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297 lines
9.6 KiB
297 lines
9.6 KiB
// Copyright 2010 The Go Authors. All rights reserved.
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// Copyright 2011 ThePiachu. All rights reserved.
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// Copyright 2015 Jeffrey Wilcke, Felix Lange, Gustav Simonsson. All rights reserved.
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//
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// Redistribution and use in source and binary forms, with or without
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// modification, are permitted provided that the following conditions are
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// met:
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//
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// * Redistributions of source code must retain the above copyright
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// notice, this list of conditions and the following disclaimer.
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// * Redistributions in binary form must reproduce the above
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// copyright notice, this list of conditions and the following disclaimer
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// in the documentation and/or other materials provided with the
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// distribution.
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// * Neither the name of Google Inc. nor the names of its
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// contributors may be used to endorse or promote products derived from
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// this software without specific prior written permission.
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// * The name of ThePiachu may not be used to endorse or promote products
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// derived from this software without specific prior written permission.
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//
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// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
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// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
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// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
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// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
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// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
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// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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package secp256k1
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import (
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"crypto/elliptic"
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"math/big"
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)
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const (
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// number of bits in a big.Word
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wordBits = 32 << (uint64(^big.Word(0)) >> 63)
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// number of bytes in a big.Word
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wordBytes = wordBits / 8
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)
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// readBits encodes the absolute value of bigint as big-endian bytes. Callers
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// must ensure that buf has enough space. If buf is too short the result will
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// be incomplete.
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func readBits(bigint *big.Int, buf []byte) {
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i := len(buf)
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for _, d := range bigint.Bits() {
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for j := 0; j < wordBytes && i > 0; j++ {
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i--
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buf[i] = byte(d)
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d >>= 8
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}
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}
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}
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// This code is from https://github.com/ThePiachu/GoBit and implements
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// several Koblitz elliptic curves over prime fields.
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//
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// The curve methods, internally, on Jacobian coordinates. For a given
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// (x, y) position on the curve, the Jacobian coordinates are (x1, y1,
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// z1) where x = x1/z1² and y = y1/z1³. The greatest speedups come
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// when the whole calculation can be performed within the transform
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// (as in ScalarMult and ScalarBaseMult). But even for Add and Double,
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// it's faster to apply and reverse the transform than to operate in
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// affine coordinates.
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// A BitCurve represents a Koblitz Curve with a=0.
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// See http://www.hyperelliptic.org/EFD/g1p/auto-shortw.html
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type BitCurve struct {
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P *big.Int // the order of the underlying field
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N *big.Int // the order of the base point
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B *big.Int // the constant of the BitCurve equation
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Gx, Gy *big.Int // (x,y) of the base point
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BitSize int // the size of the underlying field
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}
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func (BitCurve *BitCurve) Params() *elliptic.CurveParams {
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return &elliptic.CurveParams{
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P: BitCurve.P,
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N: BitCurve.N,
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B: BitCurve.B,
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Gx: BitCurve.Gx,
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Gy: BitCurve.Gy,
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BitSize: BitCurve.BitSize,
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}
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}
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// IsOnCurve returns true if the given (x,y) lies on the BitCurve.
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func (BitCurve *BitCurve) IsOnCurve(x, y *big.Int) bool {
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// y² = x³ + b
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y2 := new(big.Int).Mul(y, y) //y²
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y2.Mod(y2, BitCurve.P) //y²%P
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x3 := new(big.Int).Mul(x, x) //x²
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x3.Mul(x3, x) //x³
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x3.Add(x3, BitCurve.B) //x³+B
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x3.Mod(x3, BitCurve.P) //(x³+B)%P
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return x3.Cmp(y2) == 0
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}
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// affineFromJacobian reverses the Jacobian transform. See the comment at the
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// top of the file.
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func (BitCurve *BitCurve) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) {
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if z.Sign() == 0 {
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return new(big.Int), new(big.Int)
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}
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zinv := new(big.Int).ModInverse(z, BitCurve.P)
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zinvsq := new(big.Int).Mul(zinv, zinv)
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xOut = new(big.Int).Mul(x, zinvsq)
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xOut.Mod(xOut, BitCurve.P)
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zinvsq.Mul(zinvsq, zinv)
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yOut = new(big.Int).Mul(y, zinvsq)
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yOut.Mod(yOut, BitCurve.P)
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return
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}
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// Add returns the sum of (x1,y1) and (x2,y2)
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func (BitCurve *BitCurve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
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// If one point is at infinity, return the other point.
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// Adding the point at infinity to any point will preserve the other point.
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if x1.Sign() == 0 && y1.Sign() == 0 {
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return x2, y2
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}
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if x2.Sign() == 0 && y2.Sign() == 0 {
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return x1, y1
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}
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z := new(big.Int).SetInt64(1)
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if x1.Cmp(x2) == 0 && y1.Cmp(y2) == 0 {
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return BitCurve.affineFromJacobian(BitCurve.doubleJacobian(x1, y1, z))
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}
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return BitCurve.affineFromJacobian(BitCurve.addJacobian(x1, y1, z, x2, y2, z))
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}
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// addJacobian takes two points in Jacobian coordinates, (x1, y1, z1) and
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// (x2, y2, z2) and returns their sum, also in Jacobian form.
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func (BitCurve *BitCurve) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) {
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// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
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z1z1 := new(big.Int).Mul(z1, z1)
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z1z1.Mod(z1z1, BitCurve.P)
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z2z2 := new(big.Int).Mul(z2, z2)
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z2z2.Mod(z2z2, BitCurve.P)
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u1 := new(big.Int).Mul(x1, z2z2)
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u1.Mod(u1, BitCurve.P)
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u2 := new(big.Int).Mul(x2, z1z1)
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u2.Mod(u2, BitCurve.P)
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h := new(big.Int).Sub(u2, u1)
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if h.Sign() == -1 {
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h.Add(h, BitCurve.P)
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}
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i := new(big.Int).Lsh(h, 1)
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i.Mul(i, i)
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j := new(big.Int).Mul(h, i)
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s1 := new(big.Int).Mul(y1, z2)
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s1.Mul(s1, z2z2)
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s1.Mod(s1, BitCurve.P)
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s2 := new(big.Int).Mul(y2, z1)
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s2.Mul(s2, z1z1)
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s2.Mod(s2, BitCurve.P)
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r := new(big.Int).Sub(s2, s1)
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if r.Sign() == -1 {
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r.Add(r, BitCurve.P)
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}
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r.Lsh(r, 1)
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v := new(big.Int).Mul(u1, i)
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x3 := new(big.Int).Set(r)
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x3.Mul(x3, x3)
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x3.Sub(x3, j)
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x3.Sub(x3, v)
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x3.Sub(x3, v)
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x3.Mod(x3, BitCurve.P)
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y3 := new(big.Int).Set(r)
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v.Sub(v, x3)
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y3.Mul(y3, v)
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s1.Mul(s1, j)
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s1.Lsh(s1, 1)
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y3.Sub(y3, s1)
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y3.Mod(y3, BitCurve.P)
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z3 := new(big.Int).Add(z1, z2)
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z3.Mul(z3, z3)
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z3.Sub(z3, z1z1)
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if z3.Sign() == -1 {
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z3.Add(z3, BitCurve.P)
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}
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z3.Sub(z3, z2z2)
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if z3.Sign() == -1 {
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z3.Add(z3, BitCurve.P)
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}
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z3.Mul(z3, h)
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z3.Mod(z3, BitCurve.P)
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return x3, y3, z3
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}
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// Double returns 2*(x,y)
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func (BitCurve *BitCurve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
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z1 := new(big.Int).SetInt64(1)
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return BitCurve.affineFromJacobian(BitCurve.doubleJacobian(x1, y1, z1))
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}
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// doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and
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// returns its double, also in Jacobian form.
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func (BitCurve *BitCurve) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) {
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// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
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a := new(big.Int).Mul(x, x) //X1²
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b := new(big.Int).Mul(y, y) //Y1²
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c := new(big.Int).Mul(b, b) //B²
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d := new(big.Int).Add(x, b) //X1+B
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d.Mul(d, d) //(X1+B)²
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d.Sub(d, a) //(X1+B)²-A
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d.Sub(d, c) //(X1+B)²-A-C
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d.Mul(d, big.NewInt(2)) //2*((X1+B)²-A-C)
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e := new(big.Int).Mul(big.NewInt(3), a) //3*A
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f := new(big.Int).Mul(e, e) //E²
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x3 := new(big.Int).Mul(big.NewInt(2), d) //2*D
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x3.Sub(f, x3) //F-2*D
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x3.Mod(x3, BitCurve.P)
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y3 := new(big.Int).Sub(d, x3) //D-X3
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y3.Mul(e, y3) //E*(D-X3)
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y3.Sub(y3, new(big.Int).Mul(big.NewInt(8), c)) //E*(D-X3)-8*C
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y3.Mod(y3, BitCurve.P)
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z3 := new(big.Int).Mul(y, z) //Y1*Z1
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z3.Mul(big.NewInt(2), z3) //3*Y1*Z1
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z3.Mod(z3, BitCurve.P)
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return x3, y3, z3
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}
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// ScalarBaseMult returns k*G, where G is the base point of the group and k is
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// an integer in big-endian form.
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func (BitCurve *BitCurve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
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return BitCurve.ScalarMult(BitCurve.Gx, BitCurve.Gy, k)
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}
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// Marshal converts a point into the form specified in section 4.3.6 of ANSI
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// X9.62.
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func (BitCurve *BitCurve) Marshal(x, y *big.Int) []byte {
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byteLen := (BitCurve.BitSize + 7) >> 3
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ret := make([]byte, 1+2*byteLen)
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ret[0] = 4 // uncompressed point flag
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readBits(x, ret[1:1+byteLen])
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readBits(y, ret[1+byteLen:])
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return ret
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}
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// Unmarshal converts a point, serialised by Marshal, into an x, y pair. On
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// error, x = nil.
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func (BitCurve *BitCurve) Unmarshal(data []byte) (x, y *big.Int) {
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byteLen := (BitCurve.BitSize + 7) >> 3
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if len(data) != 1+2*byteLen {
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return
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}
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if data[0] != 4 { // uncompressed form
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return
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}
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x = new(big.Int).SetBytes(data[1 : 1+byteLen])
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y = new(big.Int).SetBytes(data[1+byteLen:])
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return
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}
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var theCurve = new(BitCurve)
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func init() {
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// See SEC 2 section 2.7.1
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// curve parameters taken from:
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// http://www.secg.org/sec2-v2.pdf
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theCurve.P, _ = new(big.Int).SetString("0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F", 0)
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theCurve.N, _ = new(big.Int).SetString("0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141", 0)
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theCurve.B, _ = new(big.Int).SetString("0x0000000000000000000000000000000000000000000000000000000000000007", 0)
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theCurve.Gx, _ = new(big.Int).SetString("0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798", 0)
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theCurve.Gy, _ = new(big.Int).SetString("0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8", 0)
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theCurve.BitSize = 256
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}
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// S256 returns a BitCurve which implements secp256k1.
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func S256() *BitCurve {
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return theCurve
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}
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