|
|
|
@ -79,52 +79,52 @@ type BitCurve struct { |
|
|
|
|
BitSize int // the size of the underlying field
|
|
|
|
|
} |
|
|
|
|
|
|
|
|
|
func (BitCurve *BitCurve) Params() *elliptic.CurveParams { |
|
|
|
|
func (bitCurve *BitCurve) Params() *elliptic.CurveParams { |
|
|
|
|
return &elliptic.CurveParams{ |
|
|
|
|
P: BitCurve.P, |
|
|
|
|
N: BitCurve.N, |
|
|
|
|
B: BitCurve.B, |
|
|
|
|
Gx: BitCurve.Gx, |
|
|
|
|
Gy: BitCurve.Gy, |
|
|
|
|
BitSize: BitCurve.BitSize, |
|
|
|
|
P: bitCurve.P, |
|
|
|
|
N: bitCurve.N, |
|
|
|
|
B: bitCurve.B, |
|
|
|
|
Gx: bitCurve.Gx, |
|
|
|
|
Gy: bitCurve.Gy, |
|
|
|
|
BitSize: bitCurve.BitSize, |
|
|
|
|
} |
|
|
|
|
} |
|
|
|
|
|
|
|
|
|
// IsOnCurve returns true if the given (x,y) lies on the BitCurve.
|
|
|
|
|
func (BitCurve *BitCurve) IsOnCurve(x, y *big.Int) bool { |
|
|
|
|
func (bitCurve *BitCurve) IsOnCurve(x, y *big.Int) bool { |
|
|
|
|
// y² = x³ + b
|
|
|
|
|
y2 := new(big.Int).Mul(y, y) //y²
|
|
|
|
|
y2.Mod(y2, BitCurve.P) //y²%P
|
|
|
|
|
y2.Mod(y2, bitCurve.P) //y²%P
|
|
|
|
|
|
|
|
|
|
x3 := new(big.Int).Mul(x, x) //x²
|
|
|
|
|
x3.Mul(x3, x) //x³
|
|
|
|
|
|
|
|
|
|
x3.Add(x3, BitCurve.B) //x³+B
|
|
|
|
|
x3.Mod(x3, BitCurve.P) //(x³+B)%P
|
|
|
|
|
x3.Add(x3, bitCurve.B) //x³+B
|
|
|
|
|
x3.Mod(x3, bitCurve.P) //(x³+B)%P
|
|
|
|
|
|
|
|
|
|
return x3.Cmp(y2) == 0 |
|
|
|
|
} |
|
|
|
|
|
|
|
|
|
// affineFromJacobian reverses the Jacobian transform. See the comment at the
|
|
|
|
|
// top of the file.
|
|
|
|
|
func (BitCurve *BitCurve) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) { |
|
|
|
|
func (bitCurve *BitCurve) 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, BitCurve.P) |
|
|
|
|
zinv := new(big.Int).ModInverse(z, bitCurve.P) |
|
|
|
|
zinvsq := new(big.Int).Mul(zinv, zinv) |
|
|
|
|
|
|
|
|
|
xOut = new(big.Int).Mul(x, zinvsq) |
|
|
|
|
xOut.Mod(xOut, BitCurve.P) |
|
|
|
|
xOut.Mod(xOut, bitCurve.P) |
|
|
|
|
zinvsq.Mul(zinvsq, zinv) |
|
|
|
|
yOut = new(big.Int).Mul(y, zinvsq) |
|
|
|
|
yOut.Mod(yOut, BitCurve.P) |
|
|
|
|
yOut.Mod(yOut, bitCurve.P) |
|
|
|
|
return |
|
|
|
|
} |
|
|
|
|
|
|
|
|
|
// Add returns the sum of (x1,y1) and (x2,y2)
|
|
|
|
|
func (BitCurve *BitCurve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) { |
|
|
|
|
func (bitCurve *BitCurve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) { |
|
|
|
|
// If one point is at infinity, return the other point.
|
|
|
|
|
// Adding the point at infinity to any point will preserve the other point.
|
|
|
|
|
if x1.Sign() == 0 && y1.Sign() == 0 { |
|
|
|
@ -135,27 +135,27 @@ func (BitCurve *BitCurve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) { |
|
|
|
|
} |
|
|
|
|
z := new(big.Int).SetInt64(1) |
|
|
|
|
if x1.Cmp(x2) == 0 && y1.Cmp(y2) == 0 { |
|
|
|
|
return BitCurve.affineFromJacobian(BitCurve.doubleJacobian(x1, y1, z)) |
|
|
|
|
return bitCurve.affineFromJacobian(bitCurve.doubleJacobian(x1, y1, z)) |
|
|
|
|
} |
|
|
|
|
return BitCurve.affineFromJacobian(BitCurve.addJacobian(x1, y1, z, x2, y2, z)) |
|
|
|
|
return bitCurve.affineFromJacobian(bitCurve.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 (BitCurve *BitCurve) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) { |
|
|
|
|
func (bitCurve *BitCurve) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) { |
|
|
|
|
// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
|
|
|
|
|
z1z1 := new(big.Int).Mul(z1, z1) |
|
|
|
|
z1z1.Mod(z1z1, BitCurve.P) |
|
|
|
|
z1z1.Mod(z1z1, bitCurve.P) |
|
|
|
|
z2z2 := new(big.Int).Mul(z2, z2) |
|
|
|
|
z2z2.Mod(z2z2, BitCurve.P) |
|
|
|
|
z2z2.Mod(z2z2, bitCurve.P) |
|
|
|
|
|
|
|
|
|
u1 := new(big.Int).Mul(x1, z2z2) |
|
|
|
|
u1.Mod(u1, BitCurve.P) |
|
|
|
|
u1.Mod(u1, bitCurve.P) |
|
|
|
|
u2 := new(big.Int).Mul(x2, z1z1) |
|
|
|
|
u2.Mod(u2, BitCurve.P) |
|
|
|
|
u2.Mod(u2, bitCurve.P) |
|
|
|
|
h := new(big.Int).Sub(u2, u1) |
|
|
|
|
if h.Sign() == -1 { |
|
|
|
|
h.Add(h, BitCurve.P) |
|
|
|
|
h.Add(h, bitCurve.P) |
|
|
|
|
} |
|
|
|
|
i := new(big.Int).Lsh(h, 1) |
|
|
|
|
i.Mul(i, i) |
|
|
|
@ -163,13 +163,13 @@ func (BitCurve *BitCurve) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int |
|
|
|
|
|
|
|
|
|
s1 := new(big.Int).Mul(y1, z2) |
|
|
|
|
s1.Mul(s1, z2z2) |
|
|
|
|
s1.Mod(s1, BitCurve.P) |
|
|
|
|
s1.Mod(s1, bitCurve.P) |
|
|
|
|
s2 := new(big.Int).Mul(y2, z1) |
|
|
|
|
s2.Mul(s2, z1z1) |
|
|
|
|
s2.Mod(s2, BitCurve.P) |
|
|
|
|
s2.Mod(s2, bitCurve.P) |
|
|
|
|
r := new(big.Int).Sub(s2, s1) |
|
|
|
|
if r.Sign() == -1 { |
|
|
|
|
r.Add(r, BitCurve.P) |
|
|
|
|
r.Add(r, bitCurve.P) |
|
|
|
|
} |
|
|
|
|
r.Lsh(r, 1) |
|
|
|
|
v := new(big.Int).Mul(u1, i) |
|
|
|
@ -179,7 +179,7 @@ func (BitCurve *BitCurve) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int |
|
|
|
|
x3.Sub(x3, j) |
|
|
|
|
x3.Sub(x3, v) |
|
|
|
|
x3.Sub(x3, v) |
|
|
|
|
x3.Mod(x3, BitCurve.P) |
|
|
|
|
x3.Mod(x3, bitCurve.P) |
|
|
|
|
|
|
|
|
|
y3 := new(big.Int).Set(r) |
|
|
|
|
v.Sub(v, x3) |
|
|
|
@ -187,33 +187,33 @@ func (BitCurve *BitCurve) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int |
|
|
|
|
s1.Mul(s1, j) |
|
|
|
|
s1.Lsh(s1, 1) |
|
|
|
|
y3.Sub(y3, s1) |
|
|
|
|
y3.Mod(y3, BitCurve.P) |
|
|
|
|
y3.Mod(y3, bitCurve.P) |
|
|
|
|
|
|
|
|
|
z3 := new(big.Int).Add(z1, z2) |
|
|
|
|
z3.Mul(z3, z3) |
|
|
|
|
z3.Sub(z3, z1z1) |
|
|
|
|
if z3.Sign() == -1 { |
|
|
|
|
z3.Add(z3, BitCurve.P) |
|
|
|
|
z3.Add(z3, bitCurve.P) |
|
|
|
|
} |
|
|
|
|
z3.Sub(z3, z2z2) |
|
|
|
|
if z3.Sign() == -1 { |
|
|
|
|
z3.Add(z3, BitCurve.P) |
|
|
|
|
z3.Add(z3, bitCurve.P) |
|
|
|
|
} |
|
|
|
|
z3.Mul(z3, h) |
|
|
|
|
z3.Mod(z3, BitCurve.P) |
|
|
|
|
z3.Mod(z3, bitCurve.P) |
|
|
|
|
|
|
|
|
|
return x3, y3, z3 |
|
|
|
|
} |
|
|
|
|
|
|
|
|
|
// Double returns 2*(x,y)
|
|
|
|
|
func (BitCurve *BitCurve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) { |
|
|
|
|
func (bitCurve *BitCurve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) { |
|
|
|
|
z1 := new(big.Int).SetInt64(1) |
|
|
|
|
return BitCurve.affineFromJacobian(BitCurve.doubleJacobian(x1, y1, z1)) |
|
|
|
|
return bitCurve.affineFromJacobian(bitCurve.doubleJacobian(x1, y1, z1)) |
|
|
|
|
} |
|
|
|
|
|
|
|
|
|
// doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and
|
|
|
|
|
// returns its double, also in Jacobian form.
|
|
|
|
|
func (BitCurve *BitCurve) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) { |
|
|
|
|
func (bitCurve *BitCurve) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) { |
|
|
|
|
// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
|
|
|
|
|
|
|
|
|
|
a := new(big.Int).Mul(x, x) //X1²
|
|
|
|
@ -231,30 +231,30 @@ func (BitCurve *BitCurve) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, |
|
|
|
|
|
|
|
|
|
x3 := new(big.Int).Mul(big.NewInt(2), d) //2*D
|
|
|
|
|
x3.Sub(f, x3) //F-2*D
|
|
|
|
|
x3.Mod(x3, BitCurve.P) |
|
|
|
|
x3.Mod(x3, bitCurve.P) |
|
|
|
|
|
|
|
|
|
y3 := new(big.Int).Sub(d, x3) //D-X3
|
|
|
|
|
y3.Mul(e, y3) //E*(D-X3)
|
|
|
|
|
y3.Sub(y3, new(big.Int).Mul(big.NewInt(8), c)) //E*(D-X3)-8*C
|
|
|
|
|
y3.Mod(y3, BitCurve.P) |
|
|
|
|
y3.Mod(y3, bitCurve.P) |
|
|
|
|
|
|
|
|
|
z3 := new(big.Int).Mul(y, z) //Y1*Z1
|
|
|
|
|
z3.Mul(big.NewInt(2), z3) //3*Y1*Z1
|
|
|
|
|
z3.Mod(z3, BitCurve.P) |
|
|
|
|
z3.Mod(z3, bitCurve.P) |
|
|
|
|
|
|
|
|
|
return x3, y3, z3 |
|
|
|
|
} |
|
|
|
|
|
|
|
|
|
// ScalarBaseMult returns k*G, where G is the base point of the group and k is
|
|
|
|
|
// an integer in big-endian form.
|
|
|
|
|
func (BitCurve *BitCurve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) { |
|
|
|
|
return BitCurve.ScalarMult(BitCurve.Gx, BitCurve.Gy, k) |
|
|
|
|
func (bitCurve *BitCurve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) { |
|
|
|
|
return bitCurve.ScalarMult(bitCurve.Gx, bitCurve.Gy, k) |
|
|
|
|
} |
|
|
|
|
|
|
|
|
|
// Marshal converts a point into the form specified in section 4.3.6 of ANSI
|
|
|
|
|
// X9.62.
|
|
|
|
|
func (BitCurve *BitCurve) Marshal(x, y *big.Int) []byte { |
|
|
|
|
byteLen := (BitCurve.BitSize + 7) >> 3 |
|
|
|
|
func (bitCurve *BitCurve) Marshal(x, y *big.Int) []byte { |
|
|
|
|
byteLen := (bitCurve.BitSize + 7) >> 3 |
|
|
|
|
ret := make([]byte, 1+2*byteLen) |
|
|
|
|
ret[0] = 4 // uncompressed point flag
|
|
|
|
|
readBits(x, ret[1:1+byteLen]) |
|
|
|
@ -264,8 +264,8 @@ func (BitCurve *BitCurve) Marshal(x, y *big.Int) []byte { |
|
|
|
|
|
|
|
|
|
// Unmarshal converts a point, serialised by Marshal, into an x, y pair. On
|
|
|
|
|
// error, x = nil.
|
|
|
|
|
func (BitCurve *BitCurve) Unmarshal(data []byte) (x, y *big.Int) { |
|
|
|
|
byteLen := (BitCurve.BitSize + 7) >> 3 |
|
|
|
|
func (bitCurve *BitCurve) Unmarshal(data []byte) (x, y *big.Int) { |
|
|
|
|
byteLen := (bitCurve.BitSize + 7) >> 3 |
|
|
|
|
if len(data) != 1+2*byteLen { |
|
|
|
|
return |
|
|
|
|
} |
|
|
|
|