Bracken, Carl and Byrne, Eimear and McGuire, Gary and Nebe, Gabriele
On the Equivalence of Quadratic APN Functions.
Designs, Codes and Cryptography, 61 (3).
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 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.
||Preprint of published article. The original publication is available at www.springerlink.com. Research supported by the Claude Shannon Institute, Science Foundation Ireland Grant
06/MI/006 and the Irish Research Council for Science, Engineering and Technology
||almost perfect nonlinear; APN; automorphism group; CCZ-equivalence; EA-equivalence; Gold function;
||Science & Engineering > Mathematics
||01 Sep 2011 11:27
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||Designs, Codes and Cryptography
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