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// Copyright 2010 The Go Authors. All rights reserved.
// Copyright 2011 ThePiachu. All rights reserved.
// Copyright 2013-2014 The btcsuite developers
// Use of this source code is governed by an ISC
// license that can be found in the LICENSE file.
package btcec
// References:
// [SECG]: Recommended Elliptic Curve Domain Parameters
// http://www.secg.org/sec2-v2.pdf
//
// [GECC]: Guide to Elliptic Curve Cryptography (Hankerson, Menezes, Vanstone)
// This package operates, internally, on Jacobian coordinates. For a given
// (x, y) position on the curve, the Jacobian coordinates are (x1, y1, z1)
// where x = x1/z1² and y = y1/z1³. The greatest speedups come when the whole
// calculation can be performed within the transform (as in ScalarMult and
// ScalarBaseMult). But even for Add and Double, it's faster to apply and
// reverse the transform than to operate in affine coordinates.
import (
"crypto/elliptic"
"math/big"
"sync"
)
var (
// fieldOne is simply the integer 1 in field representation. It is
// used to avoid needing to create it multiple times during the internal
// arithmetic.
fieldOne = new(fieldVal).SetInt(1)
)
// KoblitzCurve supports a koblitz curve implementation that fits the ECC Curve
// interface from crypto/elliptic.
type KoblitzCurve struct {
*elliptic.CurveParams
q *big.Int
H int // cofactor of the curve.
halfOrder *big.Int // half the order N
// byteSize is simply the bit size / 8 and is provided for convenience
// since it is calculated repeatedly.
byteSize int
// bytePoints
bytePoints *[32][256][3]fieldVal
// The next 6 values are used specifically for endomorphism
// optimizations in ScalarMult.
// lambda must fulfill lambda^3 = 1 mod N where N is the order of G.
lambda *big.Int
// beta must fulfill beta^3 = 1 mod P where P is the prime field of the
// curve.
beta *fieldVal
// See the EndomorphismVectors in gensecp256k1.go to see how these are
// derived.
a1 *big.Int
b1 *big.Int
a2 *big.Int
b2 *big.Int
}
// Params returns the parameters for the curve.
func (curve *KoblitzCurve) Params() *elliptic.CurveParams {
return curve.CurveParams
}
// bigAffineToField takes an affine point (x, y) as big integers and converts
// it to an affine point as field values.
func (curve *KoblitzCurve) bigAffineToField(x, y *big.Int) (*fieldVal, *fieldVal) {
x3, y3 := new(fieldVal), new(fieldVal)
x3.SetByteSlice(x.Bytes())
y3.SetByteSlice(y.Bytes())
return x3, y3
}
// fieldJacobianToBigAffine takes a Jacobian point (x, y, z) as field values and
// converts it to an affine point as big integers.
func (curve *KoblitzCurve) fieldJacobianToBigAffine(x, y, z *fieldVal) (*big.Int, *big.Int) {
// Inversions are expensive and both point addition and point doubling
// are faster when working with points that have a z value of one. So,
// if the point needs to be converted to affine, go ahead and normalize
// the point itself at the same time as the calculation is the same.
var zInv, tempZ fieldVal
zInv.Set(z).Inverse() // zInv = Z^-1
tempZ.SquareVal(&zInv) // tempZ = Z^-2
x.Mul(&tempZ) // X = X/Z^2 (mag: 1)
y.Mul(tempZ.Mul(&zInv)) // Y = Y/Z^3 (mag: 1)
z.SetInt(1) // Z = 1 (mag: 1)
// Normalize the x and y values.
x.Normalize()
y.Normalize()
// Convert the field values for the now affine point to big.Ints.
x3, y3 := new(big.Int), new(big.Int)
x3.SetBytes(x.Bytes()[:])
y3.SetBytes(y.Bytes()[:])
return x3, y3
}
// IsOnCurve returns boolean if the point (x,y) is on the curve.
// Part of the elliptic.Curve interface. This function differs from the
// crypto/elliptic algorithm since a = 0 not -3.
func (curve *KoblitzCurve) IsOnCurve(x, y *big.Int) bool {
// Convert big ints to field values for faster arithmetic.
fx, fy := curve.bigAffineToField(x, y)
// Elliptic curve equation for secp256k1 is: y^2 = x^3 + 7
y2 := new(fieldVal).SquareVal(fy).Normalize()
result := new(fieldVal).SquareVal(fx).Mul(fx).AddInt(7).Normalize()
return y2.Equals(result)
}
// addZ1AndZ2EqualsOne adds two Jacobian points that are already known to have
// z values of 1 and stores the result in (x3, y3, z3). That is to say
// (x1, y1, 1) + (x2, y2, 1) = (x3, y3, z3). It performs faster addition than
// the generic add routine since less arithmetic is needed due to the ability to
// avoid the z value multiplications.
func (curve *KoblitzCurve) addZ1AndZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3 *fieldVal) {
// To compute the point addition efficiently, this implementation splits
// the equation into intermediate elements which are used to minimize
// the number of field multiplications using the method shown at:
// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-mmadd-2007-bl
//
// In particular it performs the calculations using the following:
// H = X2-X1, HH = H^2, I = 4*HH, J = H*I, r = 2*(Y2-Y1), V = X1*I
// X3 = r^2-J-2*V, Y3 = r*(V-X3)-2*Y1*J, Z3 = 2*H
//
// This results in a cost of 4 field multiplications, 2 field squarings,
// 6 field additions, and 5 integer multiplications.
// When the x coordinates are the same for two points on the curve, the
// y coordinates either must be the same, in which case it is point
// doubling, or they are opposite and the result is the point at
// infinity per the group law for elliptic curve cryptography.
x1.Normalize()
y1.Normalize()
x2.Normalize()
y2.Normalize()
if x1.Equals(x2) {
if y1.Equals(y2) {
// Since x1 == x2 and y1 == y2, point doubling must be
// done, otherwise the addition would end up dividing
// by zero.
curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
return
}
// Since x1 == x2 and y1 == -y2, the sum is the point at
// infinity per the group law.
x3.SetInt(0)
y3.SetInt(0)
z3.SetInt(0)
return
}
// Calculate X3, Y3, and Z3 according to the intermediate elements
// breakdown above.
var h, i, j, r, v fieldVal
var negJ, neg2V, negX3 fieldVal
h.Set(x1).Negate(1).Add(x2) // H = X2-X1 (mag: 3)
i.SquareVal(&h).MulInt(4) // I = 4*H^2 (mag: 4)
j.Mul2(&h, &i) // J = H*I (mag: 1)
r.Set(y1).Negate(1).Add(y2).MulInt(2) // r = 2*(Y2-Y1) (mag: 6)
v.Mul2(x1, &i) // V = X1*I (mag: 1)
negJ.Set(&j).Negate(1) // negJ = -J (mag: 2)
neg2V.Set(&v).MulInt(2).Negate(2) // neg2V = -(2*V) (mag: 3)
x3.Set(&r).Square().Add(&negJ).Add(&neg2V) // X3 = r^2-J-2*V (mag: 6)
negX3.Set(x3).Negate(6) // negX3 = -X3 (mag: 7)
j.Mul(y1).MulInt(2).Negate(2) // J = -(2*Y1*J) (mag: 3)
y3.Set(&v).Add(&negX3).Mul(&r).Add(&j) // Y3 = r*(V-X3)-2*Y1*J (mag: 4)
z3.Set(&h).MulInt(2) // Z3 = 2*H (mag: 6)
// Normalize the resulting field values to a magnitude of 1 as needed.
x3.Normalize()
y3.Normalize()
z3.Normalize()
}
// addZ1EqualsZ2 adds two Jacobian points that are already known to have the
// same z value and stores the result in (x3, y3, z3). That is to say
// (x1, y1, z1) + (x2, y2, z1) = (x3, y3, z3). It performs faster addition than
// the generic add routine since less arithmetic is needed due to the known
// equivalence.
func (curve *KoblitzCurve) addZ1EqualsZ2(x1, y1, z1, x2, y2, x3, y3, z3 *fieldVal) {
// To compute the point addition efficiently, this implementation splits
// the equation into intermediate elements which are used to minimize
// the number of field multiplications using a slightly modified version
// of the method shown at:
// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-mmadd-2007-bl
//
// In particular it performs the calculations using the following:
// A = X2-X1, B = A^2, C=Y2-Y1, D = C^2, E = X1*B, F = X2*B
// X3 = D-E-F, Y3 = C*(E-X3)-Y1*(F-E), Z3 = Z1*A
//
// This results in a cost of 5 field multiplications, 2 field squarings,
// 9 field additions, and 0 integer multiplications.
// When the x coordinates are the same for two points on the curve, the
// y coordinates either must be the same, in which case it is point
// doubling, or they are opposite and the result is the point at
// infinity per the group law for elliptic curve cryptography.
x1.Normalize()
y1.Normalize()
x2.Normalize()
y2.Normalize()
if x1.Equals(x2) {
if y1.Equals(y2) {
// Since x1 == x2 and y1 == y2, point doubling must be
// done, otherwise the addition would end up dividing
// by zero.
curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
return
}
// Since x1 == x2 and y1 == -y2, the sum is the point at
// infinity per the group law.
x3.SetInt(0)
y3.SetInt(0)
z3.SetInt(0)
return
}
// Calculate X3, Y3, and Z3 according to the intermediate elements
// breakdown above.
var a, b, c, d, e, f fieldVal
var negX1, negY1, negE, negX3 fieldVal
negX1.Set(x1).Negate(1) // negX1 = -X1 (mag: 2)
negY1.Set(y1).Negate(1) // negY1 = -Y1 (mag: 2)
a.Set(&negX1).Add(x2) // A = X2-X1 (mag: 3)
b.SquareVal(&a) // B = A^2 (mag: 1)
c.Set(&negY1).Add(y2) // C = Y2-Y1 (mag: 3)
d.SquareVal(&c) // D = C^2 (mag: 1)
e.Mul2(x1, &b) // E = X1*B (mag: 1)
negE.Set(&e).Negate(1) // negE = -E (mag: 2)
f.Mul2(x2, &b) // F = X2*B (mag: 1)
x3.Add2(&e, &f).Negate(3).Add(&d) // X3 = D-E-F (mag: 5)
negX3.Set(x3).Negate(5).Normalize() // negX3 = -X3 (mag: 1)
y3.Set(y1).Mul(f.Add(&negE)).Negate(3) // Y3 = -(Y1*(F-E)) (mag: 4)
y3.Add(e.Add(&negX3).Mul(&c)) // Y3 = C*(E-X3)+Y3 (mag: 5)
z3.Mul2(z1, &a) // Z3 = Z1*A (mag: 1)
// Normalize the resulting field values to a magnitude of 1 as needed.
x3.Normalize()
y3.Normalize()
}
// addZ2EqualsOne adds two Jacobian points when the second point is already
// known to have a z value of 1 (and the z value for the first point is not 1)
// and stores the result in (x3, y3, z3). That is to say (x1, y1, z1) +
// (x2, y2, 1) = (x3, y3, z3). It performs faster addition than the generic
// add routine since less arithmetic is needed due to the ability to avoid
// multiplications by the second point's z value.
func (curve *KoblitzCurve) addZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3 *fieldVal) {
// To compute the point addition efficiently, this implementation splits
// the equation into intermediate elements which are used to minimize
// the number of field multiplications using the method shown at:
// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-madd-2007-bl
//
// In particular it performs the calculations using the following:
// Z1Z1 = Z1^2, U2 = X2*Z1Z1, S2 = Y2*Z1*Z1Z1, H = U2-X1, HH = H^2,
// I = 4*HH, J = H*I, r = 2*(S2-Y1), V = X1*I
// X3 = r^2-J-2*V, Y3 = r*(V-X3)-2*Y1*J, Z3 = (Z1+H)^2-Z1Z1-HH
//
// This results in a cost of 7 field multiplications, 4 field squarings,
// 9 field additions, and 4 integer multiplications.
// When the x coordinates are the same for two points on the curve, the
// y coordinates either must be the same, in which case it is point
// doubling, or they are opposite and the result is the point at
// infinity per the group law for elliptic curve cryptography. Since
// any number of Jacobian coordinates can represent the same affine
// point, the x and y values need to be converted to like terms. Due to
// the assumption made for this function that the second point has a z
// value of 1 (z2=1), the first point is already "converted".
var z1z1, u2, s2 fieldVal
x1.Normalize()
y1.Normalize()
z1z1.SquareVal(z1) // Z1Z1 = Z1^2 (mag: 1)
u2.Set(x2).Mul(&z1z1).Normalize() // U2 = X2*Z1Z1 (mag: 1)
s2.Set(y2).Mul(&z1z1).Mul(z1).Normalize() // S2 = Y2*Z1*Z1Z1 (mag: 1)
if x1.Equals(&u2) {
if y1.Equals(&s2) {
// Since x1 == x2 and y1 == y2, point doubling must be
// done, otherwise the addition would end up dividing
// by zero.
curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
return
}
// Since x1 == x2 and y1 == -y2, the sum is the point at
// infinity per the group law.
x3.SetInt(0)
y3.SetInt(0)
z3.SetInt(0)
return
}
// Calculate X3, Y3, and Z3 according to the intermediate elements
// breakdown above.
var h, hh, i, j, r, rr, v fieldVal
var negX1, negY1, negX3 fieldVal
negX1.Set(x1).Negate(1) // negX1 = -X1 (mag: 2)
h.Add2(&u2, &negX1) // H = U2-X1 (mag: 3)
hh.SquareVal(&h) // HH = H^2 (mag: 1)
i.Set(&hh).MulInt(4) // I = 4 * HH (mag: 4)
j.Mul2(&h, &i) // J = H*I (mag: 1)
negY1.Set(y1).Negate(1) // negY1 = -Y1 (mag: 2)
r.Set(&s2).Add(&negY1).MulInt(2) // r = 2*(S2-Y1) (mag: 6)
rr.SquareVal(&r) // rr = r^2 (mag: 1)
v.Mul2(x1, &i) // V = X1*I (mag: 1)
x3.Set(&v).MulInt(2).Add(&j).Negate(3) // X3 = -(J+2*V) (mag: 4)
x3.Add(&rr) // X3 = r^2+X3 (mag: 5)
negX3.Set(x3).Negate(5) // negX3 = -X3 (mag: 6)
y3.Set(y1).Mul(&j).MulInt(2).Negate(2) // Y3 = -(2*Y1*J) (mag: 3)
y3.Add(v.Add(&negX3).Mul(&r)) // Y3 = r*(V-X3)+Y3 (mag: 4)
z3.Add2(z1, &h).Square() // Z3 = (Z1+H)^2 (mag: 1)
z3.Add(z1z1.Add(&hh).Negate(2)) // Z3 = Z3-(Z1Z1+HH) (mag: 4)
// Normalize the resulting field values to a magnitude of 1 as needed.
x3.Normalize()
y3.Normalize()
z3.Normalize()
}
// addGeneric adds two Jacobian points (x1, y1, z1) and (x2, y2, z2) without any
// assumptions about the z values of the two points and stores the result in
// (x3, y3, z3). That is to say (x1, y1, z1) + (x2, y2, z2) = (x3, y3, z3). It
// is the slowest of the add routines due to requiring the most arithmetic.
func (curve *KoblitzCurve) addGeneric(x1, y1, z1, x2, y2, z2, x3, y3, z3 *fieldVal) {
// To compute the point addition efficiently, this implementation splits
// the equation into intermediate elements which are used to minimize
// the number of field multiplications using the method shown at:
// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
//
// In particular it performs the calculations using the following:
// Z1Z1 = Z1^2, Z2Z2 = Z2^2, U1 = X1*Z2Z2, U2 = X2*Z1Z1, S1 = Y1*Z2*Z2Z2
// S2 = Y2*Z1*Z1Z1, H = U2-U1, I = (2*H)^2, J = H*I, r = 2*(S2-S1)
// V = U1*I
// X3 = r^2-J-2*V, Y3 = r*(V-X3)-2*S1*J, Z3 = ((Z1+Z2)^2-Z1Z1-Z2Z2)*H
//
// This results in a cost of 11 field multiplications, 5 field squarings,
// 9 field additions, and 4 integer multiplications.
// When the x coordinates are the same for two points on the curve, the
// y coordinates either must be the same, in which case it is point
// doubling, or they are opposite and the result is the point at
// infinity. Since any number of Jacobian coordinates can represent the
// same affine point, the x and y values need to be converted to like
// terms.
var z1z1, z2z2, u1, u2, s1, s2 fieldVal
z1z1.SquareVal(z1) // Z1Z1 = Z1^2 (mag: 1)
z2z2.SquareVal(z2) // Z2Z2 = Z2^2 (mag: 1)
u1.Set(x1).Mul(&z2z2).Normalize() // U1 = X1*Z2Z2 (mag: 1)
u2.Set(x2).Mul(&z1z1).Normalize() // U2 = X2*Z1Z1 (mag: 1)
s1.Set(y1).Mul(&z2z2).Mul(z2).Normalize() // S1 = Y1*Z2*Z2Z2 (mag: 1)
s2.Set(y2).Mul(&z1z1).Mul(z1).Normalize() // S2 = Y2*Z1*Z1Z1 (mag: 1)
if u1.Equals(&u2) {
if s1.Equals(&s2) {
// Since x1 == x2 and y1 == y2, point doubling must be
// done, otherwise the addition would end up dividing
// by zero.
curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
return
}
// Since x1 == x2 and y1 == -y2, the sum is the point at
// infinity per the group law.
x3.SetInt(0)
y3.SetInt(0)
z3.SetInt(0)
return
}
// Calculate X3, Y3, and Z3 according to the intermediate elements
// breakdown above.
var h, i, j, r, rr, v fieldVal
var negU1, negS1, negX3 fieldVal
negU1.Set(&u1).Negate(1) // negU1 = -U1 (mag: 2)
h.Add2(&u2, &negU1) // H = U2-U1 (mag: 3)
i.Set(&h).MulInt(2).Square() // I = (2*H)^2 (mag: 2)
j.Mul2(&h, &i) // J = H*I (mag: 1)
negS1.Set(&s1).Negate(1) // negS1 = -S1 (mag: 2)
r.Set(&s2).Add(&negS1).MulInt(2) // r = 2*(S2-S1) (mag: 6)
rr.SquareVal(&r) // rr = r^2 (mag: 1)
v.Mul2(&u1, &i) // V = U1*I (mag: 1)
x3.Set(&v).MulInt(2).Add(&j).Negate(3) // X3 = -(J+2*V) (mag: 4)
x3.Add(&rr) // X3 = r^2+X3 (mag: 5)
negX3.Set(x3).Negate(5) // negX3 = -X3 (mag: 6)
y3.Mul2(&s1, &j).MulInt(2).Negate(2) // Y3 = -(2*S1*J) (mag: 3)
y3.Add(v.Add(&negX3).Mul(&r)) // Y3 = r*(V-X3)+Y3 (mag: 4)
z3.Add2(z1, z2).Square() // Z3 = (Z1+Z2)^2 (mag: 1)
z3.Add(z1z1.Add(&z2z2).Negate(2)) // Z3 = Z3-(Z1Z1+Z2Z2) (mag: 4)
z3.Mul(&h) // Z3 = Z3*H (mag: 1)
// Normalize the resulting field values to a magnitude of 1 as needed.
x3.Normalize()
y3.Normalize()
}
// addJacobian adds the passed Jacobian points (x1, y1, z1) and (x2, y2, z2)
// together and stores the result in (x3, y3, z3).
func (curve *KoblitzCurve) addJacobian(x1, y1, z1, x2, y2, z2, x3, y3, z3 *fieldVal) {
// A point at infinity is the identity according to the group law for
// elliptic curve cryptography. Thus, ∞ + P = P and P + ∞ = P.
if (x1.IsZero() && y1.IsZero()) || z1.IsZero() {
x3.Set(x2)
y3.Set(y2)
z3.Set(z2)
return
}
if (x2.IsZero() && y2.IsZero()) || z2.IsZero() {
x3.Set(x1)
y3.Set(y1)
z3.Set(z1)
return
}
// Faster point addition can be achieved when certain assumptions are
// met. For example, when both points have the same z value, arithmetic
// on the z values can be avoided. This section thus checks for these
// conditions and calls an appropriate add function which is accelerated
// by using those assumptions.
z1.Normalize()
z2.Normalize()
isZ1One := z1.Equals(fieldOne)
isZ2One := z2.Equals(fieldOne)
switch {
case isZ1One && isZ2One:
curve.addZ1AndZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3)
return
case z1.Equals(z2):
curve.addZ1EqualsZ2(x1, y1, z1, x2, y2, x3, y3, z3)
return
case isZ2One:
curve.addZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3)
return
}
// None of the above assumptions are true, so fall back to generic
// point addition.
curve.addGeneric(x1, y1, z1, x2, y2, z2, x3, y3, z3)
}
// Add returns the sum of (x1,y1) and (x2,y2). Part of the elliptic.Curve
// interface.
func (curve *KoblitzCurve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
// A point at infinity is the identity according to the group law for
// elliptic curve cryptography. Thus, ∞ + P = P and P + ∞ = P.
if x1.Sign() == 0 && y1.Sign() == 0 {
return x2, y2
}
if x2.Sign() == 0 && y2.Sign() == 0 {
return x1, y1
}
// Convert the affine coordinates from big integers to field values
// and do the point addition in Jacobian projective space.
fx1, fy1 := curve.bigAffineToField(x1, y1)
fx2, fy2 := curve.bigAffineToField(x2, y2)
fx3, fy3, fz3 := new(fieldVal), new(fieldVal), new(fieldVal)
fOne := new(fieldVal).SetInt(1)
curve.addJacobian(fx1, fy1, fOne, fx2, fy2, fOne, fx3, fy3, fz3)
// Convert the Jacobian coordinate field values back to affine big
// integers.
return curve.fieldJacobianToBigAffine(fx3, fy3, fz3)
}
// doubleZ1EqualsOne performs point doubling on the passed Jacobian point
// when the point is already known to have a z value of 1 and stores
// the result in (x3, y3, z3). That is to say (x3, y3, z3) = 2*(x1, y1, 1). It
// performs faster point doubling than the generic routine since less arithmetic
// is needed due to the ability to avoid multiplication by the z value.
func (curve *KoblitzCurve) doubleZ1EqualsOne(x1, y1, x3, y3, z3 *fieldVal) {
// This function uses the assumptions that z1 is 1, thus the point
// doubling formulas reduce to:
//
// X3 = (3*X1^2)^2 - 8*X1*Y1^2
// Y3 = (3*X1^2)*(4*X1*Y1^2 - X3) - 8*Y1^4
// Z3 = 2*Y1
//
// To compute the above efficiently, this implementation splits the
// equation into intermediate elements which are used to minimize the
// number of field multiplications in favor of field squarings which
// are roughly 35% faster than field multiplications with the current
// implementation at the time this was written.
//
// This uses a slightly modified version of the method shown at:
// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-mdbl-2007-bl
//
// In particular it performs the calculations using the following:
// A = X1^2, B = Y1^2, C = B^2, D = 2*((X1+B)^2-A-C)
// E = 3*A, F = E^2, X3 = F-2*D, Y3 = E*(D-X3)-8*C
// Z3 = 2*Y1
//
// This results in a cost of 1 field multiplication, 5 field squarings,
// 6 field additions, and 5 integer multiplications.
var a, b, c, d, e, f fieldVal
z3.Set(y1).MulInt(2) // Z3 = 2*Y1 (mag: 2)
a.SquareVal(x1) // A = X1^2 (mag: 1)
b.SquareVal(y1) // B = Y1^2 (mag: 1)
c.SquareVal(&b) // C = B^2 (mag: 1)
b.Add(x1).Square() // B = (X1+B)^2 (mag: 1)
d.Set(&a).Add(&c).Negate(2) // D = -(A+C) (mag: 3)
d.Add(&b).MulInt(2) // D = 2*(B+D)(mag: 8)
e.Set(&a).MulInt(3) // E = 3*A (mag: 3)
f.SquareVal(&e) // F = E^2 (mag: 1)
x3.Set(&d).MulInt(2).Negate(16) // X3 = -(2*D) (mag: 17)
x3.Add(&f) // X3 = F+X3 (mag: 18)
f.Set(x3).Negate(18).Add(&d).Normalize() // F = D-X3 (mag: 1)
y3.Set(&c).MulInt(8).Negate(8) // Y3 = -(8*C) (mag: 9)
y3.Add(f.Mul(&e)) // Y3 = E*F+Y3 (mag: 10)
// Normalize the field values back to a magnitude of 1.
x3.Normalize()
y3.Normalize()
z3.Normalize()
}
// doubleGeneric performs point doubling on the passed Jacobian point without
// any assumptions about the z value and stores the result in (x3, y3, z3).
// That is to say (x3, y3, z3) = 2*(x1, y1, z1). It is the slowest of the point
// doubling routines due to requiring the most arithmetic.
func (curve *KoblitzCurve) doubleGeneric(x1, y1, z1, x3, y3, z3 *fieldVal) {
// Point doubling formula for Jacobian coordinates for the secp256k1
// curve:
// X3 = (3*X1^2)^2 - 8*X1*Y1^2
// Y3 = (3*X1^2)*(4*X1*Y1^2 - X3) - 8*Y1^4
// Z3 = 2*Y1*Z1
//
// To compute the above efficiently, this implementation splits the
// equation into intermediate elements which are used to minimize the
// number of field multiplications in favor of field squarings which
// are roughly 35% faster than field multiplications with the current
// implementation at the time this was written.
//
// This uses a slightly modified version of the method shown at:
// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
//
// In particular it performs the calculations using the following:
// A = X1^2, B = Y1^2, C = B^2, D = 2*((X1+B)^2-A-C)
// E = 3*A, F = E^2, X3 = F-2*D, Y3 = E*(D-X3)-8*C
// Z3 = 2*Y1*Z1
//
// This results in a cost of 1 field multiplication, 5 field squarings,
// 6 field additions, and 5 integer multiplications.
var a, b, c, d, e, f fieldVal
z3.Mul2(y1, z1).MulInt(2) // Z3 = 2*Y1*Z1 (mag: 2)
a.SquareVal(x1) // A = X1^2 (mag: 1)
b.SquareVal(y1) // B = Y1^2 (mag: 1)
c.SquareVal(&b) // C = B^2 (mag: 1)
b.Add(x1).Square() // B = (X1+B)^2 (mag: 1)
d.Set(&a).Add(&c).Negate(2) // D = -(A+C) (mag: 3)
d.Add(&b).MulInt(2) // D = 2*(B+D)(mag: 8)
e.Set(&a).MulInt(3) // E = 3*A (mag: 3)
f.SquareVal(&e) // F = E^2 (mag: 1)
x3.Set(&d).MulInt(2).Negate(16) // X3 = -(2*D) (mag: 17)
x3.Add(&f) // X3 = F+X3 (mag: 18)
f.Set(x3).Negate(18).Add(&d).Normalize() // F = D-X3 (mag: 1)
y3.Set(&c).MulInt(8).Negate(8) // Y3 = -(8*C) (mag: 9)
y3.Add(f.Mul(&e)) // Y3 = E*F+Y3 (mag: 10)
// Normalize the field values back to a magnitude of 1.
x3.Normalize()
y3.Normalize()
z3.Normalize()
}
// doubleJacobian doubles the passed Jacobian point (x1, y1, z1) and stores the
// result in (x3, y3, z3).
func (curve *KoblitzCurve) doubleJacobian(x1, y1, z1, x3, y3, z3 *fieldVal) {
// Doubling a point at infinity is still infinity.
if y1.IsZero() || z1.IsZero() {
x3.SetInt(0)
y3.SetInt(0)
z3.SetInt(0)
return
}
// Slightly faster point doubling can be achieved when the z value is 1
// by avoiding the multiplication on the z value. This section calls
// a point doubling function which is accelerated by using that
// assumption when possible.
if z1.Normalize().Equals(fieldOne) {
curve.doubleZ1EqualsOne(x1, y1, x3, y3, z3)
return
}
// Fall back to generic point doubling which works with arbitrary z
// values.
curve.doubleGeneric(x1, y1, z1, x3, y3, z3)
}
// Double returns 2*(x1,y1). Part of the elliptic.Curve interface.
func (curve *KoblitzCurve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
if y1.Sign() == 0 {
return new(big.Int), new(big.Int)
}
// Convert the affine coordinates from big integers to field values
// and do the point doubling in Jacobian projective space.
fx1, fy1 := curve.bigAffineToField(x1, y1)
fx3, fy3, fz3 := new(fieldVal), new(fieldVal), new(fieldVal)
fOne := new(fieldVal).SetInt(1)
curve.doubleJacobian(fx1, fy1, fOne, fx3, fy3, fz3)
// Convert the Jacobian coordinate field values back to affine big
// integers.
return curve.fieldJacobianToBigAffine(fx3, fy3, fz3)
}
// splitK returns a balanced length-two representation of k and their signs.
// This is algorithm 3.74 from [GECC].
//
// One thing of note about this algorithm is that no matter what c1 and c2 are,
// the final equation of k = k1 + k2 * lambda (mod n) will hold. This is
// provable mathematically due to how a1/b1/a2/b2 are computed.
//
// c1 and c2 are chosen to minimize the max(k1,k2).
func (curve *KoblitzCurve) splitK(k []byte) ([]byte, []byte, int, int) {
// All math here is done with big.Int, which is slow.
// At some point, it might be useful to write something similar to
// fieldVal but for N instead of P as the prime field if this ends up
// being a bottleneck.
bigIntK := new(big.Int)
c1, c2 := new(big.Int), new(big.Int)
tmp1, tmp2 := new(big.Int), new(big.Int)
k1, k2 := new(big.Int), new(big.Int)
bigIntK.SetBytes(k)
// c1 = round(b2 * k / n) from step 4.
// Rounding isn't really necessary and costs too much, hence skipped
c1.Mul(curve.b2, bigIntK)
c1.Div(c1, curve.N)
// c2 = round(b1 * k / n) from step 4 (sign reversed to optimize one step)
// Rounding isn't really necessary and costs too much, hence skipped
c2.Mul(curve.b1, bigIntK)
c2.Div(c2, curve.N)
// k1 = k - c1 * a1 - c2 * a2 from step 5 (note c2's sign is reversed)
tmp1.Mul(c1, curve.a1)
tmp2.Mul(c2, curve.a2)
k1.Sub(bigIntK, tmp1)
k1.Add(k1, tmp2)
// k2 = - c1 * b1 - c2 * b2 from step 5 (note c2's sign is reversed)
tmp1.Mul(c1, curve.b1)
tmp2.Mul(c2, curve.b2)
k2.Sub(tmp2, tmp1)
// Note Bytes() throws out the sign of k1 and k2. This matters
// since k1 and/or k2 can be negative. Hence, we pass that
// back separately.
return k1.Bytes(), k2.Bytes(), k1.Sign(), k2.Sign()
}
// moduloReduce reduces k from more than 32 bytes to 32 bytes and under. This
// is done by doing a simple modulo curve.N. We can do this since G^N = 1 and
// thus any other valid point on the elliptic curve has the same order.
func (curve *KoblitzCurve) moduloReduce(k []byte) []byte {
// Since the order of G is curve.N, we can use a much smaller number
// by doing modulo curve.N
if len(k) > curve.byteSize {
// Reduce k by performing modulo curve.N.
tmpK := new(big.Int).SetBytes(k)
tmpK.Mod(tmpK, curve.N)
return tmpK.Bytes()
}
return k
}
// NAF takes a positive integer k and returns the Non-Adjacent Form (NAF) as two
// byte slices. The first is where 1s will be. The second is where -1s will
// be. NAF is convenient in that on average, only 1/3rd of its values are
// non-zero. This is algorithm 3.30 from [GECC].
//
// Essentially, this makes it possible to minimize the number of operations
// since the resulting ints returned will be at least 50% 0s.
func NAF(k []byte) ([]byte, []byte) {
// The essence of this algorithm is that whenever we have consecutive 1s
// in the binary, we want to put a -1 in the lowest bit and get a bunch
// of 0s up to the highest bit of consecutive 1s. This is due to this
// identity:
// 2^n + 2^(n-1) + 2^(n-2) + ... + 2^(n-k) = 2^(n+1) - 2^(n-k)
//
// The algorithm thus may need to go 1 more bit than the length of the
// bits we actually have, hence bits being 1 bit longer than was
// necessary. Since we need to know whether adding will cause a carry,
// we go from right-to-left in this addition.
var carry, curIsOne, nextIsOne bool
// these default to zero
retPos := make([]byte, len(k)+1)
retNeg := make([]byte, len(k)+1)
for i := len(k) - 1; i >= 0; i-- {
curByte := k[i]
for j := uint(0); j < 8; j++ {
curIsOne = curByte&1 == 1
if j == 7 {
if i == 0 {
nextIsOne = false
} else {
nextIsOne = k[i-1]&1 == 1
}
} else {
nextIsOne = curByte&2 == 2
}
if carry {
if curIsOne {
// This bit is 1, so continue to carry
// and don't need to do anything.
} else {
// We've hit a 0 after some number of
// 1s.
if nextIsOne {
// Start carrying again since
// a new sequence of 1s is
// starting.
retNeg[i+1] += 1 << j
} else {
// Stop carrying since 1s have
// stopped.
carry = false
retPos[i+1] += 1 << j
}
}
} else if curIsOne {
if nextIsOne {
// If this is the start of at least 2
// consecutive 1s, set the current one
// to -1 and start carrying.
retNeg[i+1] += 1 << j
carry = true
} else {
// This is a singleton, not consecutive
// 1s.
retPos[i+1] += 1 << j
}
}
curByte >>= 1
}
}
if carry {
retPos[0] = 1
return retPos, retNeg
}
return retPos[1:], retNeg[1:]
}
// ScalarMult returns k*(Bx, By) where k is a big endian integer.
// Part of the elliptic.Curve interface.
func (curve *KoblitzCurve) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
// Point Q = ∞ (point at infinity).
qx, qy, qz := new(fieldVal), new(fieldVal), new(fieldVal)
// Decompose K into k1 and k2 in order to halve the number of EC ops.
// See Algorithm 3.74 in [GECC].
k1, k2, signK1, signK2 := curve.splitK(curve.moduloReduce(k))
// The main equation here to remember is:
// k * P = k1 * P + k2 * ϕ(P)
//
// P1 below is P in the equation, P2 below is ϕ(P) in the equation
p1x, p1y := curve.bigAffineToField(Bx, By)
p1yNeg := new(fieldVal).NegateVal(p1y, 1)
p1z := new(fieldVal).SetInt(1)
// NOTE: ϕ(x,y) = (βx,y). The Jacobian z coordinate is 1, so this math
// goes through.
p2x := new(fieldVal).Mul2(p1x, curve.beta)
p2y := new(fieldVal).Set(p1y)
p2yNeg := new(fieldVal).NegateVal(p2y, 1)
p2z := new(fieldVal).SetInt(1)
// Flip the positive and negative values of the points as needed
// depending on the signs of k1 and k2. As mentioned in the equation
// above, each of k1 and k2 are multiplied by the respective point.
// Since -k * P is the same thing as k * -P, and the group law for
// elliptic curves states that P(x, y) = -P(x, -y), it's faster and
// simplifies the code to just make the point negative.
if signK1 == -1 {
p1y, p1yNeg = p1yNeg, p1y
}
if signK2 == -1 {
p2y, p2yNeg = p2yNeg, p2y
}
// NAF versions of k1 and k2 should have a lot more zeros.
//
// The Pos version of the bytes contain the +1s and the Neg versions
// contain the -1s.
k1PosNAF, k1NegNAF := NAF(k1)
k2PosNAF, k2NegNAF := NAF(k2)
k1Len := len(k1PosNAF)
k2Len := len(k2PosNAF)
m := k1Len
if m < k2Len {
m = k2Len
}
// Add left-to-right using the NAF optimization. See algorithm 3.77
// from [GECC]. This should be faster overall since there will be a lot
// more instances of 0, hence reducing the number of Jacobian additions
// at the cost of 1 possible extra doubling.
var k1BytePos, k1ByteNeg, k2BytePos, k2ByteNeg byte
for i := 0; i < m; i++ {
// Since we're going left-to-right, pad the front with 0s.
if i < m-k1Len {
k1BytePos = 0
k1ByteNeg = 0
} else {
k1BytePos = k1PosNAF[i-(m-k1Len)]
k1ByteNeg = k1NegNAF[i-(m-k1Len)]
}
if i < m-k2Len {
k2BytePos = 0
k2ByteNeg = 0
} else {
k2BytePos = k2PosNAF[i-(m-k2Len)]
k2ByteNeg = k2NegNAF[i-(m-k2Len)]
}
for j := 7; j >= 0; j-- {
// Q = 2 * Q
curve.doubleJacobian(qx, qy, qz, qx, qy, qz)
if k1BytePos&0x80 == 0x80 {
curve.addJacobian(qx, qy, qz, p1x, p1y, p1z,
qx, qy, qz)
} else if k1ByteNeg&0x80 == 0x80 {
curve.addJacobian(qx, qy, qz, p1x, p1yNeg, p1z,
qx, qy, qz)
}
if k2BytePos&0x80 == 0x80 {
curve.addJacobian(qx, qy, qz, p2x, p2y, p2z,
qx, qy, qz)
} else if k2ByteNeg&0x80 == 0x80 {
curve.addJacobian(qx, qy, qz, p2x, p2yNeg, p2z,
qx, qy, qz)
}
k1BytePos <<= 1
k1ByteNeg <<= 1
k2BytePos <<= 1
k2ByteNeg <<= 1
}
}
// Convert the Jacobian coordinate field values back to affine big.Ints.
return curve.fieldJacobianToBigAffine(qx, qy, qz)
}
// ScalarBaseMult returns k*G where G is the base point of the group and k is a
// big endian integer.
// Part of the elliptic.Curve interface.
func (curve *KoblitzCurve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
newK := curve.moduloReduce(k)
diff := len(curve.bytePoints) - len(newK)
// Point Q = ∞ (point at infinity).
qx, qy, qz := new(fieldVal), new(fieldVal), new(fieldVal)
// curve.bytePoints has all 256 byte points for each 8-bit window. The
// strategy is to add up the byte points. This is best understood by
// expressing k in base-256 which it already sort of is.
// Each "digit" in the 8-bit window can be looked up using bytePoints
// and added together.
for i, byteVal := range newK {
p := curve.bytePoints[diff+i][byteVal]
curve.addJacobian(qx, qy, qz, &p[0], &p[1], &p[2], qx, qy, qz)
}
return curve.fieldJacobianToBigAffine(qx, qy, qz)
}
// QPlus1Div4 returns the Q+1/4 constant for the curve for use in calculating
// square roots via exponention.
func (curve *KoblitzCurve) QPlus1Div4() *big.Int {
return curve.q
}
var initonce sync.Once
var secp256k1 KoblitzCurve
func initAll() {
initS256()
}
// fromHex converts the passed hex string into a big integer pointer and will
// panic is there is an error. This is only provided for the hard-coded
// constants so errors in the source code can bet detected. It will only (and
// must only) be called for initialization purposes.
func fromHex(s string) *big.Int {
r, ok := new(big.Int).SetString(s, 16)
if !ok {
panic("invalid hex in source file: " + s)
}
return r
}
func initS256() {
// Curve parameters taken from [SECG] section 2.4.1.
secp256k1.CurveParams = new(elliptic.CurveParams)
secp256k1.P = fromHex("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F")
secp256k1.N = fromHex("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141")
secp256k1.B = fromHex("0000000000000000000000000000000000000000000000000000000000000007")
secp256k1.Gx = fromHex("79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798")
secp256k1.Gy = fromHex("483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8")
secp256k1.BitSize = 256
secp256k1.q = new(big.Int).Div(new(big.Int).Add(secp256k1.P,
big.NewInt(1)), big.NewInt(4))
secp256k1.H = 1
secp256k1.halfOrder = new(big.Int).Rsh(secp256k1.N, 1)
// Provided for convenience since this gets computed repeatedly.
secp256k1.byteSize = secp256k1.BitSize / 8
// Deserialize and set the pre-computed table used to accelerate scalar
// base multiplication. This is hard-coded data, so any errors are
// panics because it means something is wrong in the source code.
if err := loadS256BytePoints(); err != nil {
panic(err)
}
// Next 6 constants are from Hal Finney's bitcointalk.org post:
// https://bitcointalk.org/index.php?topic=3238.msg45565#msg45565
// May he rest in peace.
//
// They have also been independently derived from the code in the
// EndomorphismVectors function in gensecp256k1.go.
secp256k1.lambda = fromHex("5363AD4CC05C30E0A5261C028812645A122E22EA20816678DF02967C1B23BD72")
secp256k1.beta = new(fieldVal).SetHex("7AE96A2B657C07106E64479EAC3434E99CF0497512F58995C1396C28719501EE")
secp256k1.a1 = fromHex("3086D221A7D46BCDE86C90E49284EB15")
secp256k1.b1 = fromHex("-E4437ED6010E88286F547FA90ABFE4C3")
secp256k1.a2 = fromHex("114CA50F7A8E2F3F657C1108D9D44CFD8")
secp256k1.b2 = fromHex("3086D221A7D46BCDE86C90E49284EB15")
// Alternatively, we can use the parameters below, however, they seem
// to be about 8% slower.
// secp256k1.lambda = fromHex("AC9C52B33FA3CF1F5AD9E3FD77ED9BA4A880B9FC8EC739C2E0CFC810B51283CE")
// secp256k1.beta = new(fieldVal).SetHex("851695D49A83F8EF919BB86153CBCB16630FB68AED0A766A3EC693D68E6AFA40")
// secp256k1.a1 = fromHex("E4437ED6010E88286F547FA90ABFE4C3")
// secp256k1.b1 = fromHex("-3086D221A7D46BCDE86C90E49284EB15")
// secp256k1.a2 = fromHex("3086D221A7D46BCDE86C90E49284EB15")
// secp256k1.b2 = fromHex("114CA50F7A8E2F3F657C1108D9D44CFD8")
}
// S256 returns a Curve which implements secp256k1.
func S256() *KoblitzCurve {
initonce.Do(initAll)
return &secp256k1
}