Linearly Self-Equivalent APN Permutations in Small Dimension

Abstract

All almost perfect nonlinear (APN) permutations that we know to date admit a special kind of linear self-equivalence, i.e., there exists a permutation G in their CCZ-equivalence class and two linear permutations A and B, such that G °A = B °G. After providing a survey on the known APN functions with a focus on the existence of self-equivalences, we search for APN permutations in dimension 6, 7, and 8 that admit such a linear self-equivalence. In dimension six, we were able to conduct an exhaustive search and obtain that there is only one such APN permutation up to CCZ-equivalence. In dimensions 7 and 8, we performed an exhaustive search for all but a few classes of linear self-equivalences and we did not find any new APN permutation. As one interesting result in dimension 7, we obtain that all APN permutation polynomials with coefficients in \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> must be (up to CCZ-equivalence) monomial functions.