On CCZ-inequivalence of some families of almost perfect nonlinear functions to permutations

Abstract

Browning et al. (2010) exhibited almost perfect nonlinear (APN) permutations on $\mathbb {F}_{2^{6}}$ . This was the first example of an APN permutation on an even degree extension of $\mathbb {F}_{2}$ . In their approach of finding an APN permutation, Browning et al. made use of a necessary and sufficient condition based on the Walsh transform. In this paper, we give an algorithm based on a related necessary condition which checks whether a vectorial Boolean function is CCZ-inequivalent to a permutation. Using this algorithm, we are able to show that no function belonging to a known family of APN functions is equivalent to a permutation on $\mathbb {F}_{2^{2m}}$ , where m ≤ 6 (except for the known case on $\mathbb {F}_{2^{6}}$ ). We also give an EA-invariant based on the condition. Finally, we give a theoretical proof of the fact that no member of a specific family of APN functions is equivalent to a permutation on doubly-even degree extensions of $\mathbb {F}_{2}$ .