// 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 1 forward-map from representative to Edwards curve point.
// Currently a straightforward, unoptimized implementation.
// See section 3.2 of the Elligator paper.
func (el *el1param) HideDecode(P point, rep []byte) {
ec := el.ec
var t, u, u2, v, Chiv, X, Y, x, y, t1, t2 nist.Int
l := ec.PointLen()
if len(rep) != l {
panic("el1Map: wrong representative length")
}
// Take the appropriate number of bits from the representative.
b := make([]byte, l)
copy(b, rep)
b[0] &^= el.padmask() // mask off the padding bits
t.InitBytes(b, &ec.P)
// u = (1-t)/(1+t)
u.Div(t1.Sub(&ec.one, &t), t2.Add(&ec.one, &t))
// v = u^5 + (r^2-2)u^3 + u
u2.Mul(&u, &u) // u2 = u^2
v.Mul(&u2, &u2) // v = u^4
v.Add(&v, t1.Mul(&el.r2m2, &u2)) // v = u^4 + (r^2-2)u^2
v.Add(&v, &ec.one).Mul(&v, &u) // v = u^5 + (r^2-2)u^3 + u
// X = Chi(v)u
chi(&Chiv, &v)
X.Mul(&Chiv, &u)
// Y = (Chi(v)v)^((q+1)/4) Chi(v) Chi(u^2+1/c^2)
t1.Add(&u2, &el.invc2)
chi(&t1, &t1) // t1 = Chi(u^2+1/c^2)
Y.Mul(&Chiv, &v)
Y.Exp(&Y, &el.pp1d4).Mul(&Y, &Chiv).Mul(&Y, &t1)
// x = (c-1)sX(1+X)/Y
x.Add(&ec.one, &X).Mul(&X, &x).Mul(&el.cm1s, &x).Div(&x, &Y)
// y = (rX-(1+X)^2)/(rX+(1+X)^2)
t1.Mul(&el.r, &X) // t1 = rX
t2.Add(&ec.one, &X).Mul(&t2, &t2) // t2 = (1+X)^2
y.Div(u.Sub(&t1, &t2), v.Add(&t1, &t2))
// Sanity-check
if !ec.onCurve(&x, &y) {
panic("elligator1 produced invalid point")
}
P.initXY(&x.V, &y.V, ec.self)
}
// 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)
}