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package bn256
// For details of the algorithms used, see "Multiplication and Squaring on
// Pairing-Friendly Fields, Devegili et al.
// http://eprint.iacr.org/2006/471.pdf.
import (
"math/big"
)
// gfP12 implements the field of size p¹² as a quadratic extension of gfP6
// where ω²=τ.
type gfP12 struct {
x, y gfP6 // value is xω + y
}
func (e *gfP12) String() string {
return "(" + e.x.String() + "," + e.y.String() + ")"
}
func (e *gfP12) Set(a *gfP12) *gfP12 {
e.x.Set(&a.x)
e.y.Set(&a.y)
return e
}
func (e *gfP12) SetZero() *gfP12 {
e.x.SetZero()
e.y.SetZero()
return e
}
func (e *gfP12) SetOne() *gfP12 {
e.x.SetZero()
e.y.SetOne()
return e
}
func (e *gfP12) IsZero() bool {
return e.x.IsZero() && e.y.IsZero()
}
func (e *gfP12) IsOne() bool {
return e.x.IsZero() && e.y.IsOne()
}
func (e *gfP12) Conjugate(a *gfP12) *gfP12 {
e.x.Neg(&a.x)
e.y.Set(&a.y)
return e
}
func (e *gfP12) Neg(a *gfP12) *gfP12 {
e.x.Neg(&a.x)
e.y.Neg(&a.y)
return e
}
// Frobenius computes (xω+y)^p = x^p ω·ξ^((p-1)/6) + y^p
func (e *gfP12) Frobenius(a *gfP12) *gfP12 {
e.x.Frobenius(&a.x)
e.y.Frobenius(&a.y)
e.x.MulScalar(&e.x, xiToPMinus1Over6)
return e
}
// FrobeniusP2 computes (xω+y)^p² = x^p² ω·ξ^((p²-1)/6) + y^p²
func (e *gfP12) FrobeniusP2(a *gfP12) *gfP12 {
e.x.FrobeniusP2(&a.x)
e.x.MulGFP(&e.x, xiToPSquaredMinus1Over6)
e.y.FrobeniusP2(&a.y)
return e
}
func (e *gfP12) FrobeniusP4(a *gfP12) *gfP12 {
e.x.FrobeniusP4(&a.x)
e.x.MulGFP(&e.x, xiToPSquaredMinus1Over3)
e.y.FrobeniusP4(&a.y)
return e
}
func (e *gfP12) Add(a, b *gfP12) *gfP12 {
e.x.Add(&a.x, &b.x)
e.y.Add(&a.y, &b.y)
return e
}
func (e *gfP12) Sub(a, b *gfP12) *gfP12 {
e.x.Sub(&a.x, &b.x)
e.y.Sub(&a.y, &b.y)
return e
}
func (e *gfP12) Mul(a, b *gfP12) *gfP12 {
tx := (&gfP6{}).Mul(&a.x, &b.y)
t := (&gfP6{}).Mul(&b.x, &a.y)
tx.Add(tx, t)
ty := (&gfP6{}).Mul(&a.y, &b.y)
t.Mul(&a.x, &b.x).MulTau(t)
e.x.Set(tx)
e.y.Add(ty, t)
return e
}
func (e *gfP12) MulScalar(a *gfP12, b *gfP6) *gfP12 {
e.x.Mul(&e.x, b)
e.y.Mul(&e.y, b)
return e
}
func (c *gfP12) Exp(a *gfP12, power *big.Int) *gfP12 {
sum := (&gfP12{}).SetOne()
t := &gfP12{}
for i := power.BitLen() - 1; i >= 0; i-- {
t.Square(sum)
if power.Bit(i) != 0 {
sum.Mul(t, a)
} else {
sum.Set(t)
}
}
c.Set(sum)
return c
}
func (e *gfP12) Square(a *gfP12) *gfP12 {
// Complex squaring algorithm
v0 := (&gfP6{}).Mul(&a.x, &a.y)
t := (&gfP6{}).MulTau(&a.x)
t.Add(&a.y, t)
ty := (&gfP6{}).Add(&a.x, &a.y)
ty.Mul(ty, t).Sub(ty, v0)
t.MulTau(v0)
ty.Sub(ty, t)
e.x.Add(v0, v0)
e.y.Set(ty)
return e
}
func (e *gfP12) Invert(a *gfP12) *gfP12 {
// See "Implementing cryptographic pairings", M. Scott, section 3.2.
// ftp://136.206.11.249/pub/crypto/pairings.pdf
t1, t2 := &gfP6{}, &gfP6{}
t1.Square(&a.x)
t2.Square(&a.y)
t1.MulTau(t1).Sub(t2, t1)
t2.Invert(t1)
e.x.Neg(&a.x)
e.y.Set(&a.y)
e.MulScalar(e, t2)
return e
}
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