/// <summary>
/// Impplements Cipolla's algorithm
/// </summary>
/// <param name="N"></param>
/// <param name="p"></param>
/// <returns></returns>
public static BigInteger ModSqrt(BigInteger N, BigInteger p)
{
BigInteger a = 0,
w2 = 0;
BigInteger ls = 0;
// Pick up any value 'a' such that 'w^2 = a^2 - N' is a non-quadratic
// residue in Fp
do
{
++a;
w2 = (a * a + p - N) % p;
ls = LS(w2, p);
} while (ls != p - 1);
// In Fp^2, compute '(a + w)^((p + 1)/2) (mod p)'
var r = new Fp2Point(1, 0);
var s = new Fp2Point(a, 1);
for (var n = (p + 1) / 2 % p; n > 0; n >>= 1)
{
if (!n.IsEven)
{
r = Fp2Point.Mul(r, s, p, w2);
}
s = Fp2Point.Mul(s, s, p, w2);
}
// Check for errors
if (r.x * r.x % p != N)
{
return(0);
}
if (r.y != 0)
{
return(0);
}
return(r.x);
} // ModSqrt
internal static Fp2Point Mul(Fp2Point a, Fp2Point b,
BigInteger p, BigInteger w2)
{
return(new Fp2Point((a.x * b.x + a.y * b.y * w2) % p,
(a.x * b.y + b.x * a.y) % p));
}
