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On the equivalence of quadratic APN functions

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Abstract

Establishing the CCZ-equivalence of a pair of APN functions is generally quite difficult. In some cases, when seeking to show that a putative new infinite family of APN functions is CCZ inequivalent to an already known family, we rely on computer calculation for small values of n. In this paper we present a method to prove the inequivalence of quadratic APN functions with the Gold functions. Our main result is that a quadratic function is CCZ-equivalent to the APN Gold function \({x^{2^r+1}}\) if and only if it is EA-equivalent to that Gold function. As an application of this result, we prove that a trinomial family of APN functions that exist on finite fields of order 2n where n ≡ 2 mod 4 are CCZ inequivalent to the Gold functions. The proof relies on some knowledge of the automorphism group of a code associated with such a function.

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Authors and Affiliations

  1. School of Mathematical Sciences, University College, Dublin, Ireland

    Carl Bracken, Eimear Byrne & Gary McGuire

  2. Lehrstuhl D für Mathematik, RWTH Aachen University, 52056, Aachen, Germany

    Gabriele Nebe

Authors
  1. Carl Bracken
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  2. Eimear Byrne
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  3. Gary McGuire
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  4. Gabriele Nebe
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Corresponding author

Correspondence to Eimear Byrne.

Additional information

Communicated by J. Bierbrauer.

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Bracken, C., Byrne, E., McGuire, G. et al. On the equivalence of quadratic APN functions. Des. Codes Cryptogr. 61, 261–272 (2011). https://doi.org/10.1007/s10623-010-9475-8

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  • DOI: https://doi.org/10.1007/s10623-010-9475-8

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