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382 lines
12 KiB
382 lines
12 KiB
package bitcurves
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// Copyright 2010 The Go Authors. All rights reserved.
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// Copyright 2011 ThePiachu. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// Package bitelliptic implements several Koblitz elliptic curves over prime
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// fields.
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// This package operates, internally, on Jacobian coordinates. For a given
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// (x, y) position on the curve, the Jacobian coordinates are (x1, y1, z1)
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// where x = x1/z1² and y = y1/z1³. The greatest speedups come when the whole
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// calculation can be performed within the transform (as in ScalarMult and
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// ScalarBaseMult). But even for Add and Double, it's faster to apply and
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// reverse the transform than to operate in affine coordinates.
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import (
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"crypto/elliptic"
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"io"
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"math/big"
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"sync"
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)
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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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Name string
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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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// Params returns the parameters of the given BitCurve (see BitCurve struct)
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func (bitCurve *BitCurve) Params() (cp *elliptic.CurveParams) {
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cp = new(elliptic.CurveParams)
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cp.Name = bitCurve.Name
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cp.P = bitCurve.P
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cp.N = bitCurve.N
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cp.Gx = bitCurve.Gx
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cp.Gy = bitCurve.Gy
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cp.BitSize = bitCurve.BitSize
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return cp
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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.Cmp(big.NewInt(0)) == 0 {
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panic("bitcurve: Can't convert to affine with Jacobian Z = 0")
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}
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// x = YZ^2 mod P
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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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// y = YZ^3 mod 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 xOut, yOut
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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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z := new(big.Int).SetInt64(1)
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x, y, z := bitCurve.addJacobian(x1, y1, z, x2, y2, z)
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return bitCurve.affineFromJacobian(x, y, 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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//TODO: double check if it is okay
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// ScalarMult returns k*(Bx,By) where k is a number in big-endian form.
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func (bitCurve *BitCurve) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
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// We have a slight problem in that the identity of the group (the
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// point at infinity) cannot be represented in (x, y) form on a finite
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// machine. Thus the standard add/double algorithm has to be tweaked
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// slightly: our initial state is not the identity, but x, and we
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// ignore the first true bit in |k|. If we don't find any true bits in
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// |k|, then we return nil, nil, because we cannot return the identity
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// element.
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Bz := new(big.Int).SetInt64(1)
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x := Bx
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y := By
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z := Bz
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seenFirstTrue := false
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for _, byte := range k {
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for bitNum := 0; bitNum < 8; bitNum++ {
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if seenFirstTrue {
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x, y, z = bitCurve.doubleJacobian(x, y, z)
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}
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if byte&0x80 == 0x80 {
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if !seenFirstTrue {
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seenFirstTrue = true
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} else {
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x, y, z = bitCurve.addJacobian(Bx, By, Bz, x, y, z)
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}
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}
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byte <<= 1
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}
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}
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if !seenFirstTrue {
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return nil, nil
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}
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return bitCurve.affineFromJacobian(x, y, z)
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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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var mask = []byte{0xff, 0x1, 0x3, 0x7, 0xf, 0x1f, 0x3f, 0x7f}
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//TODO: double check if it is okay
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// GenerateKey returns a public/private key pair. The private key is generated
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// using the given reader, which must return random data.
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func (bitCurve *BitCurve) GenerateKey(rand io.Reader) (priv []byte, x, y *big.Int, err error) {
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byteLen := (bitCurve.BitSize + 7) >> 3
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priv = make([]byte, byteLen)
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for x == nil {
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_, err = io.ReadFull(rand, priv)
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if err != nil {
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return
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}
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// We have to mask off any excess bits in the case that the size of the
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// underlying field is not a whole number of bytes.
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priv[0] &= mask[bitCurve.BitSize%8]
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// This is because, in tests, rand will return all zeros and we don't
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// want to get the point at infinity and loop forever.
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priv[1] ^= 0x42
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x, y = bitCurve.ScalarBaseMult(priv)
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}
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return
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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
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xBytes := x.Bytes()
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copy(ret[1+byteLen-len(xBytes):], xBytes)
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yBytes := y.Bytes()
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copy(ret[1+2*byteLen-len(yBytes):], yBytes)
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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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//curve parameters taken from:
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//http://www.secg.org/collateral/sec2_final.pdf
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var initonce sync.Once
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var secp160k1 *BitCurve
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var secp192k1 *BitCurve
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var secp224k1 *BitCurve
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var secp256k1 *BitCurve
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func initAll() {
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initS160()
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initS192()
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initS224()
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initS256()
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}
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func initS160() {
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// See SEC 2 section 2.4.1
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secp160k1 = new(BitCurve)
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secp160k1.Name = "secp160k1"
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secp160k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFAC73", 16)
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secp160k1.N, _ = new(big.Int).SetString("0100000000000000000001B8FA16DFAB9ACA16B6B3", 16)
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secp160k1.B, _ = new(big.Int).SetString("0000000000000000000000000000000000000007", 16)
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secp160k1.Gx, _ = new(big.Int).SetString("3B4C382CE37AA192A4019E763036F4F5DD4D7EBB", 16)
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secp160k1.Gy, _ = new(big.Int).SetString("938CF935318FDCED6BC28286531733C3F03C4FEE", 16)
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secp160k1.BitSize = 160
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}
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func initS192() {
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// See SEC 2 section 2.5.1
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secp192k1 = new(BitCurve)
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secp192k1.Name = "secp192k1"
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secp192k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFEE37", 16)
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secp192k1.N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFE26F2FC170F69466A74DEFD8D", 16)
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secp192k1.B, _ = new(big.Int).SetString("000000000000000000000000000000000000000000000003", 16)
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secp192k1.Gx, _ = new(big.Int).SetString("DB4FF10EC057E9AE26B07D0280B7F4341DA5D1B1EAE06C7D", 16)
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secp192k1.Gy, _ = new(big.Int).SetString("9B2F2F6D9C5628A7844163D015BE86344082AA88D95E2F9D", 16)
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secp192k1.BitSize = 192
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}
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func initS224() {
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// See SEC 2 section 2.6.1
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secp224k1 = new(BitCurve)
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secp224k1.Name = "secp224k1"
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secp224k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFE56D", 16)
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secp224k1.N, _ = new(big.Int).SetString("010000000000000000000000000001DCE8D2EC6184CAF0A971769FB1F7", 16)
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secp224k1.B, _ = new(big.Int).SetString("00000000000000000000000000000000000000000000000000000005", 16)
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secp224k1.Gx, _ = new(big.Int).SetString("A1455B334DF099DF30FC28A169A467E9E47075A90F7E650EB6B7A45C", 16)
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secp224k1.Gy, _ = new(big.Int).SetString("7E089FED7FBA344282CAFBD6F7E319F7C0B0BD59E2CA4BDB556D61A5", 16)
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secp224k1.BitSize = 224
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}
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func initS256() {
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// See SEC 2 section 2.7.1
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secp256k1 = new(BitCurve)
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secp256k1.Name = "secp256k1"
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secp256k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F", 16)
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secp256k1.N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141", 16)
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secp256k1.B, _ = new(big.Int).SetString("0000000000000000000000000000000000000000000000000000000000000007", 16)
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secp256k1.Gx, _ = new(big.Int).SetString("79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798", 16)
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secp256k1.Gy, _ = new(big.Int).SetString("483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8", 16)
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secp256k1.BitSize = 256
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}
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// S160 returns a BitCurve which implements secp160k1 (see SEC 2 section 2.4.1)
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func S160() *BitCurve {
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initonce.Do(initAll)
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return secp160k1
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}
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// S192 returns a BitCurve which implements secp192k1 (see SEC 2 section 2.5.1)
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func S192() *BitCurve {
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initonce.Do(initAll)
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return secp192k1
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}
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// S224 returns a BitCurve which implements secp224k1 (see SEC 2 section 2.6.1)
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func S224() *BitCurve {
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initonce.Do(initAll)
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return secp224k1
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}
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// S256 returns a BitCurve which implements bitcurves (see SEC 2 section 2.7.1)
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func S256() *BitCurve {
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initonce.Do(initAll)
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return secp256k1
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}
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