Designs, Codes and Cryptography

, Volume 31, Issue 3, pp 301–312

On the Computation of Square Roots in Finite Fields

  • Siguna Müller

DOI: 10.1023/B:DESI.0000015890.44831.e2

Cite this article as:
Müller, S. Designs, Codes and Cryptography (2004) 31: 301. doi:10.1023/B:DESI.0000015890.44831.e2


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 FMq, 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.

finite fieldssquare rootsefficient computationcomplexity

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Siguna Müller
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada