On Shanks' Algorithm for Modular Square Roots

TL;DR

This paper presents two improved modifications of Shanks' algorithm for computing modular square roots, reducing the number of multiplications and enabling parallel computation for faster results.

Contribution

The paper introduces two novel modifications of Shanks' algorithm, improving efficiency and parallelizability for computing modular square roots.

Findings

First modification reduces multiplications to O(log q + n^{3/2})

Second parallel algorithm achieves O(log q + n) time with n processors

Enhances computational efficiency for modular square root calculations

Abstract

Let $p$ be a prime number, $p=2^nq+1$, where $q$ is odd. D. Shanks described an algorithm to compute square roots $\pmod{p}$ which needs $O(\log q + n^2)$ modular multiplications. In this note we describe two modifications of this algorithm. The first needs only $O(\log q + n^{3/2})$ modular multiplications, while the second is a parallel algorithm which needs $n$ processors and takes $O(\log q+n)$ time.