Permutation polynomials of degree 8 over finite fields of characteristic 2

Up to linear transformations, we obtain a classification of permutation polynomials (PPs) of degree 8 over F-2r with r > 3. By Bartoli et al. (2017) [1], a polynomial f of degree 8 over F-2r is exceptional if and only if f - f (0) is a linearized PP, which has already been classified. So it suffices to search for non-exceptional PPs of degree 8 over F-2r, which exist only when r <= 9 by a previous result. This can be exhausted by the SageMath software running on a personal computer. To facilitate the computation, some requirements after linear transformations and explicit equations by Hermite's criterion are provided for the polynomial coefficients. The main result is that a non-exceptional PP f of degree 8 over F-2r (with r > 3) exists if and only if r is an element of {4, 5, 6}, and such f is explicitly listed up to linear transformations. (C) 2020 Elsevier Inc. All rights reserved.