Linearly Self-Equivalent APN Permutations in Small Dimension

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