mirror of
https://github.com/golang/go
synced 2024-11-22 01:14:40 -07:00
crypto/elliptic: use a Jacobian transform
(Speeds up the code about 25x) R=r CC=golang-dev https://golang.org/cl/3359042
This commit is contained in:
Adam Langley
parent
6540c85c7f
commit
287045085d
@ -6,9 +6,12 @@
|
||||
// fields
|
||||
package elliptic
|
||||
|
||||
// WARNING: this implementation is simple but slow and not constant time.
|
||||
// A significant speedup could be obtained by using either a projective or
|
||||
// Jacobian transform.
|
||||
// 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 (
|
||||
"big"
|
||||
@ -42,77 +45,155 @@ func (curve *Curve) IsOnCurve(x, y *big.Int) bool {
|
||||
return x3.Cmp(y2) == 0
|
||||
}
|
||||
|
||||
// affineFromJacobian reverses the Jacobian transform. See the comment at the
|
||||
// top of the file.
|
||||
func (curve *Curve) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) {
|
||||
zinv := new(big.Int).ModInverse(z, curve.P)
|
||||
zinvsq := new(big.Int).Mul(zinv, zinv)
|
||||
|
||||
xOut = new(big.Int).Mul(x, zinvsq)
|
||||
xOut.Mod(xOut, curve.P)
|
||||
zinvsq.Mul(zinvsq, zinv)
|
||||
yOut = new(big.Int).Mul(y, zinvsq)
|
||||
yOut.Mod(yOut, curve.P)
|
||||
return
|
||||
}
|
||||
|
||||
// Add returns the sum of (x1,y1) and (x2,y2)
|
||||
func (curve *Curve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
|
||||
// x = (y2-y1)²/(x2-x1)²-x1-x2
|
||||
y2my1 := new(big.Int).Sub(y2, y1)
|
||||
if y2my1.Sign() < 0 {
|
||||
y2my1.Add(y2my1, curve.P)
|
||||
z := new(big.Int).SetInt64(1)
|
||||
return curve.affineFromJacobian(curve.addJacobian(x1, y1, z, x2, y2, z))
|
||||
}
|
||||
|
||||
// addJacobian takes two points in Jacobian coordinates, (x1, y1, z1) and
|
||||
// (x2, y2, z2) and returns their sum, also in Jacobian form.
|
||||
func (curve *Curve) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) {
|
||||
// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl
|
||||
z1z1 := new(big.Int).Mul(z1, z1)
|
||||
z1z1.Mod(z1z1, curve.P)
|
||||
z2z2 := new(big.Int).Mul(z2, z2)
|
||||
z2z2.Mod(z2z2, curve.P)
|
||||
|
||||
u1 := new(big.Int).Mul(x1, z2z2)
|
||||
u1.Mod(u1, curve.P)
|
||||
u2 := new(big.Int).Mul(x2, z1z1)
|
||||
u2.Mod(u2, curve.P)
|
||||
h := new(big.Int).Sub(u2, u1)
|
||||
if h.Sign() == -1 {
|
||||
h.Add(h, curve.P)
|
||||
}
|
||||
y2my1sq := new(big.Int).Mul(y2my1, y2my1)
|
||||
x2mx1 := new(big.Int).Sub(x2, x1)
|
||||
if x2mx1.Sign() < 0 {
|
||||
x2mx1.Add(x2mx1, curve.P)
|
||||
i := new(big.Int).Lsh(h, 1)
|
||||
i.Mul(i, i)
|
||||
j := new(big.Int).Mul(h, i)
|
||||
|
||||
s1 := new(big.Int).Mul(y1, z2)
|
||||
s1.Mul(s1, z2z2)
|
||||
s1.Mod(s1, curve.P)
|
||||
s2 := new(big.Int).Mul(y2, z1)
|
||||
s2.Mul(s2, z1z1)
|
||||
s2.Mod(s2, curve.P)
|
||||
r := new(big.Int).Sub(s2, s1)
|
||||
if r.Sign() == -1 {
|
||||
r.Add(r, curve.P)
|
||||
}
|
||||
x2mx1sq := new(big.Int).Mul(x2mx1, x2mx1)
|
||||
x2mx1sqinv := new(big.Int).ModInverse(x2mx1sq, curve.P)
|
||||
r.Lsh(r, 1)
|
||||
v := new(big.Int).Mul(u1, i)
|
||||
|
||||
x := new(big.Int).Mul(y2my1sq, x2mx1sqinv)
|
||||
x.Sub(x, x1)
|
||||
x.Sub(x, x2)
|
||||
x.Mod(x, curve.P)
|
||||
x3 := new(big.Int).Set(r)
|
||||
x3.Mul(x3, x3)
|
||||
x3.Sub(x3, j)
|
||||
x3.Sub(x3, v)
|
||||
x3.Sub(x3, v)
|
||||
x3.Mod(x3, curve.P)
|
||||
|
||||
// y = (2x1+x2)*(y2-y1)/(x2-x1)-(y2-y1)³/(x2-x1)³-y1
|
||||
y := new(big.Int).Lsh(x1, 1)
|
||||
y.Add(y, x2)
|
||||
x2mx1inv := new(big.Int).ModInverse(x2mx1, curve.P)
|
||||
x2mx1inv.Mul(y2my1, x2mx1inv)
|
||||
y.Mul(y, x2mx1inv)
|
||||
y3 := new(big.Int).Set(r)
|
||||
v.Sub(v, x3)
|
||||
y3.Mul(y3, v)
|
||||
s1.Mul(s1, j)
|
||||
s1.Lsh(s1, 1)
|
||||
y3.Sub(y3, s1)
|
||||
y3.Mod(y3, curve.P)
|
||||
|
||||
y2my1sq.Mul(y2my1sq, y2my1)
|
||||
x2mx1sq.Mul(x2mx1sq, x2mx1)
|
||||
x2mx1sqinv.ModInverse(x2mx1sq, curve.P)
|
||||
y2my1sq.Mul(y2my1sq, x2mx1sqinv)
|
||||
y.Sub(y, y2my1sq)
|
||||
y.Sub(y, y1)
|
||||
y.Mod(y, curve.P)
|
||||
z3 := new(big.Int).Add(z1, z2)
|
||||
z3.Mul(z3, z3)
|
||||
z3.Sub(z3, z1z1)
|
||||
if z3.Sign() == -1 {
|
||||
z3.Add(z3, curve.P)
|
||||
}
|
||||
z3.Sub(z3, z2z2)
|
||||
if z3.Sign() == -1 {
|
||||
z3.Add(z3, curve.P)
|
||||
}
|
||||
z3.Mul(z3, h)
|
||||
z3.Mod(z3, curve.P)
|
||||
|
||||
return x, y
|
||||
return x3, y3, z3
|
||||
}
|
||||
|
||||
// Double returns 2*(x,y)
|
||||
func (curve *Curve) Double(x, y *big.Int) (*big.Int, *big.Int) {
|
||||
// x = (3x²-3)²/(2y)²-x-x
|
||||
threexsqm3 := new(big.Int).Mul(x, x)
|
||||
three := new(big.Int).SetInt64(3)
|
||||
threexsqm3.Mul(threexsqm3, three)
|
||||
threexsqm3.Sub(threexsqm3, three)
|
||||
threexsqm3sq := new(big.Int).Mul(threexsqm3, threexsqm3)
|
||||
func (curve *Curve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
|
||||
z1 := new(big.Int).SetInt64(1)
|
||||
return curve.affineFromJacobian(curve.doubleJacobian(x1, y1, z1))
|
||||
}
|
||||
|
||||
twoy := new(big.Int).Lsh(y, 1)
|
||||
twoysq := new(big.Int).Mul(twoy, twoy)
|
||||
twoysqinv := new(big.Int).ModInverse(twoysq, curve.P)
|
||||
// doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and
|
||||
// returns its double, also in Jacobian form.
|
||||
func (curve *Curve) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) {
|
||||
// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
|
||||
delta := new(big.Int).Mul(z, z)
|
||||
delta.Mod(delta, curve.P)
|
||||
gamma := new(big.Int).Mul(y, y)
|
||||
gamma.Mod(gamma, curve.P)
|
||||
alpha := new(big.Int).Sub(x, delta)
|
||||
if alpha.Sign() == -1 {
|
||||
alpha.Add(alpha, curve.P)
|
||||
}
|
||||
alpha2 := new(big.Int).Add(x, delta)
|
||||
alpha.Mul(alpha, alpha2)
|
||||
alpha2.Set(alpha)
|
||||
alpha.Lsh(alpha, 1)
|
||||
alpha.Add(alpha, alpha2)
|
||||
|
||||
outx := new(big.Int).Mul(threexsqm3sq, twoysqinv)
|
||||
outx.Sub(outx, x)
|
||||
outx.Sub(outx, x)
|
||||
outx.Mod(outx, curve.P)
|
||||
beta := alpha2.Mul(x, gamma)
|
||||
|
||||
// y = 3x*(3x²-3)/(2y)-(3x²-3)³/(2y)³-y
|
||||
outy := new(big.Int).Mul(x, three)
|
||||
outy.Mul(outy, threexsqm3)
|
||||
twoyinv := new(big.Int).ModInverse(twoy, curve.P)
|
||||
outy.Mul(outy, twoyinv)
|
||||
x3 := new(big.Int).Mul(alpha, alpha)
|
||||
beta8 := new(big.Int).Lsh(beta, 3)
|
||||
x3.Sub(x3, beta8)
|
||||
for x3.Sign() == -1 {
|
||||
x3.Add(x3, curve.P)
|
||||
}
|
||||
x3.Mod(x3, curve.P)
|
||||
|
||||
threexsqm3sq.Mul(threexsqm3sq, threexsqm3)
|
||||
twoysq.Mul(twoysq, twoy)
|
||||
twoysqinv.ModInverse(twoysq, curve.P)
|
||||
threexsqm3sq.Mul(threexsqm3sq, twoysqinv)
|
||||
outy.Sub(outy, threexsqm3sq)
|
||||
outy.Sub(outy, y)
|
||||
outy.Mod(outy, curve.P)
|
||||
z3 := new(big.Int).Add(y, z)
|
||||
z3.Mul(z3, z3)
|
||||
z3.Sub(z3, gamma)
|
||||
if z3.Sign() == -1 {
|
||||
z3.Add(z3, curve.P)
|
||||
}
|
||||
z3.Sub(z3, delta)
|
||||
if z3.Sign() == -1 {
|
||||
z3.Add(z3, curve.P)
|
||||
}
|
||||
z3.Mod(z3, curve.P)
|
||||
|
||||
return outx, outy
|
||||
beta.Lsh(beta, 2)
|
||||
beta.Sub(beta, x3)
|
||||
if beta.Sign() == -1 {
|
||||
beta.Add(beta, curve.P)
|
||||
}
|
||||
y3 := alpha.Mul(alpha, beta)
|
||||
|
||||
gamma.Mul(gamma, gamma)
|
||||
gamma.Lsh(gamma, 3)
|
||||
gamma.Mod(gamma, curve.P)
|
||||
|
||||
y3.Sub(y3, gamma)
|
||||
if y3.Sign() == -1 {
|
||||
y3.Add(y3, curve.P)
|
||||
}
|
||||
y3.Mod(y3, curve.P)
|
||||
|
||||
return x3, y3, z3
|
||||
}
|
||||
|
||||
// ScalarMult returns k*(Bx,By) where k is a number in big-endian form.
|
||||
@ -125,20 +206,22 @@ func (curve *Curve) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
|
||||
// |k|, then we return nil, nil, because we cannot return the identity
|
||||
// element.
|
||||
|
||||
Bz := new(big.Int).SetInt64(1)
|
||||
x := Bx
|
||||
y := By
|
||||
z := Bz
|
||||
|
||||
seenFirstTrue := false
|
||||
for _, byte := range k {
|
||||
for bitNum := 0; bitNum < 8; bitNum++ {
|
||||
if seenFirstTrue {
|
||||
x, y = curve.Double(x, y)
|
||||
x, y, z = curve.doubleJacobian(x, y, z)
|
||||
}
|
||||
if byte&0x80 == 0x80 {
|
||||
if !seenFirstTrue {
|
||||
seenFirstTrue = true
|
||||
} else {
|
||||
x, y = curve.Add(Bx, By, x, y)
|
||||
x, y, z = curve.addJacobian(Bx, By, Bz, x, y, z)
|
||||
}
|
||||
}
|
||||
byte <<= 1
|
||||
@ -149,7 +232,7 @@ func (curve *Curve) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
|
||||
return nil, nil
|
||||
}
|
||||
|
||||
return x, y
|
||||
return curve.affineFromJacobian(x, y, z)
|
||||
}
|
||||
|
||||
// ScalarBaseMult returns k*G, where G is the base point of the group and k is
|
||||
|
||||
@ -298,3 +298,32 @@ func TestBaseMult(t *testing.T) {
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
func BenchmarkBaseMult(b *testing.B) {
|
||||
b.ResetTimer()
|
||||
p224 := P224()
|
||||
e := p224BaseMultTests[25]
|
||||
k, _ := new(big.Int).SetString(e.k, 10)
|
||||
b.StartTimer()
|
||||
for i := 0; i < b.N; i++ {
|
||||
p224.ScalarBaseMult(k.Bytes())
|
||||
}
|
||||
}
|
||||
|
||||
func TestMultiples(t *testing.T) {
|
||||
p256 := P256()
|
||||
x := p256.Gx
|
||||
y := p256.Gy
|
||||
Gz := new(big.Int).SetInt64(1)
|
||||
z := Gz
|
||||
|
||||
for i := 1; i <= 16; i++ {
|
||||
fmt.Printf("i: %d\n", i)
|
||||
fmt.Printf(" %s\n %s\n %s\n", x.String(), y.String(), z.String())
|
||||
if i == 1 {
|
||||
x, y, z = p256.doubleJacobian(x, y, z)
|
||||
} else {
|
||||
x, y, z = p256.addJacobian(x, y, z, p256.Gx, p256.Gy, Gz)
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user