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// Copyright (c) 2014-2015 The btcsuite developers
// Use of this source code is governed by an ISC
// license that can be found in the LICENSE file.
// This file is ignored during the regular build due to the following build tag.
// This build tag is set during go generate.
// +build gensecp256k1
package btcec
// References:
// [GECC]: Guide to Elliptic Curve Cryptography (Hankerson, Menezes, Vanstone)
import (
"encoding/binary"
"math/big"
)
// secp256k1BytePoints are dummy points used so the code which generates the
// real values can compile.
var secp256k1BytePoints = ""
// getDoublingPoints returns all the possible G^(2^i) for i in
// 0..n-1 where n is the curve's bit size (256 in the case of secp256k1)
// the coordinates are recorded as Jacobian coordinates.
func (curve *KoblitzCurve) getDoublingPoints() [][3]fieldVal {
doublingPoints := make([][3]fieldVal, curve.BitSize)
// initialize px, py, pz to the Jacobian coordinates for the base point
px, py := curve.bigAffineToField(curve.Gx, curve.Gy)
pz := new(fieldVal).SetInt(1)
for i := 0; i < curve.BitSize; i++ {
doublingPoints[i] = [3]fieldVal{*px, *py, *pz}
// P = 2*P
curve.doubleJacobian(px, py, pz, px, py, pz)
}
return doublingPoints
}
// SerializedBytePoints returns a serialized byte slice which contains all of
// the possible points per 8-bit window. This is used to when generating
// secp256k1.go.
func (curve *KoblitzCurve) SerializedBytePoints() []byte {
doublingPoints := curve.getDoublingPoints()
// Segregate the bits into byte-sized windows
serialized := make([]byte, curve.byteSize*256*3*10*4)
offset := 0
for byteNum := 0; byteNum < curve.byteSize; byteNum++ {
// Grab the 8 bits that make up this byte from doublingPoints.
startingBit := 8 * (curve.byteSize - byteNum - 1)
computingPoints := doublingPoints[startingBit : startingBit+8]
// Compute all points in this window and serialize them.
for i := 0; i < 256; i++ {
px, py, pz := new(fieldVal), new(fieldVal), new(fieldVal)
for j := 0; j < 8; j++ {
if i>>uint(j)&1 == 1 {
curve.addJacobian(px, py, pz, &computingPoints[j][0],
&computingPoints[j][1], &computingPoints[j][2], px, py, pz)
}
}
for i := 0; i < 10; i++ {
binary.LittleEndian.PutUint32(serialized[offset:], px.n[i])
offset += 4
}
for i := 0; i < 10; i++ {
binary.LittleEndian.PutUint32(serialized[offset:], py.n[i])
offset += 4
}
for i := 0; i < 10; i++ {
binary.LittleEndian.PutUint32(serialized[offset:], pz.n[i])
offset += 4
}
}
}
return serialized
}
// sqrt returns the square root of the provided big integer using Newton's
// method. It's only compiled and used during generation of pre-computed
// values, so speed is not a huge concern.
func sqrt(n *big.Int) *big.Int {
// Initial guess = 2^(log_2(n)/2)
guess := big.NewInt(2)
guess.Exp(guess, big.NewInt(int64(n.BitLen()/2)), nil)
// Now refine using Newton's method.
big2 := big.NewInt(2)
prevGuess := big.NewInt(0)
for {
prevGuess.Set(guess)
guess.Add(guess, new(big.Int).Div(n, guess))
guess.Div(guess, big2)
if guess.Cmp(prevGuess) == 0 {
break
}
}
return guess
}
// EndomorphismVectors runs the first 3 steps of algorithm 3.74 from [GECC] to
// generate the linearly independent vectors needed to generate a balanced
// length-two representation of a multiplier such that k = k1 + k2λ (mod N) and
// returns them. Since the values will always be the same given the fact that N
// and λ are fixed, the final results can be accelerated by storing the
// precomputed values with the curve.
func (curve *KoblitzCurve) EndomorphismVectors() (a1, b1, a2, b2 *big.Int) {
bigMinus1 := big.NewInt(-1)
// This section uses an extended Euclidean algorithm to generate a
// sequence of equations:
// s[i] * N + t[i] * λ = r[i]
nSqrt := sqrt(curve.N)
u, v := new(big.Int).Set(curve.N), new(big.Int).Set(curve.lambda)
x1, y1 := big.NewInt(1), big.NewInt(0)
x2, y2 := big.NewInt(0), big.NewInt(1)
q, r := new(big.Int), new(big.Int)
qu, qx1, qy1 := new(big.Int), new(big.Int), new(big.Int)
s, t := new(big.Int), new(big.Int)
ri, ti := new(big.Int), new(big.Int)
a1, b1, a2, b2 = new(big.Int), new(big.Int), new(big.Int), new(big.Int)
found, oneMore := false, false
for u.Sign() != 0 {
// q = v/u
q.Div(v, u)
// r = v - q*u
qu.Mul(q, u)
r.Sub(v, qu)
// s = x2 - q*x1
qx1.Mul(q, x1)
s.Sub(x2, qx1)
// t = y2 - q*y1
qy1.Mul(q, y1)
t.Sub(y2, qy1)
// v = u, u = r, x2 = x1, x1 = s, y2 = y1, y1 = t
v.Set(u)
u.Set(r)
x2.Set(x1)
x1.Set(s)
y2.Set(y1)
y1.Set(t)
// As soon as the remainder is less than the sqrt of n, the
// values of a1 and b1 are known.
if !found && r.Cmp(nSqrt) < 0 {
// When this condition executes ri and ti represent the
// r[i] and t[i] values such that i is the greatest
// index for which r >= sqrt(n). Meanwhile, the current
// r and t values are r[i+1] and t[i+1], respectively.
// a1 = r[i+1], b1 = -t[i+1]
a1.Set(r)
b1.Mul(t, bigMinus1)
found = true
oneMore = true
// Skip to the next iteration so ri and ti are not
// modified.
continue
} else if oneMore {
// When this condition executes ri and ti still
// represent the r[i] and t[i] values while the current
// r and t are r[i+2] and t[i+2], respectively.
// sum1 = r[i]^2 + t[i]^2
rSquared := new(big.Int).Mul(ri, ri)
tSquared := new(big.Int).Mul(ti, ti)
sum1 := new(big.Int).Add(rSquared, tSquared)
// sum2 = r[i+2]^2 + t[i+2]^2
r2Squared := new(big.Int).Mul(r, r)
t2Squared := new(big.Int).Mul(t, t)
sum2 := new(big.Int).Add(r2Squared, t2Squared)
// if (r[i]^2 + t[i]^2) <= (r[i+2]^2 + t[i+2]^2)
if sum1.Cmp(sum2) <= 0 {
// a2 = r[i], b2 = -t[i]
a2.Set(ri)
b2.Mul(ti, bigMinus1)
} else {
// a2 = r[i+2], b2 = -t[i+2]
a2.Set(r)
b2.Mul(t, bigMinus1)
}
// All done.
break
}
ri.Set(r)
ti.Set(t)
}
return a1, b1, a2, b2
}
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