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package goldilocks
import (
"errors"
"fmt"
fp "github.com/cloudflare/circl/math/fp448"
)
// Point is a point on the Goldilocks Curve.
type Point struct{ x, y, z, ta, tb fp.Elt }
func (P Point) String() string {
return fmt.Sprintf("x: %v\ny: %v\nz: %v\nta: %v\ntb: %v", P.x, P.y, P.z, P.ta, P.tb)
}
// FromAffine creates a point from affine coordinates.
func FromAffine(x, y *fp.Elt) (*Point, error) {
P := &Point{
x: *x,
y: *y,
z: fp.One(),
ta: *x,
tb: *y,
}
if !(Curve{}).IsOnCurve(P) {
return P, errors.New("point not on curve")
}
return P, nil
}
// isLessThan returns true if 0 <= x < y, and assumes that slices are of the
// same length and are interpreted in little-endian order.
func isLessThan(x, y []byte) bool {
i := len(x) - 1
for i > 0 && x[i] == y[i] {
i--
}
return x[i] < y[i]
}
// FromBytes returns a point from the input buffer.
func FromBytes(in []byte) (*Point, error) {
if len(in) < fp.Size+1 {
return nil, errors.New("wrong input length")
}
var err = errors.New("invalid decoding")
P := &Point{}
signX := in[fp.Size] >> 7
copy(P.y[:], in[:fp.Size])
p := fp.P()
if !isLessThan(P.y[:], p[:]) {
return nil, err
}
u, v := &fp.Elt{}, &fp.Elt{}
one := fp.One()
fp.Sqr(u, &P.y) // u = y^2
fp.Mul(v, u, ¶mD) // v = dy^2
fp.Sub(u, u, &one) // u = y^2-1
fp.Sub(v, v, &one) // v = dy^2-1
isQR := fp.InvSqrt(&P.x, u, v) // x = sqrt(u/v)
if !isQR {
return nil, err
}
fp.Modp(&P.x) // x = x mod p
if fp.IsZero(&P.x) && signX == 1 {
return nil, err
}
if signX != (P.x[0] & 1) {
fp.Neg(&P.x, &P.x)
}
P.ta = P.x
P.tb = P.y
P.z = fp.One()
return P, nil
}
// IsIdentity returns true is P is the identity Point.
func (P *Point) IsIdentity() bool {
return fp.IsZero(&P.x) && !fp.IsZero(&P.y) && !fp.IsZero(&P.z) && P.y == P.z
}
// IsEqual returns true if P is equivalent to Q.
func (P *Point) IsEqual(Q *Point) bool {
l, r := &fp.Elt{}, &fp.Elt{}
fp.Mul(l, &P.x, &Q.z)
fp.Mul(r, &Q.x, &P.z)
fp.Sub(l, l, r)
b := fp.IsZero(l)
fp.Mul(l, &P.y, &Q.z)
fp.Mul(r, &Q.y, &P.z)
fp.Sub(l, l, r)
b = b && fp.IsZero(l)
fp.Mul(l, &P.ta, &P.tb)
fp.Mul(l, l, &Q.z)
fp.Mul(r, &Q.ta, &Q.tb)
fp.Mul(r, r, &P.z)
fp.Sub(l, l, r)
b = b && fp.IsZero(l)
return b
}
// Neg obtains the inverse of the Point.
func (P *Point) Neg() { fp.Neg(&P.x, &P.x); fp.Neg(&P.ta, &P.ta) }
// ToAffine returns the x,y affine coordinates of P.
func (P *Point) ToAffine() (x, y fp.Elt) {
fp.Inv(&P.z, &P.z) // 1/z
fp.Mul(&P.x, &P.x, &P.z) // x/z
fp.Mul(&P.y, &P.y, &P.z) // y/z
fp.Modp(&P.x)
fp.Modp(&P.y)
fp.SetOne(&P.z)
P.ta = P.x
P.tb = P.y
return P.x, P.y
}
// ToBytes stores P into a slice of bytes.
func (P *Point) ToBytes(out []byte) error {
if len(out) < fp.Size+1 {
return errors.New("invalid decoding")
}
x, y := P.ToAffine()
out[fp.Size] = (x[0] & 1) << 7
return fp.ToBytes(out[:fp.Size], &y)
}
// MarshalBinary encodes the receiver into a binary form and returns the result.
func (P *Point) MarshalBinary() (data []byte, err error) {
data = make([]byte, fp.Size+1)
err = P.ToBytes(data[:fp.Size+1])
return data, err
}
// UnmarshalBinary must be able to decode the form generated by MarshalBinary.
func (P *Point) UnmarshalBinary(data []byte) error { Q, err := FromBytes(data); *P = *Q; return err }
// Double sets P = 2Q.
func (P *Point) Double() { P.Add(P) }
// Add sets P =P+Q..
func (P *Point) Add(Q *Point) {
// This is formula (5) from "Twisted Edwards Curves Revisited" by
// Hisil H., Wong K.KH., Carter G., Dawson E. (2008)
// https://doi.org/10.1007/978-3-540-89255-7_20
x1, y1, z1, ta1, tb1 := &P.x, &P.y, &P.z, &P.ta, &P.tb
x2, y2, z2, ta2, tb2 := &Q.x, &Q.y, &Q.z, &Q.ta, &Q.tb
x3, y3, z3, E, H := &P.x, &P.y, &P.z, &P.ta, &P.tb
A, B, C, D := &fp.Elt{}, &fp.Elt{}, &fp.Elt{}, &fp.Elt{}
t1, t2, F, G := C, D, &fp.Elt{}, &fp.Elt{}
fp.Mul(t1, ta1, tb1) // t1 = ta1*tb1
fp.Mul(t2, ta2, tb2) // t2 = ta2*tb2
fp.Mul(A, x1, x2) // A = x1*x2
fp.Mul(B, y1, y2) // B = y1*y2
fp.Mul(C, t1, t2) // t1*t2
fp.Mul(C, C, ¶mD) // C = d*t1*t2
fp.Mul(D, z1, z2) // D = z1*z2
fp.Add(F, x1, y1) // x1+y1
fp.Add(E, x2, y2) // x2+y2
fp.Mul(E, E, F) // (x1+y1)*(x2+y2)
fp.Sub(E, E, A) // (x1+y1)*(x2+y2)-A
fp.Sub(E, E, B) // E = (x1+y1)*(x2+y2)-A-B
fp.Sub(F, D, C) // F = D-C
fp.Add(G, D, C) // G = D+C
fp.Sub(H, B, A) // H = B-A
fp.Mul(z3, F, G) // Z = F * G
fp.Mul(x3, E, F) // X = E * F
fp.Mul(y3, G, H) // Y = G * H, T = E * H
}
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