# Linearly Self-Equivalent APN Permutations in Small Dimension

Christof Beierle, Marcus Brinkmann, Gregor Leander

IEEE Transactions on Information Theory

## 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 G⃘A = B⃘G. 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 F2 must be (up to CCZ-equivalence) monomial functions.

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