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// Copyright 2013 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package elliptic
import (
"crypto/elliptic/internal/fiat"
"math/big"
)
type p521Curve struct {
*CurveParams
}
var p521 p521Curve
var p521Params *CurveParams
func initP521() {
// See FIPS 186-3, section D.2.5
p521.CurveParams = &CurveParams{Name: "P-521"}
p521.P, _ = new(big.Int).SetString("6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151", 10)
p521.N, _ = new(big.Int).SetString("6864797660130609714981900799081393217269435300143305409394463459185543183397655394245057746333217197532963996371363321113864768612440380340372808892707005449", 10)
p521.B, _ = new(big.Int).SetString("051953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b489918ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451fd46b503f00", 16)
p521.Gx, _ = new(big.Int).SetString("c6858e06b70404e9cd9e3ecb662395b4429c648139053fb521f828af606b4d3dbaa14b5e77efe75928fe1dc127a2ffa8de3348b3c1856a429bf97e7e31c2e5bd66", 16)
p521.Gy, _ = new(big.Int).SetString("11839296a789a3bc0045c8a5fb42c7d1bd998f54449579b446817afbd17273e662c97ee72995ef42640c550b9013fad0761353c7086a272c24088be94769fd16650", 16)
p521.BitSize = 521
}
func (curve p521Curve) Params() *CurveParams {
return curve.CurveParams
}
func (curve p521Curve) IsOnCurve(x, y *big.Int) bool {
if x.Sign() < 0 || x.Cmp(curve.P) >= 0 ||
y.Sign() < 0 || y.Cmp(curve.P) >= 0 {
return false
}
x1 := bigIntToFiatP521(x)
y1 := bigIntToFiatP521(y)
b := bigIntToFiatP521(curve.B) // TODO: precompute this value.
// x³ - 3x + b.
x3 := new(fiat.P521Element).Square(x1)
x3.Mul(x3, x1)
threeX := new(fiat.P521Element).Add(x1, x1)
threeX.Add(threeX, x1)
x3.Sub(x3, threeX)
x3.Add(x3, b)
// y² = x³ - 3x + b
y2 := new(fiat.P521Element).Square(y1)
return x3.Equal(y2) == 1
}
type p521Point struct {
x, y, z *fiat.P521Element
}
func fiatP521ToBigInt(x *fiat.P521Element) *big.Int {
xBytes := x.Bytes()
for i := range xBytes[:len(xBytes)/2] {
xBytes[i], xBytes[len(xBytes)-i-1] = xBytes[len(xBytes)-i-1], xBytes[i]
}
return new(big.Int).SetBytes(xBytes)
}
// affineFromJacobian brings a point in Jacobian coordinates back to affine
// coordinates, with (0, 0) representing infinity by convention. It also goes
// back to big.Int values to match the exposed API.
func (curve p521Curve) affineFromJacobian(p *p521Point) (x, y *big.Int) {
if p.z.IsZero() == 1 {
return new(big.Int), new(big.Int)
}
zinv := new(fiat.P521Element).Invert(p.z)
zinvsq := new(fiat.P521Element).Mul(zinv, zinv)
xx := new(fiat.P521Element).Mul(p.x, zinvsq)
zinvsq.Mul(zinvsq, zinv)
yy := new(fiat.P521Element).Mul(p.y, zinvsq)
return fiatP521ToBigInt(xx), fiatP521ToBigInt(yy)
}
func bigIntToFiatP521(x *big.Int) *fiat.P521Element {
xBytes := new(big.Int).Mod(x, p521.P).FillBytes(make([]byte, 66))
for i := range xBytes[:len(xBytes)/2] {
xBytes[i], xBytes[len(xBytes)-i-1] = xBytes[len(xBytes)-i-1], xBytes[i]
}
x1, err := new(fiat.P521Element).SetBytes(xBytes)
if err != nil {
// The input is reduced modulo P and encoded in a fixed size bytes
// slice, this should be impossible.
panic("internal error: bigIntToFiatP521")
}
return x1
}
// jacobianFromAffine converts (x, y) affine coordinates into (x, y, z) Jacobian
// coordinates. It also converts from big.Int to fiat, which is necessarily a
// messy and variable-time operation, which we can't avoid due to the exposed API.
func (curve p521Curve) jacobianFromAffine(x, y *big.Int) *p521Point {
// (0, 0) is by convention the point at infinity, which can't be represented
// in affine coordinates, but is (0, 0, 0) in Jacobian.
if x.Sign() == 0 && y.Sign() == 0 {
return &p521Point{
x: new(fiat.P521Element),
y: new(fiat.P521Element),
z: new(fiat.P521Element),
}
}
return &p521Point{
x: bigIntToFiatP521(x),
y: bigIntToFiatP521(y),
z: new(fiat.P521Element).One(),
}
}
func (curve p521Curve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
p1 := curve.jacobianFromAffine(x1, y1)
p2 := curve.jacobianFromAffine(x2, y2)
return curve.affineFromJacobian(p1.addJacobian(p1, p2))
}
// addJacobian sets q = p1 + p2, and returns q. The points may overlap.
func (q *p521Point) addJacobian(p1, p2 *p521Point) *p521Point {
// https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl
z1IsZero := p1.z.IsZero()
z2IsZero := p2.z.IsZero()
z1z1 := new(fiat.P521Element).Square(p1.z)
z2z2 := new(fiat.P521Element).Square(p2.z)
u1 := new(fiat.P521Element).Mul(p1.x, z2z2)
u2 := new(fiat.P521Element).Mul(p2.x, z1z1)
h := new(fiat.P521Element).Sub(u2, u1)
xEqual := h.IsZero() == 1
i := new(fiat.P521Element).Add(h, h)
i.Square(i)
j := new(fiat.P521Element).Mul(h, i)
s1 := new(fiat.P521Element).Mul(p1.y, p2.z)
s1.Mul(s1, z2z2)
s2 := new(fiat.P521Element).Mul(p2.y, p1.z)
s2.Mul(s2, z1z1)
r := new(fiat.P521Element).Sub(s2, s1)
yEqual := r.IsZero() == 1
if xEqual && yEqual && z1IsZero == 0 && z2IsZero == 0 {
return q.doubleJacobian(p1)
}
r.Add(r, r)
v := new(fiat.P521Element).Mul(u1, i)
x := new(fiat.P521Element).Set(r)
x.Square(x)
x.Sub(x, j)
x.Sub(x, v)
x.Sub(x, v)
y := new(fiat.P521Element).Set(r)
v.Sub(v, x)
y.Mul(y, v)
s1.Mul(s1, j)
s1.Add(s1, s1)
y.Sub(y, s1)
z := new(fiat.P521Element).Add(p1.z, p2.z)
z.Square(z)
z.Sub(z, z1z1)
z.Sub(z, z2z2)
z.Mul(z, h)
x.Select(p2.x, x, z1IsZero)
x.Select(p1.x, x, z2IsZero)
y.Select(p2.y, y, z1IsZero)
y.Select(p1.y, y, z2IsZero)
z.Select(p2.z, z, z1IsZero)
z.Select(p1.z, z, z2IsZero)
q.x.Set(x)
q.y.Set(y)
q.z.Set(z)
return q
}
func (curve p521Curve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
p := curve.jacobianFromAffine(x1, y1)
return curve.affineFromJacobian(p.doubleJacobian(p))
}
// doubleJacobian sets q = p + p, and returns q. The points may overlap.
func (q *p521Point) doubleJacobian(p *p521Point) *p521Point {
// https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
delta := new(fiat.P521Element).Square(p.z)
gamma := new(fiat.P521Element).Square(p.y)
alpha := new(fiat.P521Element).Sub(p.x, delta)
alpha2 := new(fiat.P521Element).Add(p.x, delta)
alpha.Mul(alpha, alpha2)
alpha2.Set(alpha)
alpha.Add(alpha, alpha)
alpha.Add(alpha, alpha2)
beta := alpha2.Mul(p.x, gamma)
q.x.Square(alpha)
beta8 := new(fiat.P521Element).Add(beta, beta)
beta8.Add(beta8, beta8)
beta8.Add(beta8, beta8)
q.x.Sub(q.x, beta8)
q.z.Add(p.y, p.z)
q.z.Square(q.z)
q.z.Sub(q.z, gamma)
q.z.Sub(q.z, delta)
beta.Add(beta, beta)
beta.Add(beta, beta)
beta.Sub(beta, q.x)
q.y.Mul(alpha, beta)
gamma.Square(gamma)
gamma.Add(gamma, gamma)
gamma.Add(gamma, gamma)
gamma.Add(gamma, gamma)
q.y.Sub(q.y, gamma)
return q
}
func (curve p521Curve) ScalarMult(Bx, By *big.Int, scalar []byte) (*big.Int, *big.Int) {
B := curve.jacobianFromAffine(Bx, By)
p, t := &p521Point{
x: new(fiat.P521Element),
y: new(fiat.P521Element),
z: new(fiat.P521Element),
}, &p521Point{
x: new(fiat.P521Element),
y: new(fiat.P521Element),
z: new(fiat.P521Element),
}
for _, byte := range scalar {
for bitNum := 0; bitNum < 8; bitNum++ {
p.doubleJacobian(p)
bit := (byte >> (7 - bitNum)) & 1
t.addJacobian(p, B)
p.x.Select(t.x, p.x, int(bit))
p.y.Select(t.y, p.y, int(bit))
p.z.Select(t.z, p.z, int(bit))
}
}
return curve.affineFromJacobian(p)
}
func (curve p521Curve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
return curve.ScalarMult(curve.Gx, curve.Gy, k)
}
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