gmsm/sm9/gfp12.go

package sm9

import "math/big"

// For details of the algorithms used, see "Multiplication and Squaring on
// Pairing-Friendly Fields, Devegili et al.
// http://eprint.iacr.org/2006/471.pdf.
//

// gfP12 implements the field of size p¹² as a cubic extension of gfP4 where v³=u
type gfP12 struct {
	x, y, z gfP4 // value is xw² + yw + z
}

func gfP12Decode(in *gfP12) *gfP12 {
	out := &gfP12{}
	out.x = *gfP4Decode(&in.x)
	out.y = *gfP4Decode(&in.y)
	out.z = *gfP4Decode(&in.z)
	return out
}

var gfP12Gen *gfP12 = &gfP12{
	x: gfP4{
		x: gfP2{
			x: *fromBigInt(bigFromHex("256943fbdb2bf87ab91ae7fbeaff14e146cf7e2279b9d155d13461e09b22f523")),
			y: *fromBigInt(bigFromHex("0167b0280051495c6af1ec23ba2cd2ff1cdcdeca461a5ab0b5449e9091308310")),
		},
		y: gfP2{
			x: *fromBigInt(bigFromHex("5e7addaddf7fbfe16291b4e89af50b8217ddc47ba3cba833c6e77c3fb027685e")),
			y: *fromBigInt(bigFromHex("79d0c8337072c93fef482bb055f44d6247ccac8e8e12525854b3566236337ebe")),
		},
	},
	y: gfP4{
		x: gfP2{
			x: *fromBigInt(bigFromHex("082cde173022da8cd09b28a2d80a8cee53894436a52007f978dc37f36116d39b")),
			y: *fromBigInt(bigFromHex("3fa7ed741eaed99a58f53e3df82df7ccd3407bcc7b1d44a9441920ced5fb824f")),
		},
		y: gfP2{
			x: *fromBigInt(bigFromHex("7fc6eb2aa771d99c9234fddd31752edfd60723e05a4ebfdeb5c33fbd47e0cf06")),
			y: *fromBigInt(bigFromHex("6fa6b6fa6dd6b6d3b19a959a110e748154eef796dc0fc2dd766ea414de786968")),
		},
	},
	z: gfP4{
		x: gfP2{
			x: *fromBigInt(bigFromHex("8ffe1c0e9de45fd0fed790ac26be91f6b3f0a49c084fe29a3fb6ed288ad7994d")),
			y: *fromBigInt(bigFromHex("1664a1366beb3196f0443e15f5f9042a947354a5678430d45ba031cff06db927")),
		},
		y: gfP2{
			x: *fromBigInt(bigFromHex("7f7c6d52b475e6aaa827fdc5b4175ac6929320f782d998f86b6b57cda42a0426")),
			y: *fromBigInt(bigFromHex("36a699de7c136f78eee2dbac4ca9727bff0cee02ee920f5822e65ea170aa9669")),
		},
	},
}

func (e *gfP12) String() string {
	return "(" + e.x.String() + ", " + e.y.String() + ", " + e.z.String() + ")"
}

func (e *gfP12) Set(a *gfP12) *gfP12 {
	e.x.Set(&a.x)
	e.y.Set(&a.y)
	e.z.Set(&a.z)
	return e
}

func (e *gfP12) SetZero() *gfP12 {
	e.x.SetZero()
	e.y.SetZero()
	e.z.SetZero()
	return e
}

func (e *gfP12) SetOne() *gfP12 {
	e.x.SetZero()
	e.y.SetZero()
	e.z.SetOne()
	return e
}

func (e *gfP12) SetW() *gfP12 {
	e.x.SetZero()
	e.y.SetOne()
	e.z.SetZero()
	return e
}

func (e *gfP12) SetW2() *gfP12 {
	e.x.SetOne()
	e.y.SetZero()
	e.z.SetZero()
	return e
}

func (e *gfP12) IsZero() bool {
	return e.x.IsZero() && e.y.IsZero() && e.z.IsZero()
}

func (e *gfP12) IsOne() bool {
	return e.x.IsZero() && e.y.IsZero() && e.z.IsOne()
}

func (e *gfP12) Neg(a *gfP12) *gfP12 {
	e.x.Neg(&a.x)
	e.y.Neg(&a.y)
	e.z.Neg(&a.z)
	return e
}

func (e *gfP12) Add(a, b *gfP12) *gfP12 {
	e.x.Add(&a.x, &b.x)
	e.y.Add(&a.y, &b.y)
	e.z.Add(&a.z, &b.z)
	return e
}

func (e *gfP12) Sub(a, b *gfP12) *gfP12 {
	e.x.Sub(&a.x, &b.x)
	e.y.Sub(&a.y, &b.y)
	e.z.Sub(&a.z, &b.z)
	return e
}

func (e *gfP12) MulScalar(a *gfP12, b *gfP4) *gfP12 {
	e.x.Mul(&a.x, b)
	e.y.Mul(&a.y, b)
	e.z.Mul(&a.z, b)
	return e
}

func (e *gfP12) MulGFP2(a *gfP12, b *gfP2) *gfP12 {
	e.x.MulScalar(&a.x, b)
	e.y.MulScalar(&a.y, b)
	e.z.MulScalar(&a.z, b)
	return e
}

func (e *gfP12) MulGFP(a *gfP12, b *gfP) *gfP12 {
	e.x.MulGFP(&a.x, b)
	e.y.MulGFP(&a.y, b)
	e.z.MulGFP(&a.z, b)
	return e
}

func (e *gfP12) Mul(a, b *gfP12) *gfP12 {
	// (z0 + y0*w + x0*w^2)* (z1 + y1*w + x1*w^2)
	//  z0*z1 + z0*y1*w + z0*x1*w^2
	// +y0*z1*w + y0*y1*w^2 + y0*x1*v
	// +x0*z1*w^2 + x0*y1*v + x0*x1*v*w
	//=(z0*z1+y0*x1*v+x0*y1*v) + (z0*y1+y0*z1+x0*x1*v)w + (z0*x1 + y0*y1 + x0*z1)*w^2
	tx, ty, tz, t := &gfP4{}, &gfP4{}, &gfP4{}, &gfP4{}
	tz.Mul(&a.z, &b.z)
	t.MulV(&a.y, &b.x)
	tz.Add(tz, t)
	t.MulV(&a.x, &b.y)
	tz.Add(tz, t)

	ty.Mul(&a.z, &b.y)
	t.Mul(&a.y, &b.z)
	ty.Add(ty, t)
	t.MulV(&a.x, &b.x)
	ty.Add(ty, t)

	tx.Mul(&a.z, &b.x)
	t.Mul(&a.y, &b.y)
	tx.Add(tx, t)
	t.Mul(&a.x, &b.z)
	tx.Add(tx, t)

	e.x.Set(tx)
	e.y.Set(ty)
	e.z.Set(tz)
	return e
}

func (e *gfP12) Square(a *gfP12) *gfP12 {
	// (z + y*w + x*w^2)* (z + y*w + x*w^2)
	// z^2 + z*y*w + z*x*w^2 + y*z*w + y^2*w^2 + y*x*v + x*z*w^2 + x*y*v + x^2 *v *w
	// (z^2 + y*x*v + x*y*v) + (z*y + y*z + v * x^2)w + (z*x + y^2 + x*z)*w^2
	// (z^2 + 2*x*y*v) + (v*x^2 + 2*y*z) *w + (y^2 + 2*x*z) * w^2
	tx, ty, tz, t := &gfP4{}, &gfP4{}, &gfP4{}, &gfP4{}

	tz.Square(&a.z)
	t.MulV(&a.x, &a.y)
	t.Add(t, t)
	tz.Add(tz, t)

	ty.SquareV(&a.x)
	t.Mul(&a.y, &a.z)
	t.Add(t, t)
	ty.Add(ty, t)

	tx.Square(&a.y)
	t.Mul(&a.x, &a.z)
	t.Add(t, t)
	tx.Add(tx, t)

	e.x.Set(tx)
	e.y.Set(ty)
	e.z.Set(tz)
	return e
}

func (e *gfP12) Exp(f *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, f)
		} else {
			sum.Set(t)
		}
	}

	e.Set(sum)
	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

	// Here we can give a short explanation of how it works: let j be a cubic root of
	// unity in GF(p^4) so that 1+j+j²=0.
	// Then (xτ² + yτ + z)(xj²τ² + yjτ + z)(xjτ² + yj²τ + z)
	// = (xτ² + yτ + z)(Cτ²+Bτ+A)
	// = (x³ξ²+y³ξ+z³-3ξxyz) = F is an element of the base field (the norm).
	//
	// On the other hand (xj²τ² + yjτ + z)(xjτ² + yj²τ + z)
	// = τ²(y²-ξxz) + τ(ξx²-yz) + (z²-ξxy)
	//
	// So that's why A = (z²-ξxy), B = (ξx²-yz), C = (y²-ξxz)
	t1 := (&gfP4{}).MulV(&a.x, &a.y)
	A := (&gfP4{}).Square(&a.z)
	A.Sub(A, t1)

	B := (&gfP4{}).SquareV(&a.x)
	t1.Mul(&a.y, &a.z)
	B.Sub(B, t1)

	C := (&gfP4{}).Square(&a.y)
	t1.Mul(&a.x, &a.z)
	C.Sub(C, t1)

	F := (&gfP4{}).MulV(C, &a.y)
	t1.Mul(A, &a.z)
	F.Add(F, t1)
	t1.MulV(B, &a.x)
	F.Add(F, t1)

	F.Invert(F)

	e.x.Mul(C, F)
	e.y.Mul(B, F)
	e.z.Mul(A, F)
	return e
}

// (z + y*w + x*w^2)^p
//= z^p + y^p*w*w^(p-1)+x^p*w^2*(w^2)^(p-1)
// w2ToP2Minus1 = vToPMinus1 * wToPMinus1
func (e *gfP12) Frobenius(a *gfP12) *gfP12 {
	x, y := &gfP2{}, &gfP2{}

	x.Conjugate(&a.z.x)
	y.Conjugate(&a.z.y)
	x.MulScalar(x, vToPMinus1)
	e.z.x.Set(x)
	e.z.y.Set(y)

	x.Conjugate(&a.y.x)
	y.Conjugate(&a.y.y)
	x.MulScalar(x, w2ToP2Minus1)
	y.MulScalar(y, wToPMinus1)
	e.y.x.Set(x)
	e.y.y.Set(y)

	x.Conjugate(&a.x.x)
	y.Conjugate(&a.x.y)
	x.MulScalar(x, vToPMinus1Mw2ToPMinus1)
	y.MulScalar(y, w2ToPMinus1)
	e.x.x.Set(x)
	e.x.y.Set(y)

	return e
}

// (z + y*w + x*w^2)^(p^2)
//= z^(p^2) + y^(p^2)*w*w^((p^2)-1)+x^(p^2)*w^2*(w^2)^((p^2)-1)
func (e *gfP12) FrobeniusP2(a *gfP12) *gfP12 {
	tx, ty, tz := &gfP4{}, &gfP4{}, &gfP4{}

	tz.Conjugate(&a.z)

	ty.Conjugate(&a.y)
	ty.MulGFP(ty, wToP2Minus1)

	tx.Conjugate(&a.x)
	tx.MulGFP(tx, w2ToP2Minus1)

	e.x.Set(tx)
	e.y.Set(ty)
	e.z.Set(tz)

	return e
}

// (z + y*w + x*w^2)^(p^3)
//=z^(p^3) + y^(p^3)*w*w^((p^3)-1)+x^(p^3)*w^2*(w^2)^((p^3)-1)
//=z^(p^3) + y^(p^3)*w*vToPMinus1-x^(p^3)*w^2
// vToPMinus1 * vToPMinus1 = -1
func (e *gfP12) FrobeniusP3(a *gfP12) *gfP12 {
	x, y := &gfP2{}, &gfP2{}

	x.Conjugate(&a.z.x)
	y.Conjugate(&a.z.y)
	x.MulScalar(x, vToPMinus1)
	x.Neg(x)
	e.z.x.Set(x)
	e.z.y.Set(y)

	x.Conjugate(&a.y.x)
	y.Conjugate(&a.y.y)
	//x.MulScalar(x, vToPMinus1)
	//x.Neg(x)
	//x.MulScalar(x, vToPMinus1)
	y.MulScalar(y, vToPMinus1)
	e.y.x.Set(x)
	e.y.y.Set(y)

	x.Conjugate(&a.x.x)
	y.Conjugate(&a.x.y)
	x.MulScalar(x, vToPMinus1)
	y.Neg(y)
	e.x.x.Set(x)
	e.x.y.Set(y)

	return e
}

// (z + y*w + x*w^2)^(p^6)
// = ((z + y*w + x*w^2)^(p^3))^(p^3)
func (e *gfP12) FrobeniusP6(a *gfP12) *gfP12 {
	tx, ty, tz := &gfP4{}, &gfP4{}, &gfP4{}

	tz.Conjugate(&a.z)

	ty.Conjugate(&a.y)
	ty.Neg(ty)

	tx.Conjugate(&a.x)

	e.x.Set(tx)
	e.y.Set(ty)
	e.z.Set(tz)

	return e
}

// code logic from https://github.com/miracl/MIRACL/blob/master/source/curve/pairing/zzn12a.h
func (e *gfP12) Conjugate(a *gfP12) *gfP12 {
	e.z.Conjugate(&a.z)
	e.y.Conjugate(&a.y)
	e.y.Neg(&e.y)
	e.x.Conjugate(&a.x)
	return e
}

// Select sets q to p1 if cond == 1, and to p2 if cond == 0.
func (q *gfP12) Select(p1, p2 *gfP12, cond int) *gfP12 {
	q.x.Select(&p1.x, &p2.x, cond)
	q.y.Select(&p1.y, &p2.y, cond)
	q.z.Select(&p1.z, &p2.z, cond)
	return q
}