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secp256k1.go
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secp256k1.go
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// Copyright 2010 The Go Authors. All rights reserved.
// Copyright 2011 ThePiachu. All rights reserved.
// Copyright 2013 Michael Hendricks. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package ecbdsa
// See http://www.hyperelliptic.org/EFD/g1p/auto-shortw.html
// and http://stackoverflow.com/a/8392111/174463
// for details on how this Koblitz curve math works.
import "crypto/elliptic"
import "fmt"
import "math/big"
// A Koblitz Curve with a=0.
type KoblitzCurve 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 KoblitzCurve equation
Gx, Gy *big.Int // (x,y) of the base point
BitSize int // the size of the underlying field
}
// Returns the secp256k1 curve.
var secp256k1 *KoblitzCurve
func Secp256k1() elliptic.Curve {
return secp256k1
}
func init() {
var p, n, gx, gy big.Int
fmt.Sscan("0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F", &p)
fmt.Sscan("0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141", &n)
fmt.Sscan("0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798", &gx)
fmt.Sscan("0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8", &gy)
b := big.NewInt(7)
secp256k1 = &KoblitzCurve{
P: &p,
N: &n,
B: b,
Gx: &gx,
Gy: &gy,
BitSize: 256,
}
}
func (curve *KoblitzCurve) IsOnCurve(x, y *big.Int) bool {
// y² = x³ + b
y2 := new(big.Int).Mul(y, y)
y2.Mod(y2, curve.P)
x3 := new(big.Int).Mul(x, x)
x3.Mul(x3, x)
x3.Add(x3, curve.B)
x3.Mod(x3, curve.P)
return x3.Cmp(y2) == 0
}
// affineFromJacobian reverses the Jacobian transform.
func (curve *KoblitzCurve) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *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 *KoblitzCurve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
z := new(big.Int).SetInt64(1)
return curve.affineFromJacobian(curve.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 (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
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)
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)
}
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, curve.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, curve.P)
z3 := new(big.Int).Add(z1, z2)
z3.Mul(z3, z3)
z3.Sub(z3, z1z1)
if z3.Sign() == -1 {
z3.Add(z3, curve.P)
}
z3.Sub(z3, z2z2)
if z3.Sign() == -1 {
z3.Add(z3, curve.P)
}
z3.Mul(z3, h)
z3.Mod(z3, curve.P)
return x3, y3, z3
}
func (curve *KoblitzCurve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
z1 := new(big.Int).SetInt64(1)
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
}
func (curve *KoblitzCurve) 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 = curve.doubleJacobian(x, y, z)
}
if byte&0x80 == 0x80 {
if !seenFirstTrue {
seenFirstTrue = true
} else {
x, y, z = curve.addJacobian(Bx, By, Bz, x, y, z)
}
}
byte <<= 1
}
}
if !seenFirstTrue {
return nil, nil
}
return curve.affineFromJacobian(x, y, z)
}
func (curve *KoblitzCurve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
return curve.ScalarMult(curve.Gx, curve.Gy, k)
}
func (curve *KoblitzCurve) Params() *elliptic.CurveParams {
return &elliptic.CurveParams{
P: curve.P,
N: curve.N,
B: curve.B,
Gx: curve.Gx,
Gy: curve.Gy,
BitSize: curve.BitSize,
}
}