On the Equivalence of Quadratic APN Functions
Bracken, Carl and Byrne, Eimear and McGuire, Gary and Nebe, Gabriele (2011) On the Equivalence of Quadratic APN Functions. Designs, Codes and Cryptography, 61 (3). pp. 261-272. ISSN 0925-1022
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.
Repository Staff Only: item control page