COMPUTING SQUARE ROOTS FASTER THAN THE TONELLI-SHANKS/BERNSTEIN ALGORITHM

Article Type

Research Article

Publication Title

Advances in Mathematics of Communications

Abstract

Let p be a prime such that p = 1+2n m, where n ≥ 1 and m is odd. Given a square u in Zp and a non-square z in Zp, we describe an algorithm to compute a square root of u which requires T + O(n3/2) operations (i.e., squarings and multiplications), where T is the number of operations required to exponentiate an element of Zp to the power (m−1)/2. This improves upon the Tonelli-Shanks (TS) algorithm which requires T + O(n2) operations. Bernstein had proposed a table look-up based variant of the TS algorithm which requires T + O((n/w)2) operations and O(2w n/w) storage, where w is a parameter. A table look-up variant of the new algorithm requires T+O((n/w)3/2) operations and the same storage. In concrete terms, the new algorithm is shown to require significantly fewer operations for particular values of n.

First Page

141

Last Page

162

DOI

10.3934/amc.2022007

Publication Date

2-1-2024

Comments

Open Access; Bronze Open Access

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