l0.r1, &l0.r2)
// (ℓ × ℓ) × res
result.MulBy01234(&prodLines)
case -1:
tmp.Neg(&p0[k])
// pProj1[k] ← pProj1[k]-p0[k] and
Expand All @@ -344,20 +299,6 @@ func MillerLoop(P []G1Affine, Q []G2Affine) (GT, error) {
prodLines = fptower.Mul034By034(&l.r0, &l.r1, &l.r2, &l0.r0, &l0.r1, &l0.r2)
// (ℓ × ℓ) × res
result.MulBy01234(&prodLines)
case 2:
tmp.Neg(&p10[k])
// pProj1[k] ← pProj1[k]-p10[k] and
// l the line ℓ passing pProj1[k] and -p10[k]
pProj1[k].addMixedStep(&l, &tmp)
// line evaluation at Q[k]
l.r1.Mul(&l.r1, &q[k].X)
l.r0.Mul(&l.r0, &q[k].Y)
// ℓ × ℓ
prodLines = fptower.Mul034By034(&l.r0, &l.r1, &l.r2, &l01[k].r0, &l01[k].r1, &l01[k].r2)
// ℓ × res
result.MulBy034(&l0.r0, &l0.r1, &l0.r2)
// (ℓ × ℓ) × res
result.MulBy01234(&prodLines)
case 3:
// pProj1[k] ← pProj1[k]+p1[k] and
// l the line ℓ passing pProj1[k] and p1[k]
Expand All @@ -369,19 +310,6 @@ func MillerLoop(P []G1Affine, Q []G2Affine) (GT, error) {
prodLines = fptower.Mul034By034(&l.r0, &l.r1, &l.r2, &l0.r0, &l0.r1, &l0.r2)
// (ℓ × ℓ) × res
result.MulBy01234(&prodLines)
case 4:
// pProj1[k] ← pProj1[k]+p01[k] and
// l the line ℓ passing pProj1[k] and p01[k]
pProj1[k].addMixedStep(&l, &p01[k])
// line evaluation at Q[k]
l.r1.Mul(&l.r1, &q[k].X)
l.r0.Mul(&l.r0, &q[k].Y)
// ℓ × ℓ
prodLines = fptower.Mul034By034(&l.r0, &l.r1, &l.r2, &l01[k].r0, &l01[k].r1, &l01[k].r2)
// ℓ × res
result.MulBy034(&l0.r0, &l0.r1, &l0.r2)
// (ℓ × ℓ) × res
result.MulBy01234(&prodLines)
default:
return GT{}, errors.New("invalid loopCounter")
}
Expand Down
125 changes: 93 additions & 32 deletions ecc/bw6-761/internal/fptower/e6_pairing.go
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -14,16 +14,11 @@ func (z *E6) nSquareCompressed(n int) {
}
}

// Expt set z to x^t in E6 and return z
func (z *E6) Expt(x *E6) *E6 {
// const tAbsVal uint64 = 9586122913090633729
// tAbsVal in binary: 1000010100001000110000000000000000000000000000000000000000000001
// drop the low 46 bits (all 0 except the least significant bit): 100001010000100011 = 136227
// Shortest addition chains can be found at https://wwwhomes.uni-bielefeld.de/achim/addition_chain.html
// ExptMinus1 set z to x^(t-1) in E6 and return z
// t-1 = 91893752504881257682351033800651177983
func (z *E6) ExptMinus1(x *E6) *E6 {

var result, x33 E6

// a shortest addition chain for 136227
result.Set(x)
result.nSquare(5)
result.Mul(&result, x)
Expand All @@ -34,55 +29,121 @@ func (z *E6) Expt(x *E6) *E6 {
result.Mul(&result, x)
result.CyclotomicSquare(&result)
result.Mul(&result, x)

// the remaining 46 bits
result.nSquareCompressed(46)
result.DecompressKarabina(&result)
result.Mul(&result, x)

z.Set(&result)

return z
}

// Expc2 set z to x^c2 in E6 and return z
// ht, hy = 13, 9
// c1 = ht+hy = 22 (10110)
func (z *E6) Expc2(x *E6) *E6 {
// ExptMinus1Square set z to x^{(t-1)²} in E6 and return z
// (t-1)² = 91893752504881257682351033800651177984
func (z *E6) ExptMinus1Square(x *E6) *E6 {
var result, t0, t1, t2 E6

result.Set(x)
result.CyclotomicSquare(&result)
t0.Mul(x, &result)
t1.CyclotomicSquare(&t0)
t0.Mul(&t0, &t1)
result.Mul(&result, &t0)
t1.Mul(&t1, &result)
t0.Mul(&t0, &t1)
t2.CyclotomicSquare(&t0)
t2.Mul(&t1, &t2)
t0.Mul(&t0, &t2)
t2.nSquare(7)
t1.Mul(&t1, &t2)
t1.nSquare(11)
t1.Mul(&t0, &t1)
t1.nSquare(9)
t0.Mul(&t0, &t1)
t0.CyclotomicSquare(&t0)
result.Mul(&result, &t0)
result.nSquareCompressed(92)
result.DecompressKarabina(&result)
z.Set(&result)

return z
}

// Expt set z to x^t in E6 and return z
// t = 91893752504881257682351033800651177984
func (z *E6) Expt(x *E6) *E6 {
var result E6

result.CyclotomicSquare(x)
result.CyclotomicSquare(&result)
result.ExptMinus1(x)
result.Mul(&result, x)
z.Set(&result)

return z
}

// ExptPlus1 set z to x^(t+1) in E6 and return z
// t+1 = 91893752504881257682351033800651177985
func (z *E6) ExptPlus1(x *E6) *E6 {
var result, t E6

result.ExptMinus1(x)
t.CyclotomicSquare(x)
result.Mul(&result, &t)
z.Set(&result)

return z
}

// ExptMinus1Div3 set z to x^(t-1)/3 in E6 and return z
// (t-1)/3 = 3195374304363544576
func (z *E6) ExptMinus1Div3(x *E6) *E6 {
var result, t0 E6

result.Set(x)
result.CyclotomicSquare(&result)
result.Mul(&result, x)
result.CyclotomicSquare(&result)

t0.Mul(&result, x)
t0.CyclotomicSquare(&t0)
result.Mul(&result, &t0)
t0.Set(&result)
t0.nSquare(7)
result.Mul(&result, &t0)
result.nSquare(5)
result.Mul(&result, x)
result.nSquareCompressed(46)
result.DecompressKarabina(&result)
z.Set(&result)

return z
}

// Expc1 set z to x^c1 in E6 and return z
// Expc1 set z to z^c1 in E6 and return z
// ht, hy = 13, 9
// c1 = ht**2+3*hy**2 = 412 (110011100)
// c1 = (ht+hy)/2 = 11
func (z *E6) Expc1(x *E6) *E6 {

var result E6
var result, t0 E6

result.CyclotomicSquare(x)
result.Mul(&result, x)
result.CyclotomicSquare(&result)
result.CyclotomicSquare(&result)
result.CyclotomicSquare(&result)
result.Mul(&result, x)
result.CyclotomicSquare(&result)
t0.Mul(x, &result)
t0.CyclotomicSquare(&t0)
result.Mul(&result, &t0)
z.Set(&result)

return z
}

// Expx2 set z to z^c2 in E6 and return z
// ht, hy = 13, 9
// c2 = (ht**2+3*hy**2)/4 = 103
func (z *E6) Expc2(x *E6) *E6 {
var result, t0 E6

result.CyclotomicSquare(x)
result.Mul(&result, x)
t0.Set(&result)
t0.nSquare(4)
result.Mul(&result, &t0)
result.CyclotomicSquare(&result)
result.Mul(&result, x)
result.CyclotomicSquare(&result)
result.CyclotomicSquare(&result)

z.Set(&result)

return z
Expand Down
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