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
}
