// 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)
}
// Elligator 2 forward-map from representative to Edwards curve point.
// Currently a straightforward, unoptimized implementation.
// See section 5.2 of the Elligator paper.
func (el *el2param) HideDecode(P point, rep []byte) {
ec := el.ec
var r, v, x, y, t1, edx, edy nist.Int
l := ec.PointLen()
if len(rep) != l {
panic("el2Map: wrong representative length")
}
// Take the appropriate number of bits from the representative.
buf := make([]byte, l)
copy(buf, rep)
buf[0] &^= el.padmask() // mask off the padding bits
r.InitBytes(buf, &ec.P)
// v = -A/(1+ur^2)
v.Mul(&r, &r).Mul(&el.u, &v).Add(&ec.one, &v).Div(&el.negA, &v)
// e = Chi(v^3+Av^2+Bv), where B=1 because of ed2mont equivalence
t1.Add(&v, &el.A).Mul(&t1, &v).Add(&t1, &ec.one).Mul(&t1, &v)
e := math.Jacobi(&t1.V, t1.M)
// x = ev-(1-e)A/2
if e == 1 {
x.Set(&v)
} else {
x.Add(&v, &el.A).Neg(&x)
}
// y = -e sqrt(x^3+Ax^2+Bx), where B=1
y.Add(&x, &el.A).Mul(&y, &x).Add(&y, &ec.one).Mul(&y, &x)
el.sqrt(&y, &y)
if e == 1 {
y.Neg(&y) // -e factor
}
// Convert Montgomery to Edwards coordinates
el.mont2ed(&edx, &edy, &x, &y)
// Sanity-check
if !ec.onCurve(&edx, &edy) {
panic("elligator2 produced invalid point")
}
P.initXY(&edx.V, &edy.V, ec.self)
}
