Added S256 curve

crypto/curve.go

@@ -0,0 +1,363 @@
+package crypto
+
+// Copyright 2010 The Go Authors. All rights reserved.
+// Copyright 2011 ThePiachu. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Package bitelliptic implements several Koblitz elliptic curves over prime
+// fields.
+
+// 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"
+	"io"
+	"math/big"
+	"sync"
+)
+
+// A BitCurve represents a Koblitz Curve with a=0.
+// See http://www.hyperelliptic.org/EFD/g1p/auto-shortw.html
+type BitCurve 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 BitCurve equation
+	Gx, Gy  *big.Int // (x,y) of the base point
+	BitSize int      // the size of the underlying field
+}
+
+func (BitCurve *BitCurve) Params() *elliptic.CurveParams {
+	return &elliptic.CurveParams{BitCurve.P, BitCurve.N, BitCurve.B, BitCurve.Gx, BitCurve.Gy, BitCurve.BitSize}
+}
+
+// IsOnBitCurve returns true if the given (x,y) lies on the BitCurve.
+func (BitCurve *BitCurve) IsOnCurve(x, y *big.Int) bool {
+	// y² = x³ + b
+	y2 := new(big.Int).Mul(y, y) //y²
+	y2.Mod(y2, BitCurve.P)       //y²%P
+
+	x3 := new(big.Int).Mul(x, x) //x²
+	x3.Mul(x3, x)                //x³
+
+	x3.Add(x3, BitCurve.B) //x³+B
+	x3.Mod(x3, BitCurve.P) //(x³+B)%P
+
+	return x3.Cmp(y2) == 0
+}
+
+//TODO: double check if the function is okay
+// affineFromJacobian reverses the Jacobian transform. See the comment at the
+// top of the file.
+func (BitCurve *BitCurve) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) {
+	zinv := new(big.Int).ModInverse(z, BitCurve.P)
+	zinvsq := new(big.Int).Mul(zinv, zinv)
+
+	xOut = new(big.Int).Mul(x, zinvsq)
+	xOut.Mod(xOut, BitCurve.P)
+	zinvsq.Mul(zinvsq, zinv)
+	yOut = new(big.Int).Mul(y, zinvsq)
+	yOut.Mod(yOut, BitCurve.P)
+	return
+}
+
+// Add returns the sum of (x1,y1) and (x2,y2)
+func (BitCurve *BitCurve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
+	z := new(big.Int).SetInt64(1)
+	return BitCurve.affineFromJacobian(BitCurve.addJacobian(x1, y1, z, x2, y2, z))
+}
+
+// addJacobian takes two points in Jacobian coordinates, (x1, y1, z1) and
+// (x2, y2, z2) and returns their sum, also in Jacobian form.
+func (BitCurve *BitCurve) 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
+	z1z1 := new(big.Int).Mul(z1, z1)
+	z1z1.Mod(z1z1, BitCurve.P)
+	z2z2 := new(big.Int).Mul(z2, z2)
+	z2z2.Mod(z2z2, BitCurve.P)
+
+	u1 := new(big.Int).Mul(x1, z2z2)
+	u1.Mod(u1, BitCurve.P)
+	u2 := new(big.Int).Mul(x2, z1z1)
+	u2.Mod(u2, BitCurve.P)
+	h := new(big.Int).Sub(u2, u1)
+	if h.Sign() == -1 {
+		h.Add(h, BitCurve.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, BitCurve.P)
+	s2 := new(big.Int).Mul(y2, z1)
+	s2.Mul(s2, z1z1)
+	s2.Mod(s2, BitCurve.P)
+	r := new(big.Int).Sub(s2, s1)
+	if r.Sign() == -1 {
+		r.Add(r, BitCurve.P)
+	}
+	r.Lsh(r, 1)
+	v := new(big.Int).Mul(u1, i)
+
+	x3 := new(big.Int).Set(r)
+	x3.Mul(x3, x3)
+	x3.Sub(x3, j)
+	x3.Sub(x3, v)
+	x3.Sub(x3, v)
+	x3.Mod(x3, BitCurve.P)
+
+	y3 := new(big.Int).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, BitCurve.P)
+
+	z3 := new(big.Int).Add(z1, z2)
+	z3.Mul(z3, z3)
+	z3.Sub(z3, z1z1)
+	if z3.Sign() == -1 {
+		z3.Add(z3, BitCurve.P)
+	}
+	z3.Sub(z3, z2z2)
+	if z3.Sign() == -1 {
+		z3.Add(z3, BitCurve.P)
+	}
+	z3.Mul(z3, h)
+	z3.Mod(z3, BitCurve.P)
+
+	return x3, y3, z3
+}
+
+// Double returns 2*(x,y)
+func (BitCurve *BitCurve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
+	z1 := new(big.Int).SetInt64(1)
+	return BitCurve.affineFromJacobian(BitCurve.doubleJacobian(x1, y1, z1))
+}
+
+// doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and
+// returns its double, also in Jacobian form.
+func (BitCurve *BitCurve) 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, BitCurve.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, BitCurve.P)
+
+	z3 := new(big.Int).Mul(y, z) //Y1*Z1
+	z3.Mul(big.NewInt(2), z3)    //3*Y1*Z1
+	z3.Mod(z3, BitCurve.P)
+
+	return x3, y3, z3
+}
+
+//TODO: double check if it is okay
+// ScalarMult returns k*(Bx,By) where k is a number in big-endian form.
+func (BitCurve *BitCurve) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
+	// We have a slight problem in that the identity of the group (the
+	// point at infinity) cannot be represented in (x, y) form on a finite
+	// machine. Thus the standard add/double algorithm has to be tweaked
+	// slightly: our initial state is not the identity, but x, and we
+	// ignore the first true bit in |k|.  If we don't find any true bits in
+	// |k|, then we return nil, nil, because we cannot return the identity
+	// element.
+
+	Bz := new(big.Int).SetInt64(1)
+	x := Bx
+	y := By
+	z := Bz
+
+	seenFirstTrue := false
+	for _, byte := range k {
+		for bitNum := 0; bitNum < 8; bitNum++ {
+			if seenFirstTrue {
+				x, y, z = BitCurve.doubleJacobian(x, y, z)
+			}
+			if byte&0x80 == 0x80 {
+				if !seenFirstTrue {
+					seenFirstTrue = true
+				} else {
+					x, y, z = BitCurve.addJacobian(Bx, By, Bz, x, y, z)
+				}
+			}
+			byte <<= 1
+		}
+	}
+
+	if !seenFirstTrue {
+		return nil, nil
+	}
+
+	return BitCurve.affineFromJacobian(x, y, z)
+}
+
+// ScalarBaseMult returns k*G, where G is the base point of the group and k is
+// an integer in big-endian form.
+func (BitCurve *BitCurve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
+	return BitCurve.ScalarMult(BitCurve.Gx, BitCurve.Gy, k)
+}
+
+var mask = []byte{0xff, 0x1, 0x3, 0x7, 0xf, 0x1f, 0x3f, 0x7f}
+
+//TODO: double check if it is okay
+// GenerateKey returns a public/private key pair. The private key is generated
+// using the given reader, which must return random data.
+func (BitCurve *BitCurve) GenerateKey(rand io.Reader) (priv []byte, x, y *big.Int, err error) {
+	byteLen := (BitCurve.BitSize + 7) >> 3
+	priv = make([]byte, byteLen)
+
+	for x == nil {
+		_, err = io.ReadFull(rand, priv)
+		if err != nil {
+			return
+		}
+		// We have to mask off any excess bits in the case that the size of the
+		// underlying field is not a whole number of bytes.
+		priv[0] &= mask[BitCurve.BitSize%8]
+		// This is because, in tests, rand will return all zeros and we don't
+		// want to get the point at infinity and loop forever.
+		priv[1] ^= 0x42
+		x, y = BitCurve.ScalarBaseMult(priv)
+	}
+	return
+}
+
+// Marshal converts a point into the form specified in section 4.3.6 of ANSI
+// X9.62.
+func (BitCurve *BitCurve) Marshal(x, y *big.Int) []byte {
+	byteLen := (BitCurve.BitSize + 7) >> 3
+
+	ret := make([]byte, 1+2*byteLen)
+	ret[0] = 4 // uncompressed point
+
+	xBytes := x.Bytes()
+	copy(ret[1+byteLen-len(xBytes):], xBytes)
+	yBytes := y.Bytes()
+	copy(ret[1+2*byteLen-len(yBytes):], yBytes)
+	return ret
+}
+
+// Unmarshal converts a point, serialised by Marshal, into an x, y pair. On
+// error, x = nil.
+func (BitCurve *BitCurve) Unmarshal(data []byte) (x, y *big.Int) {
+	byteLen := (BitCurve.BitSize + 7) >> 3
+	if len(data) != 1+2*byteLen {
+		return
+	}
+	if data[0] != 4 { // uncompressed form
+		return
+	}
+	x = new(big.Int).SetBytes(data[1 : 1+byteLen])
+	y = new(big.Int).SetBytes(data[1+byteLen:])
+	return
+}
+
+//curve parameters taken from:
+//http://www.secg.org/collateral/sec2_final.pdf
+
+var initonce sync.Once
+var ecp160k1 *BitCurve
+var ecp192k1 *BitCurve
+var ecp224k1 *BitCurve
+var ecp256k1 *BitCurve
+
+func initAll() {
+	initS160()
+	initS192()
+	initS224()
+	initS256()
+}
+
+func initS160() {
+	// See SEC 2 section 2.4.1
+	ecp160k1 = new(BitCurve)
+	ecp160k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFAC73", 16)
+	ecp160k1.N, _ = new(big.Int).SetString("0100000000000000000001B8FA16DFAB9ACA16B6B3", 16)
+	ecp160k1.B, _ = new(big.Int).SetString("0000000000000000000000000000000000000007", 16)
+	ecp160k1.Gx, _ = new(big.Int).SetString("3B4C382CE37AA192A4019E763036F4F5DD4D7EBB", 16)
+	ecp160k1.Gy, _ = new(big.Int).SetString("938CF935318FDCED6BC28286531733C3F03C4FEE", 16)
+	ecp160k1.BitSize = 160
+}
+
+func initS192() {
+	// See SEC 2 section 2.5.1
+	ecp192k1 = new(BitCurve)
+	ecp192k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFEE37", 16)
+	ecp192k1.N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFE26F2FC170F69466A74DEFD8D", 16)
+	ecp192k1.B, _ = new(big.Int).SetString("000000000000000000000000000000000000000000000003", 16)
+	ecp192k1.Gx, _ = new(big.Int).SetString("DB4FF10EC057E9AE26B07D0280B7F4341DA5D1B1EAE06C7D", 16)
+	ecp192k1.Gy, _ = new(big.Int).SetString("9B2F2F6D9C5628A7844163D015BE86344082AA88D95E2F9D", 16)
+	ecp192k1.BitSize = 192
+}
+
+func initS224() {
+	// See SEC 2 section 2.6.1
+	ecp224k1 = new(BitCurve)
+	ecp224k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFE56D", 16)
+	ecp224k1.N, _ = new(big.Int).SetString("010000000000000000000000000001DCE8D2EC6184CAF0A971769FB1F7", 16)
+	ecp224k1.B, _ = new(big.Int).SetString("00000000000000000000000000000000000000000000000000000005", 16)
+	ecp224k1.Gx, _ = new(big.Int).SetString("A1455B334DF099DF30FC28A169A467E9E47075A90F7E650EB6B7A45C", 16)
+	ecp224k1.Gy, _ = new(big.Int).SetString("7E089FED7FBA344282CAFBD6F7E319F7C0B0BD59E2CA4BDB556D61A5", 16)
+	ecp224k1.BitSize = 224
+}
+
+func initS256() {
+	// See SEC 2 section 2.7.1
+	ecp256k1 = new(BitCurve)
+	ecp256k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F", 16)
+	ecp256k1.N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141", 16)
+	ecp256k1.B, _ = new(big.Int).SetString("0000000000000000000000000000000000000000000000000000000000000007", 16)
+	ecp256k1.Gx, _ = new(big.Int).SetString("79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798", 16)
+	ecp256k1.Gy, _ = new(big.Int).SetString("483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8", 16)
+	ecp256k1.BitSize = 256
+}
+
+// S160 returns a BitCurve which implements secp160k1 (see SEC 2 section 2.4.1)
+func S160() *BitCurve {
+	initonce.Do(initAll)
+	return ecp160k1
+}
+
+// S192 returns a BitCurve which implements secp192k1 (see SEC 2 section 2.5.1)
+func S192() *BitCurve {
+	initonce.Do(initAll)
+	return ecp192k1
+}
+
+// S224 returns a BitCurve which implements secp224k1 (see SEC 2 section 2.6.1)
+func S224() *BitCurve {
+	initonce.Do(initAll)
+	return ecp224k1
+}
+
+// S256 returns a BitCurve which implements secp256k1 (see SEC 2 section 2.7.1)
+func S256() *BitCurve {
+	initonce.Do(initAll)
+	return ecp256k1
+}