params.go

package bn256

import "math/big"

// CurveParams contains the parameters of an elliptic curve and also provides
// a generic, non-constant time implementation of Curve.
type CurveParams struct {
	P       *big.Int // the order of the underlying field
	N       *big.Int // the order of the base point
	B       *big.Int // the constant of the curve equation
	Gx, Gy  *big.Int // (x,y) of the base point
	BitSize int      // the size of the underlying field
	Name    string   // the canonical name of the curve
}

func (curve *CurveParams) Params() *CurveParams {
	return curve
}

// CurveParams operates, internally, on Jacobian coordinates. For a given
// (x, y) position on the curve, the Jacobian coordinates are (x1, y1, z1)
// where x = x1/z1² and y = y1/z1³. The greatest speedups come when the whole
// calculation can be performed within the transform (as in ScalarMult and
// ScalarBaseMult). But even for Add and Double, it's faster to apply and
// reverse the transform than to operate in affine coordinates.

// polynomial returns x³ + b.
func (curve *CurveParams) polynomial(x *big.Int) *big.Int {
	x3 := new(big.Int).Mul(x, x)
	x3.Mul(x3, x)

	x3.Add(x3, curve.B)
	x3.Mod(x3, curve.P)

	return x3
}

func (curve *CurveParams) IsOnCurve(x, y *big.Int) bool {
	if x.Sign() < 0 || x.Cmp(curve.P) >= 0 ||
		y.Sign() < 0 || y.Cmp(curve.P) >= 0 {
		return false
	}

	// y² = x³ + b
	y2 := new(big.Int).Mul(y, y)
	y2.Mod(y2, curve.P)

	return curve.polynomial(x).Cmp(y2) == 0
}

// zForAffine returns a Jacobian Z value for the affine point (x, y). If x and
// y are zero, it assumes that they represent the point at infinity because (0,
// 0) is not on the any of the curves handled here.
func zForAffine(x, y *big.Int) *big.Int {
	z := new(big.Int)
	if x.Sign() != 0 || y.Sign() != 0 {
		z.SetInt64(1)
	}
	return z
}

// affineFromJacobian reverses the Jacobian transform. See the comment at the
// top of the file. If the point is ∞ it returns 0, 0.
func (curve *CurveParams) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) {
	if z.Sign() == 0 {
		return new(big.Int), new(big.Int)
	}

	zinv := new(big.Int).ModInverse(z, curve.P)
	zinvsq := new(big.Int).Mul(zinv, zinv)

	xOut = new(big.Int).Mul(x, zinvsq)
	xOut.Mod(xOut, curve.P)
	zinvsq.Mul(zinvsq, zinv)
	yOut = new(big.Int).Mul(y, zinvsq)
	yOut.Mod(yOut, curve.P)
	return
}

func (curve *CurveParams) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
	z1 := zForAffine(x1, y1)
	z2 := zForAffine(x2, y2)
	return curve.affineFromJacobian(curve.addJacobian(x1, y1, z1, x2, y2, z2))
}

// addJacobian takes two points in Jacobian coordinates, (x1, y1, z1) and
// (x2, y2, z2) and returns their sum, also in Jacobian form.
func (curve *CurveParams) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) {
	// See https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
	x3, y3, z3 := new(big.Int), new(big.Int), new(big.Int)
	if z1.Sign() == 0 {
		x3.Set(x2)
		y3.Set(y2)
		z3.Set(z2)
		return x3, y3, z3
	}
	if z2.Sign() == 0 {
		x3.Set(x1)
		y3.Set(y1)
		z3.Set(z1)
		return x3, y3, z3
	}

	z1z1 := new(big.Int).Mul(z1, z1)
	z1z1.Mod(z1z1, curve.P)
	z2z2 := new(big.Int).Mul(z2, z2)
	z2z2.Mod(z2z2, curve.P)

	u1 := new(big.Int).Mul(x1, z2z2)
	u1.Mod(u1, curve.P)
	u2 := new(big.Int).Mul(x2, z1z1)
	u2.Mod(u2, curve.P)
	h := new(big.Int).Sub(u2, u1)
	xEqual := h.Sign() == 0
	if h.Sign() == -1 {
		h.Add(h, curve.P)
	}
	i := new(big.Int).Lsh(h, 1)
	i.Mul(i, i)
	j := new(big.Int).Mul(h, i)

	s1 := new(big.Int).Mul(y1, z2)
	s1.Mul(s1, z2z2)
	s1.Mod(s1, curve.P)
	s2 := new(big.Int).Mul(y2, z1)
	s2.Mul(s2, z1z1)
	s2.Mod(s2, curve.P)
	r := new(big.Int).Sub(s2, s1)
	if r.Sign() == -1 {
		r.Add(r, curve.P)
	}
	yEqual := r.Sign() == 0
	if xEqual && yEqual {
		return curve.doubleJacobian(x1, y1, z1)
	}
	r.Lsh(r, 1)
	v := new(big.Int).Mul(u1, i)

	x3.Set(r)
	x3.Mul(x3, x3)
	x3.Sub(x3, j)
	x3.Sub(x3, v)
	x3.Sub(x3, v)
	x3.Mod(x3, curve.P)

	y3.Set(r)
	v.Sub(v, x3)
	y3.Mul(y3, v)
	s1.Mul(s1, j)
	s1.Lsh(s1, 1)
	y3.Sub(y3, s1)
	y3.Mod(y3, curve.P)

	z3.Add(z1, z2)
	z3.Mul(z3, z3)
	z3.Sub(z3, z1z1)
	z3.Sub(z3, z2z2)
	z3.Mul(z3, h)
	z3.Mod(z3, curve.P)

	return x3, y3, z3
}

func (curve *CurveParams) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
	z1 := zForAffine(x1, y1)
	return curve.affineFromJacobian(curve.doubleJacobian(x1, y1, z1))
}

// doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and
// returns its double, also in Jacobian form.
func (curve *CurveParams) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) {
	// See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/doubling/dbl-2009-l.op3
	a := new(big.Int).Mul(x, x)
	a.Mod(a, curve.P)
	b := new(big.Int).Mul(y, y)
	b.Mod(b, curve.P)
	c := new(big.Int).Mul(b, b)
	c.Mod(c, curve.P)

	d := new(big.Int).Add(x, b)
	d.Mul(d, d)
	d.Sub(d, a)
	d.Sub(d, c)
	d.Lsh(d, 1)
	if d.Sign() < 0 {
		d.Add(d, curve.P)
	} else {
		d.Mod(d, curve.P)
	}

	e := new(big.Int).Lsh(a, 1)
	e.Add(e, a)
	f := new(big.Int).Mul(e, e)
	x3 := new(big.Int).Lsh(d, 1)
	x3.Sub(f, x3)
	if x3.Sign() < 0 {
		x3.Add(x3, curve.P)
	} else {
		x3.Mod(x3, curve.P)
	}

	y3 := new(big.Int).Sub(d, x3)
	y3.Mul(y3, e)
	c.Lsh(c, 3)
	y3.Sub(y3, c)
	if y3.Sign() < 0 {
		y3.Add(y3, curve.P)
	} else {
		y3.Mod(y3, curve.P)
	}

	z3 := new(big.Int).Mul(y, z)
	z3.Lsh(z3, 1)
	z3.Mod(z3, curve.P)

	return x3, y3, z3
}

func (curve *CurveParams) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
	Bz := new(big.Int).SetInt64(1)
	x, y, z := new(big.Int), new(big.Int), new(big.Int)

	for _, byte := range k {
		for bitNum := 0; bitNum < 8; bitNum++ {
			x, y, z = curve.doubleJacobian(x, y, z)
			if byte&0x80 == 0x80 {
				x, y, z = curve.addJacobian(Bx, By, Bz, x, y, z)
			}
			byte <<= 1
		}
	}

	return curve.affineFromJacobian(x, y, z)
}

func (curve *CurveParams) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
	return curve.ScalarMult(curve.Gx, curve.Gy, k)
}