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// Copyright 2021 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 "math/big"
// CurveParams contains the parameters of an elliptic curve and also provides
// a generic, non-constant time implementation of Curve.
//
// The generic Curve implementation is deprecated, and using custom curves
// (those not returned by P224(), P256(), P384(), and P521()) is not guaranteed
// to provide any security property.
type CurveParams struct {
P *big.Int // the order of the underlying field
N *big.Int // the order of the base point
B *big.Int // the constant of the curve equation
Gx, Gy *big.Int // (x,y) of the base point
BitSize int // the size of the underlying field
Name string // the canonical name of the curve
}
func (curve *CurveParams) Params() *CurveParams {
return curve
}
// CurveParams 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.
// polynomial returns x³ - 3x + b.
func (curve *CurveParams) polynomial(x *big.Int) *big.Int {
x3 := new(big.Int).Mul(x, x)
x3.Mul(x3, x)
threeX := new(big.Int).Lsh(x, 1)
threeX.Add(threeX, x)
x3.Sub(x3, threeX)
x3.Add(x3, curve.B)
x3.Mod(x3, curve.P)
return x3
}
// IsOnCurve implements Curve.IsOnCurve.
//
// Deprecated: the CurveParams methods are deprecated and are not guaranteed to
// provide any security property. For ECDH, use the crypto/ecdh package.
// For ECDSA, use the crypto/ecdsa package with a Curve value returned directly
// from P224(), P256(), P384(), or P521().
func (curve *CurveParams) IsOnCurve(x, y *big.Int) bool {
// If there is a dedicated constant-time implementation for this curve operation,
// use that instead of the generic one.
if specific, ok := matchesSpecificCurve(curve); ok {
return specific.IsOnCurve(x, y)
}
if x.Sign() < 0 || x.Cmp(curve.P) >= 0 ||
y.Sign() < 0 || y.Cmp(curve.P) >= 0 {
return false
}
// y² = x³ - 3x + b
y2 := new(big.Int).Mul(y, y)
y2.Mod(y2, curve.P)
return curve.polynomial(x).Cmp(y2) == 0
}
// zForAffine returns a Jacobian Z value for the affine point (x, y). If x and
// y are zero, it assumes that they represent the point at infinity because (0,
// 0) is not on the any of the curves handled here.
func zForAffine(x, y *big.Int) *big.Int {
z := new(big.Int)
if x.Sign() != 0 || y.Sign() != 0 {
z.SetInt64(1)
}
return z
}
// affineFromJacobian reverses the Jacobian transform. See the comment at the
// top of the file. If the point is ∞ it returns 0, 0.
func (curve *CurveParams) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) {
if z.Sign() == 0 {
return new(big.Int), new(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 implements Curve.Add.
//
// Deprecated: the CurveParams methods are deprecated and are not guaranteed to
// provide any security property. For ECDH, use the crypto/ecdh package.
// For ECDSA, use the crypto/ecdsa package with a Curve value returned directly
// from P224(), P256(), P384(), or P521().
func (curve *CurveParams) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
// If there is a dedicated constant-time implementation for this curve operation,
// use that instead of the generic one.
if specific, ok := matchesSpecificCurve(curve); ok {
return specific.Add(x1, y1, x2, y2)
}
panicIfNotOnCurve(curve, x1, y1)
panicIfNotOnCurve(curve, x2, y2)
z1 := zForAffine(x1, y1)
z2 := zForAffine(x2, y2)
return curve.affineFromJacobian(curve.addJacobian(x1, y1, z1, x2, y2, z2))
}
// addJacobian takes two points in Jacobian coordinates, (x1, y1, z1) and
// (x2, y2, z2) and returns their sum, also in Jacobian form.
func (curve *CurveParams) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) {
// See https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl
x3, y3, z3 := new(big.Int), new(big.Int), new(big.Int)
if z1.Sign() == 0 {
x3.Set(x2)
y3.Set(y2)
z3.Set(z2)
return x3, y3, z3
}
if z2.Sign() == 0 {
x3.Set(x1)
y3.Set(y1)
z3.Set(z1)
return x3, y3, z3
}
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)
xEqual := h.Sign() == 0
if h.Sign() == -1 {
h.Add(h, 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)
}
yEqual := r.Sign() == 0
if xEqual && yEqual {
return curve.doubleJacobian(x1, y1, z1)
}
r.Lsh(r, 1)
v := new(big.Int).Mul(u1, i)
x3.Set(r)
x3.Mul(x3, x3)
x3.Sub(x3, j)
x3.Sub(x3, v)
x3.Sub(x3, v)
x3.Mod(x3, curve.P)
y3.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)
z3.Add(z1, z2)
z3.Mul(z3, z3)
z3.Sub(z3, z1z1)
z3.Sub(z3, z2z2)
z3.Mul(z3, h)
z3.Mod(z3, curve.P)
return x3, y3, z3
}
// Double implements Curve.Double.
//
// Deprecated: the CurveParams methods are deprecated and are not guaranteed to
// provide any security property. For ECDH, use the crypto/ecdh package.
// For ECDSA, use the crypto/ecdsa package with a Curve value returned directly
// from P224(), P256(), P384(), or P521().
func (curve *CurveParams) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
// If there is a dedicated constant-time implementation for this curve operation,
// use that instead of the generic one.
if specific, ok := matchesSpecificCurve(curve); ok {
return specific.Double(x1, y1)
}
panicIfNotOnCurve(curve, x1, y1)
z1 := zForAffine(x1, y1)
return curve.affineFromJacobian(curve.doubleJacobian(x1, y1, z1))
}
// doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and
// returns its double, also in Jacobian form.
func (curve *CurveParams) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) {
// See https://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)
beta := alpha2.Mul(x, gamma)
x3 := new(big.Int).Mul(alpha, alpha)
beta8 := new(big.Int).Lsh(beta, 3)
beta8.Mod(beta8, curve.P)
x3.Sub(x3, beta8)
if x3.Sign() == -1 {
x3.Add(x3, curve.P)
}
x3.Mod(x3, 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)
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 implements Curve.ScalarMult.
//
// Deprecated: the CurveParams methods are deprecated and are not guaranteed to
// provide any security property. For ECDH, use the crypto/ecdh package.
// For ECDSA, use the crypto/ecdsa package with a Curve value returned directly
// from P224(), P256(), P384(), or P521().
func (curve *CurveParams) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
// If there is a dedicated constant-time implementation for this curve operation,
// use that instead of the generic one.
if specific, ok := matchesSpecificCurve(curve); ok {
return specific.ScalarMult(Bx, By, k)
}
panicIfNotOnCurve(curve, Bx, By)
Bz := new(big.Int).SetInt64(1)
x, y, z := new(big.Int), new(big.Int), new(big.Int)
for _, byte := range k {
for bitNum := 0; bitNum < 8; bitNum++ {
x, y, z = curve.doubleJacobian(x, y, z)
if byte&0x80 == 0x80 {
x, y, z = curve.addJacobian(Bx, By, Bz, x, y, z)
}
byte <<= 1
}
}
return curve.affineFromJacobian(x, y, z)
}
// ScalarBaseMult implements Curve.ScalarBaseMult.
//
// Deprecated: the CurveParams methods are deprecated and are not guaranteed to
// provide any security property. For ECDH, use the crypto/ecdh package.
// For ECDSA, use the crypto/ecdsa package with a Curve value returned directly
// from P224(), P256(), P384(), or P521().
func (curve *CurveParams) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
// If there is a dedicated constant-time implementation for this curve operation,
// use that instead of the generic one.
if specific, ok := matchesSpecificCurve(curve); ok {
return specific.ScalarBaseMult(k)
}
return curve.ScalarMult(curve.Gx, curve.Gy, k)
}
func matchesSpecificCurve(params *CurveParams) (Curve, bool) {
for _, c := range []Curve{p224, p256, p384, p521} {
if params == c.Params() {
return c, true
}
}
return nil, false
}