btcd/btcec.go

// Copyright 2010 The Go Authors. All rights reserved.
// Copyright 2011 ThePiachu. All rights reserved.
// Copyright 2013 Conformal Systems LLC. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.

package btcec

// This package 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.

import (
	"crypto/elliptic"
	"math/big"
	"sync"
)

//TODO: examine if we need to care about EC optimization as descibed here
//      https://bitcointalk.org/index.php?topic=155054.0;all

// KoblitzCurve supports a koblitz curve implementation that fits the ECC Curve
// interface from crypto/elliptic.
type KoblitzCurve struct {
	*elliptic.CurveParams
	q *big.Int
}

// Params returns the parameters fro the curve.
func (curve *KoblitzCurve) Params() *elliptic.CurveParams {
	return curve.CurveParams
}

// IsOnCurve returns boolean if the point (x,y) is on the curve.
// Part of the elliptic.Curve interface. This function differs from the
// crypto/elliptic algorithm since a = 0 not -3.
func (curve *KoblitzCurve) IsOnCurve(x, y *big.Int) bool {
	// y² = x³ + b
	y2 := new(big.Int).Mul(y, y) //y²
	y2.Mod(y2, curve.P)          //y²%P

	x3 := new(big.Int).Mul(x, x) //x²
	x3.Mul(x3, x)                //x³

	x3.Add(x3, curve.B) //x³+B
	x3.Mod(x3, curve.P) //(x³+B)%P

	return x3.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 *KoblitzCurve) 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
}

// Add returns the sum of (x1,y1 and (x2,y2). Part of the elliptic.Curve
// interface.
func (curve *KoblitzCurve) 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 *KoblitzCurve) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) {
	// See http://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
}

// Double returns 2*(x1,y1). Part of the elliptic.Curve interface.
func (curve *KoblitzCurve) 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 *KoblitzCurve) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) {
	// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l

	a := new(big.Int).Mul(x, x) //X1²
	b := new(big.Int).Mul(y, y) //Y1²
	c := new(big.Int).Mul(b, b) //B²

	d := new(big.Int).Add(x, b) //X1+B
	d.Mul(d, d)                 //(X1+B)²
	d.Sub(d, a)                 //(X1+B)²-A
	d.Sub(d, c)                 //(X1+B)²-A-C
	d.Mul(d, big.NewInt(2))     //2*((X1+B)²-A-C)

	e := new(big.Int).Mul(big.NewInt(3), a) //3*A
	f := new(big.Int).Mul(e, e)             //E²

	x3 := new(big.Int).Mul(big.NewInt(2), d) //2*D
	x3.Sub(f, x3)                            //F-2*D
	x3.Mod(x3, curve.P)

	y3 := new(big.Int).Sub(d, x3)                  //D-X3
	y3.Mul(e, y3)                                  //E*(D-X3)
	y3.Sub(y3, new(big.Int).Mul(big.NewInt(8), c)) //E*(D-X3)-8*C
	y3.Mod(y3, curve.P)

	z3 := new(big.Int).Mul(y, z) //Y1*Z1
	z3.Mul(big.NewInt(2), z3)    //3*Y1*Z1
	z3.Mod(z3, curve.P)

	return x3, y3, z3
}

// ScalarMult returns k*(Bx, By) where k is a big endian integer.
// Part of the elliptic.Curve interface.
func (curve *KoblitzCurve) 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)
}

// ScalarBaseMult returns k*G where G is the base point of the group and k is a
// big endian integer.
// Part of the elliptic.Curve interface.
func (curve *KoblitzCurve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
	return curve.ScalarMult(curve.Gx, curve.Gy, k)
}

// QPlus1Div4 returns the Q+1/4 constant for the curve for use in calculating
// square roots via exponention.
func (curve *KoblitzCurve) QPlus1Div4() *big.Int {
	if curve.q == nil {
		curve.q = new(big.Int).Div(new(big.Int).Add(secp256k1.P, big.NewInt(1)), big.NewInt(4))
	}
	return curve.q
}

// Curve parameters taken from: http://www.secg.org/collateral/sec2_final.pdf
var initonce sync.Once
var secp256k1 KoblitzCurve

func initAll() {
	initS256()
}

func initS256() {
	// See SEC 2 section 2.7.1
	secp256k1.CurveParams = new(elliptic.CurveParams)
	secp256k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F", 16)
	secp256k1.N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141", 16)
	secp256k1.B, _ = new(big.Int).SetString("0000000000000000000000000000000000000000000000000000000000000007", 16)
	secp256k1.Gx, _ = new(big.Int).SetString("79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798", 16)
	secp256k1.Gy, _ = new(big.Int).SetString("483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8", 16)
	secp256k1.BitSize = 256
}

// S256 returns a Curve which implements secp256k1.
func S256() *KoblitzCurve {
	initonce.Do(initAll)
	return &secp256k1
}