Find $x$ when $x^2\equiv y\bmod p$

Question

Given $p$ is prime and $y$ is a constant. What's the fastest possible way to find $x$ where $x^2\equiv y\bmod p$?

Example: $x^2\equiv97\bmod101$ would give us $x=81$ as one of the solutions. What's the fastest way to compute any one of the solutions of $x$?

Constraints:

Note: $10^2\equiv-1\bmod101$ and $2^2\equiv4\bmod101$, so $(\pm20)^2\equiv-4\equiv97\bmod101$ — J. W. Tanner, Dec 06 '20 at 17:51

score 2 · Accepted Answer · answered Dec 06 '20 at 17:50

2

The standard approach to compute square roots modulo primes is the Tonelli–Shanks algorithm.

answered Dec 06 '20 at 17:50

Parcly Taxel

Thank you for your comment I'll look into it! – Nishant Aklecha Dec 07 '20 at 03:42

1 Answers1