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perf: BW6 pairing computation using non-native Eval #1312

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perf: BW6 pairing computation using non-native Eval #1312

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767549d
chore: generalize deferred multiplication check logic
ivokub Nov 4, 2024
ee9e247
chore: implement limbMul method
ivokub Nov 4, 2024
215fbea
chore: implement utility sum method
ivokub Nov 4, 2024
9b3b312
feat: implement Eval on non-native arithmetic
ivokub Nov 4, 2024
cd090a7
test: add Eval test
ivokub Nov 4, 2024
620c0de
test: add skipped test for negative coefficient
ivokub Nov 4, 2024
8d16bbf
perf: use nonnative eval for fp6 mul
ivokub Nov 4, 2024
6f45b6c
perf: add square with eval
ivokub Nov 4, 2024
e36bcf2
perf: add mul 023 using eval
ivokub Nov 4, 2024
5120c3f
perf: add mul023By023 using eval
ivokub Nov 4, 2024
44f52da
perf: mulby02345 using eval
ivokub Nov 4, 2024
bf5afa4
chore: fix docs
ivokub Nov 4, 2024
2524b36
perf: cyclotomic square using eval
ivokub Nov 5, 2024
3c1bba3
chore: update stats
ivokub Nov 5, 2024
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4 changes: 2 additions & 2 deletions internal/stats/latest_stats.csv
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -195,14 +195,14 @@ pairing_bn254,bls24_315,plonk,0,0
pairing_bn254,bls24_317,plonk,0,0
pairing_bn254,bw6_761,plonk,0,0
pairing_bn254,bw6_633,plonk,0,0
pairing_bw6761,bn254,groth16,3014749,4979960
pairing_bw6761,bn254,groth16,1843705,3084217
pairing_bw6761,bls12_377,groth16,0,0
pairing_bw6761,bls12_381,groth16,0,0
pairing_bw6761,bls24_315,groth16,0,0
pairing_bw6761,bls24_317,groth16,0,0
pairing_bw6761,bw6_761,groth16,0,0
pairing_bw6761,bw6_633,groth16,0,0
pairing_bw6761,bn254,plonk,11486969,10777222
pairing_bw6761,bn254,plonk,6947630,6315782
pairing_bw6761,bls12_377,plonk,0,0
pairing_bw6761,bls12_381,plonk,0,0
pairing_bw6761,bls24_315,plonk,0,0
Expand Down
137 changes: 134 additions & 3 deletions std/algebra/emulated/fields_bw6761/e6.go
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Original file line number Diff line number Diff line change
Expand Up @@ -190,9 +190,7 @@ func (e Ext6) mulFpByNonResidue(fp *curveF, x *baseEl) *baseEl {
}

func (e Ext6) Mul(x, y *E6) *E6 {
x = e.Reduce(x)
y = e.Reduce(y)
return e.mulToomCook6(x, y)
return e.mulDirect(x, y)
}

func (e Ext6) mulMontgomery6(x, y *E6) *E6 {
Expand Down Expand Up @@ -426,6 +424,37 @@ func (e Ext6) mulMontgomery6(x, y *E6) *E6 {
}
}

func (e Ext6) mulDirect(x, y *E6) *E6 {
nonResidue := e.fp.NewElement(-4)
// c0 = a0b0 + β(a1b5 + a2b4 + a3b3 + a4b2 + a5b1)
c0 := e.fp.Eval([][]*baseEl{{&x.A0, &y.A0}, {nonResidue, &x.A1, &y.A5}, {nonResidue, &x.A2, &y.A4}, {nonResidue, &x.A3, &y.A3}, {nonResidue, &x.A4, &y.A2}, {nonResidue, &x.A5, &y.A1}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c1 = a0b1 + a1b0 + β(a2b5 + a3b4 + a4b3 + a5b2)
c1 := e.fp.Eval([][]*baseEl{{&x.A0, &y.A1}, {&x.A1, &y.A0}, {nonResidue, &x.A2, &y.A5}, {nonResidue, &x.A3, &y.A4}, {nonResidue, &x.A4, &y.A3}, {nonResidue, &x.A5, &y.A2}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c2 = a0b2 + a1b1 + a2b0 + β(a3b5 + a4b4 + a5b3)
c2 := e.fp.Eval([][]*baseEl{{&x.A0, &y.A2}, {&x.A1, &y.A1}, {&x.A2, &y.A0}, {nonResidue, &x.A3, &y.A5}, {nonResidue, &x.A4, &y.A4}, {nonResidue, &x.A5, &y.A3}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c3 = a0b3 + a1b2 + a2b1 + a3b0 + β(a4b5 + a5b4)
c3 := e.fp.Eval([][]*baseEl{{&x.A0, &y.A3}, {&x.A1, &y.A2}, {&x.A2, &y.A1}, {&x.A3, &y.A0}, {nonResidue, &x.A4, &y.A5}, {nonResidue, &x.A5, &y.A4}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c4 = a0b4 + a1b3 + a2b2 + a3b1 + a4b0 + βa5b5
c4 := e.fp.Eval([][]*baseEl{{&x.A0, &y.A4}, {&x.A1, &y.A3}, {&x.A2, &y.A2}, {&x.A3, &y.A1}, {&x.A4, &y.A0}, {nonResidue, &x.A5, &y.A5}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c5 = a0b5 + a1b4 + a2b3 + a3b2 + a4b1 + a5b0,
c5 := e.fp.Eval([][]*baseEl{{&x.A0, &y.A5}, {&x.A1, &y.A4}, {&x.A2, &y.A3}, {&x.A3, &y.A2}, {&x.A4, &y.A1}, {&x.A5, &y.A0}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})

return &E6{
A0: *c0,
A1: *c1,
A2: *c2,
A3: *c3,
A4: *c4,
A5: *c5,
}
}

func (e Ext6) mulToomCook6(x, y *E6) *E6 {
// Toom-Cook 6-way multiplication:
//
Expand Down Expand Up @@ -704,6 +733,43 @@ func (e Ext6) mulToomCook6(x, y *E6) *E6 {
}

func (e Ext6) Square(x *E6) *E6 {
// return e.squarePolyWithRand(x, e.fp.NewElement(-1))
return e.squareDirect(x)
}

// squareDirect computes the square of an element in E6 using schoolbook multiplication.
func (e Ext6) squareDirect(x *E6) *E6 {
nonResidue := e.fp.NewElement(-4)
// c0 = a0b0 + β(a1b5 + a2b4 + a3b3 + a4b2 + a5b1)
c0 := e.fp.Eval([][]*baseEl{{&x.A0, &x.A0}, {nonResidue, &x.A1, &x.A5}, {nonResidue, &x.A2, &x.A4}, {nonResidue, &x.A3, &x.A3}, {nonResidue, &x.A4, &x.A2}, {nonResidue, &x.A5, &x.A1}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c1 = a0b1 + a1b0 + β(a2b5 + a3b4 + a4b3 + a5b2)
c1 := e.fp.Eval([][]*baseEl{{&x.A0, &x.A1}, {&x.A1, &x.A0}, {nonResidue, &x.A2, &x.A5}, {nonResidue, &x.A3, &x.A4}, {nonResidue, &x.A4, &x.A3}, {nonResidue, &x.A5, &x.A2}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c2 = a0b2 + a1b1 + a2b0 + β(a3b5 + a4b4 + a5b3)
c2 := e.fp.Eval([][]*baseEl{{&x.A0, &x.A2}, {&x.A1, &x.A1}, {&x.A2, &x.A0}, {nonResidue, &x.A3, &x.A5}, {nonResidue, &x.A4, &x.A4}, {nonResidue, &x.A5, &x.A3}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c3 = a0b3 + a1b2 + a2b1 + a3b0 + β(a4b5 + a5b4)
c3 := e.fp.Eval([][]*baseEl{{&x.A0, &x.A3}, {&x.A1, &x.A2}, {&x.A2, &x.A1}, {&x.A3, &x.A0}, {nonResidue, &x.A4, &x.A5}, {nonResidue, &x.A5, &x.A4}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c4 = a0b4 + a1b3 + a2b2 + a3b1 + a4b0 + βa5b5
c4 := e.fp.Eval([][]*baseEl{{&x.A0, &x.A4}, {&x.A1, &x.A3}, {&x.A2, &x.A2}, {&x.A3, &x.A1}, {&x.A4, &x.A0}, {nonResidue, &x.A5, &x.A5}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c5 = a0b5 + a1b4 + a2b3 + a3b2 + a4b1 + a5b0,
c5 := e.fp.Eval([][]*baseEl{{&x.A0, &x.A5}, {&x.A1, &x.A4}, {&x.A2, &x.A3}, {&x.A3, &x.A2}, {&x.A4, &x.A1}, {&x.A5, &x.A0}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})

return &E6{
A0: *c0,
A1: *c1,
A2: *c2,
A3: *c3,
A4: *c4,
A5: *c5,
}
}

func (e Ext6) squareEmulatedTower(x *E6) *E6 {
// We don't use Montgomery-6 or Toom-Cook-6 for the squaring but instead we
// simulate a quadratic over cubic extension tower because Karatsuba over
// Chung-Hasan SQR2 is better constraint wise.
Expand Down Expand Up @@ -792,6 +858,45 @@ func (e Ext6) Square(x *E6) *E6 {
// https://eprint.iacr.org/2010/542.pdf
// Sec. 5.6 with minor modifications to fit our tower
func (e Ext6) CyclotomicSquareKarabina12345(x *E6) *E6 {
return e.cyclotomicSquareKarabina12345Eval(x)
}

// cyclotomicSquareKarabina12345Eval computes
// [Ext6.cyclotomicSquareKarabina12345] but with the non-native Eval method.
func (e Ext6) cyclotomicSquareKarabina12345Eval(x *E6) *E6 {
c := e.fp.NewElement(-4)
mone := e.fp.NewElement(-1)
g1 := x.A2
g2 := x.A4
g3 := x.A1
g4 := x.A3
g5 := x.A5
// h1 = 3*c*g2^2 + 3*g3^2 - 2*g1
h1 := e.fp.Eval([][]*baseEl{{c, &g2, &g2}, {&g3, &g3}, {mone, &g1}}, []*big.Int{big.NewInt(3), big.NewInt(3), big.NewInt(2)})
// h2 = 3*c*g5^2 + 3*g1^2 - 2*g2
h2 := e.fp.Eval([][]*baseEl{{c, &g5, &g5}, {&g1, &g1}, {mone, &g2}}, []*big.Int{big.NewInt(3), big.NewInt(3), big.NewInt(2)})
// h3 = 6*c*g1*g5 + 2*g3
h3 := e.fp.Eval([][]*baseEl{{c, &g1, &g5}, {&g3}}, []*big.Int{big.NewInt(6), big.NewInt(2)})
// h4 = 3*c*g2*g5 + 3*g1*g3 - g4
h4 := e.fp.Eval([][]*baseEl{{c, &g2, &g5}, {&g1, &g3}, {mone, &g4}}, []*big.Int{big.NewInt(3), big.NewInt(3), big.NewInt(1)})
// h5 = 6*g2*g3 + 2*g5
h5 := e.fp.Eval([][]*baseEl{{&g2, &g3}, {&g5}}, []*big.Int{big.NewInt(6), big.NewInt(2)})

return &E6{
A0: x.A0,
A1: *h3,
A2: *h1,
A3: *h4,
A4: *h2,
A5: *h5,
}

}

// Karabina's compressed cyclotomic square SQR12345
// https://eprint.iacr.org/2010/542.pdf
// Sec. 5.6 with minor modifications to fit our tower
func (e Ext6) cyclotomicSquareKarabina12345(x *E6) *E6 {
x = e.Reduce(x)

// h4 = -g4 + 3((g3+g5)(g1+c*g2)-g1g5-c*g3g2)
Expand Down Expand Up @@ -852,6 +957,32 @@ func (e Ext6) CyclotomicSquareKarabina12345(x *E6) *E6 {

// DecompressKarabina12345 decompresses Karabina's cyclotomic square result SQR12345
func (e Ext6) DecompressKarabina12345(x *E6) *E6 {
return e.decompressKarabina12345Eval(x)
}

// decompressKarabina12345Eval computes [Ext6.DecompressKarabina12345] but with the non-native Eval method.
func (e Ext6) decompressKarabina12345Eval(x *E6) *E6 {
mone := e.fp.NewElement(-1)
c := e.fp.NewElement(-4)
g1 := x.A2
g2 := x.A4
g3 := x.A1
g4 := x.A3
g5 := x.A5
// h0 = -3*c*g1*g2 + 2*c*g4^2 + c*g3*g5 + 1
h0 := e.fp.Eval([][]*baseEl{{mone, c, &g1, &g2}, {c, &g4, &g4}, {c, &g3, &g5}, {e.fp.One()}}, []*big.Int{big.NewInt(3), big.NewInt(2), big.NewInt(1), big.NewInt(1)})
return &E6{
A0: *h0,
A1: g3,
A2: g1,
A3: g4,
A4: g2,
A5: g5,
}
}

// DecompressKarabina12345 decompresses Karabina's cyclotomic square result SQR12345
func (e Ext6) decompressKarabina12345(x *E6) *E6 {
x = e.Reduce(x)

// h0 = (2g4^2 + g3g5 - 3g2g1)*c + 1
Expand Down
105 changes: 102 additions & 3 deletions std/algebra/emulated/fields_bw6761/e6_pairing.go
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -132,6 +132,38 @@ func (e Ext6) ExpC2(z *E6) *E6 {
//
// E6{A0: c0, A1: 0, A2: c1, A3: 1, A4: 0, A5: 0}
func (e *Ext6) MulBy023(z *E6, c0, c1 *baseEl) *E6 {
return e.mulBy023Direct(z, c0, c1)
}

// mulBy023Direct multiplies z by an E6 sparse element 023 using schoolbook multiplication
func (e Ext6) mulBy023Direct(z *E6, c0, c1 *baseEl) *E6 {
nonResidue := e.fp.NewElement(-4)

// z0 = a0c0 + β(a3 + a4c1)
z0 := e.fp.Eval([][]*baseEl{{&z.A0, c0}, {nonResidue, &z.A3}, {nonResidue, &z.A4, c1}}, []*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// z1 = a1c0 + β(a4 + a5c1)
z1 := e.fp.Eval([][]*baseEl{{&z.A1, c0}, {nonResidue, &z.A4}, {nonResidue, &z.A5, c1}}, []*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// z2 = a0c1 + a2c0 + β(a5)
z2 := e.fp.Eval([][]*baseEl{{&z.A0, c1}, {&z.A2, c0}, {nonResidue, &z.A5}}, []*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c3 = a0 + a1c1 + a3c0
z3 := e.fp.Eval([][]*baseEl{{&z.A0}, {&z.A1, c1}, {&z.A3, c0}}, []*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c4 = a1 + a2c1 + a4c0
z4 := e.fp.Eval([][]*baseEl{{&z.A1}, {&z.A2, c1}, {&z.A4, c0}}, []*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c5 = a2 + a3c1 + a5c0,
z5 := e.fp.Eval([][]*baseEl{{&z.A2}, {&z.A3, c1}, {&z.A5, c0}}, []*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1)})

return &E6{
A0: *z0,
A1: *z1,
A2: *z2,
A3: *z3,
A4: *z4,
A5: *z5,
}
}

// mulBy023 multiplies z by an E6 sparse element 023
func (e Ext6) mulBy023(z *E6, c0, c1 *baseEl) *E6 {
z = e.Reduce(z)

a := e.fp.Mul(&z.A0, c0)
Expand Down Expand Up @@ -198,14 +230,42 @@ func (e *Ext6) MulBy023(z *E6, c0, c1 *baseEl) *E6 {

}

// Mul023By023 multiplies two E6 sparse element of the form:
// Mul023By023 multiplies two E6 sparse element of the form:
//
// E6{A0: c0, A1: 0, A2: c1, A3: 1, A4: 0, A5: 0}
//
// and
//
// E6{A0: c0, A1: 0, A2: c1, A3: 1, A4: 0, A5: 0}
func (e Ext6) Mul023By023(d0, d1, c0, c1 *baseEl) [5]*baseEl {
return e.mul023by023Direct(d0, d1, c0, c1)
}

// mul023by023Direct multiplies two E6 sparse element using schoolbook multiplication
func (e Ext6) mul023by023Direct(d0, d1, c0, c1 *baseEl) [5]*baseEl {
nonResidue := e.fp.NewElement(-4)
// c0 = d0c0 + β
z0 := e.fp.Eval([][]*baseEl{{d0, c0}, {nonResidue}}, []*big.Int{big.NewInt(1), big.NewInt(1)})
// c2 = d0c1 + d1c0
z2 := e.fp.Eval([][]*baseEl{{d0, c1}, {d1, c0}}, []*big.Int{big.NewInt(1), big.NewInt(1)})
// c3 = d0 + c0
z3 := e.fp.Add(d0, c0)
// c4 = d1c1
z4 := e.fp.Mul(d1, c1)
// c5 = d1 + c1,
z5 := e.fp.Add(d1, c1)

return [5]*baseEl{z0, z2, z3, z4, z5}
}

// mul023By023 multiplies two E6 sparse element of the form:
//
// E6{A0: c0, A1: 0, A2: c1, A3: 1, A4: 0, A5: 0}
//
// and
//
// E6{A0: c0, A1: 0, A2: c1, A3: 1, A4: 0, A5: 0}
func (e Ext6) mul023By023(d0, d1, c0, c1 *baseEl) [5]*baseEl {
x0 := e.fp.Mul(c0, d0)
x1 := e.fp.Mul(c1, d1)
x04 := e.fp.Add(c0, d0)
Expand All @@ -224,9 +284,48 @@ func (e Ext6) Mul023By023(d0, d1, c0, c1 *baseEl) [5]*baseEl {

// MulBy02345 multiplies z by an E6 sparse element of the form
//
// E6{A0: y0, A1: 0, A2: y1, A3: y2, A4: y3, A5: y4},
// }
// E6{A0: y0, A1: 0, A2: y1, A3: y2, A4: y3, A5: y4}
func (e *Ext6) MulBy02345(z *E6, x [5]*baseEl) *E6 {
return e.mulBy02345Direct(z, x)
}

// mulBy02345Direct multiplies z by an E6 sparse element using schoolbook multiplication
func (e Ext6) mulBy02345Direct(z *E6, x [5]*baseEl) *E6 {
nonResidue := e.fp.NewElement(-4)

// c0 = a0y0 + β(a1y4 + a2y3 + a3y2 + a4y1)
c0 := e.fp.Eval([][]*baseEl{{&z.A0, x[0]}, {nonResidue, &z.A1, x[4]}, {nonResidue, &z.A2, x[3]}, {nonResidue, &z.A3, x[2]}, {nonResidue, &z.A4, x[1]}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c1 = a1y0 + β(a2y4 + a3y3 + a4y2 + a5y1)
c1 := e.fp.Eval([][]*baseEl{{&z.A1, x[0]}, {nonResidue, &z.A2, x[4]}, {nonResidue, &z.A3, x[3]}, {nonResidue, &z.A4, x[2]}, {nonResidue, &z.A5, x[1]}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c2 = a0y1 + a2y0 + β(a3y4 + a4y3 + a5y2)
c2 := e.fp.Eval([][]*baseEl{{&z.A0, x[1]}, {&z.A2, x[0]}, {nonResidue, &z.A3, x[4]}, {nonResidue, &z.A4, x[3]}, {nonResidue, &z.A5, x[2]}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c3 = a0y2 + a1y1 + a3y0 + β(a4y4 + a5y3)
c3 := e.fp.Eval([][]*baseEl{{&z.A0, x[2]}, {&z.A1, x[1]}, {&z.A3, x[0]}, {nonResidue, &z.A4, x[4]}, {nonResidue, &z.A5, x[3]}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c4 = a0y3 + a1y2 + a2y1 + a4y0 + βa5y4
c4 := e.fp.Eval([][]*baseEl{{&z.A0, x[3]}, {&z.A1, x[2]}, {&z.A2, x[1]}, {&z.A4, x[0]}, {nonResidue, &z.A5, x[4]}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})
// c5 = a0y4 + a1y3 + a2y2 + a3y1 + a5y0,
c5 := e.fp.Eval([][]*baseEl{{&z.A0, x[4]}, {&z.A1, x[3]}, {&z.A2, x[2]}, {&z.A3, x[1]}, {&z.A5, x[0]}},
[]*big.Int{big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1)})

return &E6{
A0: *c0,
A1: *c1,
A2: *c2,
A3: *c3,
A4: *c4,
A5: *c5,
}
}

// mulBy02345 multiplies z by an E6 sparse element of the form
//
// E6{A0: y0, A1: 0, A2: y1, A3: y2, A4: y3, A5: y4},
func (e *Ext6) mulBy02345(z *E6, x [5]*baseEl) *E6 {
a0 := e.fp.Add(&z.A0, &z.A1)
a1 := e.fp.Add(&z.A2, &z.A3)
a2 := e.fp.Add(&z.A4, &z.A5)
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