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-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 explicitly 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 exhaustively searched through parts of the space and conclude that the linear self equivalences of such APN permutations must be of a special form. As one interesting result in dimension 7, we obtain that all APN permutation polynomials with coefficients in F 2 must be (up to CCZequivalence) monomial functions.