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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.
// gfP2 implements a field of size p² as a quadratic extension of the base field
// where i²=-1.
type gfP2 struct {
x, y gfP // value is xi+y.
}
func gfP2Decode(in *gfP2) *gfP2 {
out := &gfP2{}
montDecode(&out.x, &in.x)
montDecode(&out.y, &in.y)
return out
}
func (e *gfP2) String() string {
return "(" + e.x.String() + ", " + e.y.String() + ")"
}
func (e *gfP2) Set(a *gfP2) *gfP2 {
e.x.Set(&a.x)
e.y.Set(&a.y)
return e
}
func (e *gfP2) SetZero() *gfP2 {
e.x = gfP{0}
e.y = gfP{0}
return e
}
func (e *gfP2) SetOne() *gfP2 {
e.x = gfP{0}
e.y = *newGFp(1)
return e
}
func (e *gfP2) IsZero() bool {
zero := gfP{0}
return e.x == zero && e.y == zero
}
func (e *gfP2) IsOne() bool {
zero, one := gfP{0}, *newGFp(1)
return e.x == zero && e.y == one
}
func (e *gfP2) Conjugate(a *gfP2) *gfP2 {
e.y.Set(&a.y)
gfpNeg(&e.x, &a.x)
return e
}
func (e *gfP2) Neg(a *gfP2) *gfP2 {
gfpNeg(&e.x, &a.x)
gfpNeg(&e.y, &a.y)
return e
}
func (e *gfP2) Add(a, b *gfP2) *gfP2 {
gfpAdd(&e.x, &a.x, &b.x)
gfpAdd(&e.y, &a.y, &b.y)
return e
}
func (e *gfP2) Sub(a, b *gfP2) *gfP2 {
gfpSub(&e.x, &a.x, &b.x)
gfpSub(&e.y, &a.y, &b.y)
return e
}
// See "Multiplication and Squaring in Pairing-Friendly Fields",
// http://eprint.iacr.org/2006/471.pdf
func (e *gfP2) Mul(a, b *gfP2) *gfP2 {
tx, t := &gfP{}, &gfP{}
gfpMul(tx, &a.x, &b.y)
gfpMul(t, &b.x, &a.y)
gfpAdd(tx, tx, t)
ty := &gfP{}
gfpMul(ty, &a.y, &b.y)
gfpMul(t, &a.x, &b.x)
gfpSub(ty, ty, t)
e.x.Set(tx)
e.y.Set(ty)
return e
}
func (e *gfP2) MulScalar(a *gfP2, b *gfP) *gfP2 {
gfpMul(&e.x, &a.x, b)
gfpMul(&e.y, &a.y, b)
return e
}
// MulXi sets e=ξa where ξ=i+9 and then returns e.
func (e *gfP2) MulXi(a *gfP2) *gfP2 {
// (xi+y)(i+9) = (9x+y)i+(9y-x)
tx := &gfP{}
gfpAdd(tx, &a.x, &a.x)
gfpAdd(tx, tx, tx)
gfpAdd(tx, tx, tx)
gfpAdd(tx, tx, &a.x)
gfpAdd(tx, tx, &a.y)
ty := &gfP{}
gfpAdd(ty, &a.y, &a.y)
gfpAdd(ty, ty, ty)
gfpAdd(ty, ty, ty)
gfpAdd(ty, ty, &a.y)
gfpSub(ty, ty, &a.x)
e.x.Set(tx)
e.y.Set(ty)
return e
}
func (e *gfP2) Square(a *gfP2) *gfP2 {
// Complex squaring algorithm:
// (xi+y)² = (x+y)(y-x) + 2*i*x*y
tx, ty := &gfP{}, &gfP{}
gfpSub(tx, &a.y, &a.x)
gfpAdd(ty, &a.x, &a.y)
gfpMul(ty, tx, ty)
gfpMul(tx, &a.x, &a.y)
gfpAdd(tx, tx, tx)
e.x.Set(tx)
e.y.Set(ty)
return e
}
func (e *gfP2) Invert(a *gfP2) *gfP2 {
// See "Implementing cryptographic pairings", M. Scott, section 3.2.
// ftp://136.206.11.249/pub/crypto/pairings.pdf
t1, t2 := &gfP{}, &gfP{}
gfpMul(t1, &a.x, &a.x)
gfpMul(t2, &a.y, &a.y)
gfpAdd(t1, t1, t2)
inv := &gfP{}
inv.Invert(t1)
gfpNeg(t1, &a.x)
gfpMul(&e.x, t1, inv)
gfpMul(&e.y, &a.y, inv)
return e
}
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