On the equivalence of quadratic APN functions |
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Authors: | Carl Bracken Eimear Byrne Gary McGuire Gabriele Nebe |
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Institution: | (1) School of Mathematical Sciences, University College Dublin, Dublin, Ireland; |
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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 x2r+1{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 2
n
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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Keywords: | |
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