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Computing the number of finite field elements with prescribed absolute trace and co-trace

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Abstract

In this paper, we address the problem of determining the number of finite field elements with prescribed absolute trace and co-trace for arbitrary characteristic p. We show that this problem can be converted to solving a system of p − 1 linear equations with matrix of coefficients a slight modification of circulant matrix formed by the Kloosterman sums over the field \(\mathbb {F}_{p}\). Also, the asymptotic behavior of this number has been studied utilizing André Weil bound on Kloosterman sums over finite fields. The proposed approach is illustrated for characteristic p ≤ 5.

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Authors and Affiliations

  1. Faculty of Mathematics and Informatics, Sofia University, J. Baucher Str. 2, Sofia, Bulgaria

    Assen Bojilov & Lyubomir Borissov

  2. Institute of Mathematics and Informatics, BAS, G. Bontchev Str. 8, 1113, Sofia, Bulgaria

    Lyubomir Borissov & Yuri Borissov

Authors
  1. Assen Bojilov
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  2. Lyubomir Borissov
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  3. Yuri Borissov
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Corresponding author

Correspondence to Yuri Borissov.

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Dedicated to Prof. Tor Helleseth’s 70th Birthday

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This article is part of the Topical Collection on Special Issue: Mathematical Methods for Cryptography

Appendix

Appendix

Table 1 Values of Tij for 2 ≤ m ≤ 10, char = 2
Full size table
Table 2 Values of Tij for 1 ≤ m ≤ 6, char = 3
Full size table
Table 3 Values of Tij for 1 ≤ m ≤ 6, char = 5
Full size table

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Bojilov, A., Borissov, L. & Borissov, Y. Computing the number of finite field elements with prescribed absolute trace and co-trace. Cryptogr. Commun. 11, 497–507 (2019). https://doi.org/10.1007/s12095-018-0336-z

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  • DOI: https://doi.org/10.1007/s12095-018-0336-z

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