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Gauss periods as constructions of low complexity normal bases

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Abstract

Optimal normal bases are special cases of the so-called Gauss periods (Disquisitiones Arithmeticae, Articles 343–366); in particular, optimal normal bases are Gauss periods of type (n, 1) for any characteristic and of type (n, 2) for characteristic 2. We present the multiplication tables and complexities of Gauss periods of type (n, t) for all n and t = 3, 4, 5 over any finite field and give a slightly weaker result for Gauss periods of type (n, 6). In addition, we give some general results on the so-called cyclotomic numbers, which are intimately related to the structure of Gauss periods. We also present the general form of a normal basis obtained by the trace of any normal basis in a finite extension field. Then, as an application of the trace construction, we give upper bounds on the complexity of the trace of a Gauss period of type (n, 3).

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Authors and Affiliations

  1. Department of Mathematics, University of Crete, Knoussou Ave., 71409, Heraklion, Crete, Greece

    M. Christopoulou & T. Garefalakis

  2. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Dr., Ottawa, ON, K1S 5B6, Canada

    D. Panario & D. Thomson

Authors
  1. M. Christopoulou
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  2. T. Garefalakis
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  3. D. Panario
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  4. D. Thomson
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Correspondence to D. Thomson.

Additional information

Communicated by D. Hachenberger.

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Christopoulou, M., Garefalakis, T., Panario, D. et al. Gauss periods as constructions of low complexity normal bases. Des. Codes Cryptogr. 62, 43–62 (2012). https://doi.org/10.1007/s10623-011-9490-4

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  • DOI: https://doi.org/10.1007/s10623-011-9490-4

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