Permutation polynomials of degree 8 over finite fields of characteristic 2

Abstract

Up to linear transformations, we obtain a classification of permutation polynomials (PPs) of degree 8 over F2r with r>3. By Bartoli et al. (2017) [1], a polynomial f of degree 8 over F2r 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 F2r, 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 F2r (with r>3) exists if and only if r∈{4,5,6}, and such f is explicitly listed up to linear transformations.