On Shanks' Algorithm for Modular Square Roots

Details On Shanks' Algorithm for Modular Square Roots On Shanks' Algorithm for Modular Square Roots

2011-05-07

arxiv.org/abs/1105.1456v1

Appl. Math. E-Notes 5 (2005), 84--88

Mathematics

Number Theory

Scientific paper

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.

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