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On the Computation of Square Roots in Finite Fields

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Abstract

In this paper, two improvements for computing square roots in finite fields are presented. Firstly, we give a simple extension of a method by O. Atkin, which requires two exponentiations in FM q , when q≡9 mod 16. Our second method gives a major improvement to the Cipolla–Lehmer algorithm, which is both easier to implement and also much faster. While our method is independent of the power of 2 in q−1, its expected running time is equivalent to 1.33 as many multiplications as exponentiation via square and multiply. Several numerical examples are given that show the speed-up of the proposed methods, compared to the routines employed by Mathematica, Maple, respectively Magma.

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  1. Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W, Calgary, AB, T2N 1N4, Canada

    Siguna Müller

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Müller, S. On the Computation of Square Roots in Finite Fields. Designs, Codes and Cryptography 31, 301–312 (2004). https://doi.org/10.1023/B:DESI.0000015890.44831.e2

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