// Test if a supposed point is on the curve,
// by checking the characteristic equation for Edwards curves:
//
// a*x^2 + y^2 = 1 + d*x^2*y^2
//
func (c *curve) onCurve(x, y *nist.Int) bool {
var xx, yy, l, r nist.Int
xx.Mul(x, x) // xx = x^2
yy.Mul(y, y) // yy = y^2
l.Mul(&c.a, &xx).Add(&l, &yy) // l = a*x^2 + y^2
r.Mul(&c.d, &xx).Mul(&r, &yy).Add(&c.one, &r)
// r = 1 + d*x^2*y^2
return l.Equal(&r)
}
// Elligator 1 reverse-map from point to uniform representative.
// Returns nil if point has no uniform representative.
// See section 3.3 of the Elligator paper.
func (el *el1param) HideEncode(P point, rand cipher.Stream) []byte {
ec := el.ec
x, y := P.getXY()
var a, b, etar, etarp1, X, z, u, t, t1 nist.Int
// condition 1: a = y+1 is nonzero
a.Add(y, &ec.one)
if a.V.Sign() == 0 {
return nil // y+1 = 0, no representative
}
// etar = r(y-1)/2(y+1)
t1.Add(y, &ec.one).Add(&t1, &t1) // 2(y+1)
etar.Sub(y, &ec.one).Mul(&etar, &el.r).Div(&etar, &t1)
// condition 2: b = (1 + eta r)^2 - 1 is a square
etarp1.Add(&ec.one, &etar) // etarp1 = (1 + eta r)
b.Mul(&etarp1, &etarp1).Sub(&b, &ec.one)
if math.Jacobi(&b.V, b.M) < 0 {
return nil // b not a square, no representative
}
// condition 3: if etar = -2 then x=2s(c-1)Chi(c)/r
if etar.Equal(&el.m2) && !x.Equal(&el.c3x) {
return nil
}
// X = -(1+eta r)+((1+eta r)^2-1)^((q+1)/4)
X.Exp(&b, &el.pp1d4).Sub(&X, &etarp1)
// z = Chi((c-1)sX(1+X)x(X^2+1/c^2))
z.Mul(&el.cm1s, &X).Mul(&z, t.Add(&ec.one, &X)).Mul(&z, x)
z.Mul(&z, t.Mul(&X, &X).Add(&t, &el.invc2))
chi(&z, &z)
// u = zX
u.Mul(&z, &X)
// t = (1-u)/(1+u)
t.Div(a.Sub(&ec.one, &u), b.Add(&ec.one, &u))
// Map representative to a byte-string by padding the upper byte.
// This assumes that the prime c.P is close enough to a power of 2
// that the adversary will never notice the "missing" values;
// this is true for the class of curves Elligator1 was designed for.
rep, _ := t.MarshalBinary()
padmask := el.padmask()
if padmask != 0 {
var pad [1]byte
rand.XORKeyStream(pad[:], pad[:])
rep[0] |= pad[0] & padmask
}
return rep
}
// Elligator 2 reverse-map from point to uniform representative.
// Returns nil if point has no uniform representative.
// See section 5.3 of the Elligator paper.
func (el *el2param) HideEncode(P point, rand cipher.Stream) []byte {
edx, edy := P.getXY()
var x, y, r, xpA, t1 nist.Int
// convert Edwards to Montgomery coordinates
el.ed2mont(&x, &y, edx, edy)
// condition 1: x != -A
if x.Equal(&el.negA) {
return nil // x = -A, no representative
}
// condition 2: if y=0, then x=0
if y.V.Sign() == 0 && x.V.Sign() != 0 {
return nil // y=0 but x!=0, no representative
}
// condition 3: -ux(x+A) is a square
xpA.Add(&x, &el.A)
t1.Mul(&el.u, &x).Mul(&t1, &xpA).Neg(&t1)
if math.Jacobi(&t1.V, t1.M) < 0 {
return nil // not a square, no representative
}
if y.V.Cmp(&el.pm1d2) <= 0 { // y in image of sqrt function
r.Mul(&xpA, &el.u).Div(&x, &r)
} else { // y not in image of sqrt function
r.Mul(&el.u, &x).Div(&xpA, &r)
}
r.Neg(&r)
el.sqrt(&r, &r)
// Sanity check on result
if r.V.Cmp(&el.pm1d2) > 0 {
panic("el2: r too big")
}
// Map representative to a byte-string by padding the upper byte.
// This assumes that the prime c.P is close enough to a power of 2
// that the adversary will never notice the "missing" values;
// this is true for the class of curves Elligator1 was designed for.
rep, _ := r.MarshalBinary()
padmask := el.padmask()
if padmask != 0 {
var pad [1]byte
rand.XORKeyStream(pad[:], pad[:])
rep[0] |= pad[0] & padmask
}
return rep
}
// Compute the square root function,
// specified in section 5.5 of the Elligator paper.
func (el *el2param) sqrt(r, a *nist.Int) {
var b, b2 nist.Int
b.Exp(a, &el.pp3d8) // b = a^((p+3)/8); b in {a,-a}
b2.Mul(&b, &b) // b^2 = a?
if !b2.Equal(a) {
b.Mul(&b, &el.sqrtm1) // b*sqrt(-1)
}
if b.V.Cmp(&el.pm1d2) > 0 { // |b|
b.Neg(&b)
}
r.Set(&b)
}
