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\(\boldsymbol{x^{2^l+1}+x+a}\) and related affine polynomials over GF (2k)

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Abstract

In this paper, the polynomials \(P_a(x)=x^{2^l+1}+x+a\) with a ∈ GF(2k) are studied. New criteria for the number of zeros of P a (x) in GF(2k) are proved. We also study the affine polynomial \(a^{2^l}x^{2^{2l}}+x^{2^l}+ax+1\) which is closely related to P a (x). In many cases, explicit expressions for calculating zeros of these polynomials are provided.

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Acknowledgements

The authors would like to thank the anonymous reviewers for thorough reviews containing constructive comments and valuable suggestions that helped to improve the manuscript significantly.

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Authors and Affiliations

  1. The Selmer Center, Department of Informatics, University of Bergen, PB 7800, 5020, Bergen, Norway

    Tor Helleseth & Alexander Kholosha

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  1. Tor Helleseth
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  2. Alexander Kholosha
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Correspondence to Alexander Kholosha.

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This work was supported by the Norwegian Research Council and partially by the grant NIL-I-004 from Iceland, Liechtenstein and Norway through the EEA and Norwegian Financial Mechanisms. Preliminary version of this paper can be found in [7].

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Helleseth, T., Kholosha, A. \(\boldsymbol{x^{2^l+1}+x+a}\) and related affine polynomials over GF (2k). Cryptogr. Commun. 2, 85–109 (2010). https://doi.org/10.1007/s12095-009-0018-y

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  • DOI: https://doi.org/10.1007/s12095-009-0018-y

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