On the Computation of Square Roots in Finite Fields Article DOI:
Cite this article as: Müller, S. Designs, Codes and Cryptography (2004) 31: 301. doi:10.1023/B:DESI.0000015890.44831.e2 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 , when q 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. finite fields square roots efficient computation complexity References
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