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quadratic_residue.rs
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/
quadratic_residue.rs
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use super::modpow::*;
type U = u128;
/// Legendre symbol.
fn legendre(a: U, p: U) -> i128 {
let l = modpow(a, (p - 1) / 2, p);
if l == p - 1 {
-1
} else {
l as i128
}
}
/// Tonelli–Shanks algorithm.
/// calc x where x^2 = a mod p
#[allow(clippy::mut_range_bound)]
pub fn modsqrt(a: U, p: U) -> U {
let l = legendre(a, p);
if l != 1 {
return l as u128;
}
let mut s = 0;
let mut q = p - 1;
while (q & 1) == 0 {
s += 1;
q >>= 1;
}
let mut z = 1;
while legendre(z, p) == 1 {
z += 1;
}
let mut m = s;
let mut c = modpow(z, q, p);
let mut t = modpow(a, q, p);
let mut r = modpow(a, (q + 1) / 2, p);
while t != 1 {
for i in 1..m {
if modpow(t, 1 << i, p) != 1 {
continue;
}
let b = modpow(c, 1 << (m - i - 1), p);
m = i;
c = b * b % p;
t = t * b % p * b % p;
r = r * b % p;
break;
}
}
r
}
#[test]
fn test() {
assert_eq!(modsqrt(5, 41), 28);
}