Created
September 28, 2013 19:33
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A simple implementation of the Tonelli-Shanks algorithm to compute a square root in Z/pZ where p is prime. It could probably be made quite faster by using a faster pow_mod function instead of the recursive one and also by trying to avoid some of the modulus calculations.
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long pow_mod(long x, long n, long p) { | |
if (n == 0) return 1; | |
if (n & 1) | |
return (pow_mod(x, n-1, p) * x) % p; | |
x = pow_mod(x, n/2, p); | |
return (x * x) % p; | |
} | |
/* Takes as input an odd prime p and n < p and returns r | |
* such that r * r = n [mod p]. */ | |
long tonelli_shanks(long n, long p) { | |
long s = 0; | |
long q = p - 1; | |
while ((q & 1) == 0) { q /= 2; ++s; } | |
if (s == 1) { | |
long r = pow_mod(n, (p+1)/4, p); | |
if ((r * r) % p == n) return r; | |
return 0; | |
} | |
// Find the first quadratic non-residue z by brute-force search | |
long z = 1; | |
while (pow_mod(++z, (p-1)/2, p) != p - 1); | |
long c = pow_mod(z, q, p); | |
long r = pow_mod(n, (q+1)/2, p); | |
long t = pow_mod(n, q, p); | |
long m = s; | |
while (t != 1) { | |
long tt = t; | |
long i = 0; | |
while (tt != 1) { | |
tt = (tt * tt) % p; | |
++i; | |
if (i == m) return 0; | |
} | |
long b = pow_mod(c, pow_mod(2, m-i-1, p-1), p); | |
long b2 = (b * b) % p; | |
r = (r * b) % p; | |
t = (t * b2) % p; | |
c = b2; | |
m = i; | |
} | |
if ((r * r) % p == n) return r; | |
return 0; | |
} |
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Thanks!
I rewrited this to js BigInt -> https://github.com/gkucmierz/utils/blob/main/src/tonelli-shanks.mjs#L8