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e6_pairing.go
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/
e6_pairing.go
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package fields_bw6761
import (
"math/big"
"github.com/aakash4dev/gnark2/std/math/emulated"
)
func (e Ext6) nSquareKarabina12345(z *E6, n int) *E6 {
for i := 0; i < n; i++ {
z = e.CyclotomicSquareKarabina12345(z)
}
z = e.DecompressKarabina12345(z)
return z
}
// ExpX0Minus1 set z to z^{x₀-1} in E6 and return z
// x₀-1 = 9586122913090633728
func (e Ext6) ExpX0Minus1(z *E6) *E6 {
z = e.Reduce(z)
result := e.Copy(z)
result = e.nSquareKarabina12345(result, 5)
result = e.Mul(result, z)
z33 := e.Copy(result)
result = e.nSquareKarabina12345(result, 7)
result = e.Mul(result, z33)
result = e.nSquareKarabina12345(result, 4)
result = e.Mul(result, z)
result = e.CyclotomicSquare(result)
result = e.Mul(result, z)
result = e.nSquareKarabina12345(result, 46)
return result
}
// ExpX0Minus1Square set z to z^{(x₀-1)²} in E6 and return z
// (x₀-1)² = 91893752504881257682351033800651177984
func (e Ext6) ExpX0Minus1Square(z *E6) *E6 {
z = e.Reduce(z)
result := e.Copy(z)
result = e.nSquareKarabina12345(result, 3)
t0 := e.CyclotomicSquare(result)
t2 := e.Mul(z, t0)
result = e.Mul(result, t2)
t0 = e.Mul(z, result)
t1 := e.CyclotomicSquare(t0)
t1 = e.Mul(t2, t1)
t3 := e.nSquareKarabina12345(t1, 7)
t2 = e.Mul(t2, t3)
t2 = e.nSquareKarabina12345(t2, 11)
t1 = e.Mul(t1, t2)
t0 = e.Mul(t0, t1)
t0 = e.nSquareKarabina12345(t0, 7)
result = e.Mul(result, t0)
result = e.nSquareKarabina12345(result, 3)
result = e.Mul(z, result)
result = e.nSquareKarabina12345(result, 92)
return result
}
// ExpX0Plus1 set z to z^(x₀+1) in E6 and return z
// x₀+1 = 9586122913090633730
func (e Ext6) ExpX0Plus1(z *E6) *E6 {
z = e.Reduce(z)
result := e.Copy(z)
t := e.CyclotomicSquare(result)
result = e.nSquareKarabina12345(t, 4)
result = e.Mul(result, z)
z33 := e.Copy(result)
result = e.nSquareKarabina12345(result, 7)
result = e.Mul(result, z33)
result = e.nSquareKarabina12345(result, 4)
result = e.Mul(result, z)
result = e.CyclotomicSquare(result)
result = e.Mul(result, z)
result = e.nSquareKarabina12345(result, 46)
result = e.Mul(result, t)
return result
}
// ExpX0Minus1Div3 set z to z^(x₀-1)/3 in E6 and return z
// (x₀-1)/3 = 3195374304363544576
func (e Ext6) ExptMinus1Div3(z *E6) *E6 {
z = e.Reduce(z)
result := e.Copy(z)
result = e.CyclotomicSquare(result)
result = e.CyclotomicSquare(result)
result = e.Mul(result, z)
result = e.CyclotomicSquare(result)
result = e.Mul(result, z)
t0 := e.nSquareKarabina12345(result, 7)
result = e.Mul(result, t0)
result = e.nSquareKarabina12345(result, 5)
result = e.Mul(result, z)
result = e.nSquareKarabina12345(result, 46)
return result
}
// ExpC1 set z to z^C1 in E6 and return z
// ht, hy = 13, 9
// C1 = (ht+hy)/2 = 11
func (e Ext6) ExpC1(z *E6) *E6 {
z = e.Reduce(z)
result := e.Copy(z)
result = e.CyclotomicSquare(result)
result = e.CyclotomicSquare(result)
result = e.Mul(result, z)
result = e.CyclotomicSquare(result)
result = e.Mul(result, z)
return result
}
// ExpC2 set z to z^C2 in E6 and return z
// ht, hy = 13, 9
// C2 = (ht**2+3*hy**2)/4 = 103
func (e Ext6) ExpC2(z *E6) *E6 {
z = e.Reduce(z)
result := e.CyclotomicSquare(z)
result = e.Mul(result, z)
t0 := e.nSquareKarabina12345(result, 4)
result = e.Mul(result, t0)
result = e.CyclotomicSquare(result)
result = e.Mul(result, z)
return result
}
// MulBy014 multiplies z by an E6 sparse element of the form
//
// E6{
// B0: E3{A0: c0, A1: c1, A2: 0},
// B1: E3{A0: 0, A1: 1, A2: 0},
// }
func (e *Ext6) MulBy014(z *E6, c0, c1 *baseEl) *E6 {
z = e.Reduce(z)
a := e.MulBy01(&z.B0, c0, c1)
var b E3
// Mul by E3{0, 1, 0}
b.A0 = *e.fp.MulConst(&z.B1.A2, big.NewInt(4))
b.A2 = *e.fp.Neg(&z.B1.A1)
b.A1 = *e.fp.Neg(&z.B1.A0)
one := e.fp.One()
d := e.fp.Add(c1, one)
zC1 := e.Ext3.Add(&z.B1, &z.B0)
zC1 = e.Ext3.MulBy01(zC1, c0, d)
zC1 = e.Ext3.Sub(zC1, a)
zC1 = e.Ext3.Add(zC1, &b)
zC0 := &E3{
A0: *e.fp.MulConst(&b.A2, big.NewInt(4)),
A1: *e.fp.Neg(&b.A0),
A2: *e.fp.Neg(&b.A1),
}
zC0 = e.Ext3.Add(zC0, a)
return &E6{
B0: *zC0,
B1: *zC1,
}
}
// multiplies two E6 sparse element of the form:
//
// E6{
// B0: E3{A0: c0, A1: c1, A2: 0},
// B1: E3{A0: 0, A1: 1, A2: 0},
// }
//
// and
//
// E6{
// B0: E3{A0: d0, A1: d1, A2: 0},
// B1: E3{A0: 0, A1: 1, A2: 0},
// }
func (e Ext6) Mul014By014(d0, d1, c0, c1 *baseEl) [5]*baseEl {
one := e.fp.One()
x0 := e.fp.Mul(c0, d0)
x1 := e.fp.Mul(c1, d1)
tmp := e.fp.Add(c0, one)
x04 := e.fp.Add(d0, one)
x04 = e.fp.Mul(x04, tmp)
x04 = e.fp.Sub(x04, x0)
x04 = e.fp.Sub(x04, one)
tmp = e.fp.Add(c0, c1)
x01 := e.fp.Add(d0, d1)
x01 = e.fp.Mul(x01, tmp)
x01 = e.fp.Sub(x01, x0)
x01 = e.fp.Sub(x01, x1)
tmp = e.fp.Add(c1, one)
x14 := e.fp.Add(d1, one)
x14 = e.fp.Mul(x14, tmp)
x14 = e.fp.Sub(x14, x1)
x14 = e.fp.Sub(x14, one)
four := emulated.ValueOf[emulated.BW6761Fp](big.NewInt(4))
zC0B0 := e.fp.Sub(x0, &four)
return [5]*baseEl{zC0B0, x01, x1, x04, x14}
}
// Mul01245By014 multiplies two E6 sparse element of the form
//
// E6{
// C0: E3{B0: x0, B1: x1, B2: x2},
// C1: E3{B0: 0, B1: x4, B2: x5},
// }
//
// and
//
// E6{
// C0: E3{B0: d0, B1: d1, B2: 0},
// C1: E3{B0: 0, B1: 1, B2: 0},
// }
func (e *Ext6) Mul01245By014(x [5]*baseEl, d0, d1 *baseEl) *E6 {
zero := e.fp.Zero()
c0 := &E3{A0: *x[0], A1: *x[1], A2: *x[2]}
b := &E3{
A0: *x[0],
A1: *e.fp.Add(x[1], x[3]),
A2: *e.fp.Add(x[2], x[4]),
}
a := e.Ext3.MulBy01(b, d0, e.fp.Add(d1, e.fp.One()))
b = e.Ext3.MulBy01(c0, d0, d1)
c := &E3{
A0: *e.fp.MulConst(x[4], big.NewInt(4)),
A1: *e.fp.Neg(zero),
A2: *e.fp.Neg(x[3]),
}
z1 := e.Ext3.Sub(a, b)
z1 = e.Ext3.Add(z1, c)
z0 := &E3{
A0: *e.fp.MulConst(&c.A2, big.NewInt(4)),
A1: *e.fp.Neg(&c.A0),
A2: *e.fp.Neg(&c.A1),
}
z0 = e.Ext3.Add(z0, b)
return &E6{
B0: *z0,
B1: *z1,
}
}