An algorithm for finding square root modulo p

TL;DR

This paper introduces a new algorithm for computing square roots modulo odd primes, offering practical efficiency improvements over Tonelli-Shanks and requiring less restrictive conditions for non-residues.

Contribution

The paper presents a novel algorithm that improves practical performance for finding square roots modulo all odd primes, with easier non-residue requirements.

Findings

Improved practical computation time over Tonelli-Shanks

Works with relative non-residues, which are easier to find

Maintains asymptotic runtime similar to existing methods

Abstract

We propose a novel algorithm for finding square roots modulo p. Although there exists a direct formula to calculate square root of an element modulo prime (3 mod 4), but calculating square root modulo prime (1 mod 4) is non trivial. Tonelli-Shanks algorithm remains the most widely used and probably the fastest when averaged over all primes [19]. This paper proposes a new algorithm for finding square roots modulo all odd primes, which shows improvement over existing method in practical terms although asymptotically gives the same run time as Tonelli-Shanks. Apart from practically efficient computation time, the proposed method does not necessarily require availability of non-residue and can work with `relative non-residue' also. Such `relative non-residues' are much easier to find ( probability 2/3) compared to non-residues ( probability 1/2).