374 lines
11 KiB
Go
374 lines
11 KiB
Go
// Copyright 2010 The Go Authors. 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 elliptic implements several standard elliptic curves over prime
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// fields.
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package elliptic
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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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"io"
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"math/big"
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"sync"
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)
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// A Curve represents a short-form Weierstrass curve with a=-3.
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// See http://www.hyperelliptic.org/EFD/g1p/auto-shortw.html
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type Curve interface {
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// Params returns the parameters for the curve.
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Params() *CurveParams
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// IsOnCurve returns true if the given (x,y) lies on the curve.
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IsOnCurve(x, y *big.Int) bool
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// Add returns the sum of (x1,y1) and (x2,y2)
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Add(x1, y1, x2, y2 *big.Int) (x, y *big.Int)
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// Double returns 2*(x,y)
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Double(x1, y1 *big.Int) (x, y *big.Int)
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// ScalarMult returns k*(Bx,By) where k is a number in big-endian form.
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ScalarMult(x1, y1 *big.Int, k []byte) (x, y *big.Int)
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// ScalarBaseMult returns k*G, where G is the base point of the group
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// and k is an integer in big-endian form.
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ScalarBaseMult(k []byte) (x, y *big.Int)
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}
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// CurveParams contains the parameters of an elliptic curve and also provides
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// a generic, non-constant time implementation of Curve.
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type CurveParams 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 curve 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 (curve *CurveParams) Params() *CurveParams {
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return curve
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}
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func (curve *CurveParams) IsOnCurve(x, y *big.Int) bool {
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// y² = x³ - 3x + b
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y2 := new(big.Int).Mul(y, y)
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y2.Mod(y2, curve.P)
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x3 := new(big.Int).Mul(x, x)
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x3.Mul(x3, x)
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threeX := new(big.Int).Lsh(x, 1)
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threeX.Add(threeX, x)
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x3.Sub(x3, threeX)
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x3.Add(x3, curve.B)
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x3.Mod(x3, curve.P)
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return x3.Cmp(y2) == 0
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}
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// zForAffine returns a Jacobian Z value for the affine point (x, y). If x and
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// y are zero, it assumes that they represent the point at infinity because (0,
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// 0) is not on the any of the curves handled here.
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func zForAffine(x, y *big.Int) *big.Int {
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z := new(big.Int)
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if x.Sign() != 0 || y.Sign() != 0 {
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z.SetInt64(1)
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}
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return z
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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. If the point is ∞ it returns 0, 0.
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func (curve *CurveParams) 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, curve.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, curve.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, curve.P)
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return
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}
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func (curve *CurveParams) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
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z1 := zForAffine(x1, y1)
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z2 := zForAffine(x2, y2)
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return curve.affineFromJacobian(curve.addJacobian(x1, y1, z1, x2, y2, z2))
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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 (curve *CurveParams) 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-3.html#addition-add-2007-bl
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x3, y3, z3 := new(big.Int), new(big.Int), new(big.Int)
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if z1.Sign() == 0 {
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x3.Set(x2)
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y3.Set(y2)
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z3.Set(z2)
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return x3, y3, z3
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}
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if z2.Sign() == 0 {
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x3.Set(x1)
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y3.Set(y1)
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z3.Set(z1)
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return x3, y3, z3
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}
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z1z1 := new(big.Int).Mul(z1, z1)
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z1z1.Mod(z1z1, curve.P)
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z2z2 := new(big.Int).Mul(z2, z2)
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z2z2.Mod(z2z2, curve.P)
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u1 := new(big.Int).Mul(x1, z2z2)
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u1.Mod(u1, curve.P)
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u2 := new(big.Int).Mul(x2, z1z1)
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u2.Mod(u2, curve.P)
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h := new(big.Int).Sub(u2, u1)
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xEqual := h.Sign() == 0
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if h.Sign() == -1 {
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h.Add(h, curve.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, curve.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, curve.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, curve.P)
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}
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yEqual := r.Sign() == 0
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if xEqual && yEqual {
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return curve.doubleJacobian(x1, y1, z1)
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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.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, curve.P)
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y3.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, curve.P)
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z3.Add(z1, z2)
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z3.Mul(z3, z3)
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z3.Sub(z3, z1z1)
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z3.Sub(z3, z2z2)
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z3.Mul(z3, h)
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z3.Mod(z3, curve.P)
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return x3, y3, z3
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}
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func (curve *CurveParams) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
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z1 := zForAffine(x1, y1)
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return curve.affineFromJacobian(curve.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 (curve *CurveParams) 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-3.html#doubling-dbl-2001-b
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delta := new(big.Int).Mul(z, z)
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delta.Mod(delta, curve.P)
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gamma := new(big.Int).Mul(y, y)
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gamma.Mod(gamma, curve.P)
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alpha := new(big.Int).Sub(x, delta)
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if alpha.Sign() == -1 {
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alpha.Add(alpha, curve.P)
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}
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alpha2 := new(big.Int).Add(x, delta)
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alpha.Mul(alpha, alpha2)
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alpha2.Set(alpha)
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alpha.Lsh(alpha, 1)
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alpha.Add(alpha, alpha2)
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beta := alpha2.Mul(x, gamma)
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x3 := new(big.Int).Mul(alpha, alpha)
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beta8 := new(big.Int).Lsh(beta, 3)
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x3.Sub(x3, beta8)
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for x3.Sign() == -1 {
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x3.Add(x3, curve.P)
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}
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x3.Mod(x3, curve.P)
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z3 := new(big.Int).Add(y, z)
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z3.Mul(z3, z3)
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z3.Sub(z3, gamma)
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if z3.Sign() == -1 {
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z3.Add(z3, curve.P)
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}
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z3.Sub(z3, delta)
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if z3.Sign() == -1 {
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z3.Add(z3, curve.P)
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}
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z3.Mod(z3, curve.P)
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beta.Lsh(beta, 2)
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beta.Sub(beta, x3)
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if beta.Sign() == -1 {
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beta.Add(beta, curve.P)
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}
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y3 := alpha.Mul(alpha, beta)
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gamma.Mul(gamma, gamma)
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gamma.Lsh(gamma, 3)
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gamma.Mod(gamma, curve.P)
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y3.Sub(y3, gamma)
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if y3.Sign() == -1 {
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y3.Add(y3, curve.P)
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}
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y3.Mod(y3, curve.P)
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return x3, y3, z3
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}
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func (curve *CurveParams) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
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Bz := new(big.Int).SetInt64(1)
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x, y, z := new(big.Int), new(big.Int), new(big.Int)
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for _, byte := range k {
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for bitNum := 0; bitNum < 8; bitNum++ {
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x, y, z = curve.doubleJacobian(x, y, z)
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if byte&0x80 == 0x80 {
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x, y, z = curve.addJacobian(Bx, By, Bz, x, y, z)
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}
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byte <<= 1
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}
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}
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return curve.affineFromJacobian(x, y, z)
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}
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func (curve *CurveParams) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
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return curve.ScalarMult(curve.Gx, curve.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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// GenerateKey returns a public/private key pair. The private key is
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// generated using the given reader, which must return random data.
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func GenerateKey(curve Curve, rand io.Reader) (priv []byte, x, y *big.Int, err error) {
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bitSize := curve.Params().BitSize
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byteLen := (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[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 = curve.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 X9.62.
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func Marshal(curve Curve, x, y *big.Int) []byte {
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byteLen := (curve.Params().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, serialized by Marshal, into an x, y pair. On error, x = nil.
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func Unmarshal(curve Curve, data []byte) (x, y *big.Int) {
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byteLen := (curve.Params().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 initonce sync.Once
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var p384 *CurveParams
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var p521 *CurveParams
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func initAll() {
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initP224()
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initP256()
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initP384()
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initP521()
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}
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func initP384() {
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// See FIPS 186-3, section D.2.4
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p384 = new(CurveParams)
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p384.P, _ = new(big.Int).SetString("39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319", 10)
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p384.N, _ = new(big.Int).SetString("39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942643", 10)
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p384.B, _ = new(big.Int).SetString("b3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef", 16)
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p384.Gx, _ = new(big.Int).SetString("aa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7", 16)
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p384.Gy, _ = new(big.Int).SetString("3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f", 16)
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p384.BitSize = 384
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}
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func initP521() {
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// See FIPS 186-3, section D.2.5
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p521 = new(CurveParams)
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p521.P, _ = new(big.Int).SetString("6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151", 10)
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p521.N, _ = new(big.Int).SetString("6864797660130609714981900799081393217269435300143305409394463459185543183397655394245057746333217197532963996371363321113864768612440380340372808892707005449", 10)
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p521.B, _ = new(big.Int).SetString("051953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b489918ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451fd46b503f00", 16)
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p521.Gx, _ = new(big.Int).SetString("c6858e06b70404e9cd9e3ecb662395b4429c648139053fb521f828af606b4d3dbaa14b5e77efe75928fe1dc127a2ffa8de3348b3c1856a429bf97e7e31c2e5bd66", 16)
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p521.Gy, _ = new(big.Int).SetString("11839296a789a3bc0045c8a5fb42c7d1bd998f54449579b446817afbd17273e662c97ee72995ef42640c550b9013fad0761353c7086a272c24088be94769fd16650", 16)
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p521.BitSize = 521
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}
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// P256 returns a Curve which implements P-256 (see FIPS 186-3, section D.2.3)
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func P256() Curve {
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initonce.Do(initAll)
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return p256
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}
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// P384 returns a Curve which implements P-384 (see FIPS 186-3, section D.2.4)
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func P384() Curve {
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initonce.Do(initAll)
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return p384
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}
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// P521 returns a Curve which implements P-521 (see FIPS 186-3, section D.2.5)
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func P521() Curve {
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initonce.Do(initAll)
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return p521
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}
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