# Permutation polynomials of degree 8 over finite fields of characteristic 2

## TL;DR

This paper classifies permutation polynomials of degree 8 over finite fields of characteristic 2, identifying when non-exceptional cases exist and providing explicit examples for certain field sizes.

## Contribution

It provides a complete classification of degree 8 permutation polynomials over finite fields of characteristic 2, including explicit forms for non-exceptional cases.

## Key findings

- Non-exceptional degree 8 PPs exist only for r in {4,5,6}
- Explicit forms of these PPs are listed
- Classification is achieved using SageMath computations

## Abstract

Up to linear transformations, we obtain a classification of permutation polynomials (PPs) of degree $8$ over $\mathbb{F}_{2^r}$ with $r>3$. By [J. Number Theory 176 (2017) 466-66], a polynomial $f$ of degree $8$ over $\mathbb{F}_{2^r}$ is exceptional if and only if $f-f(0)$ is a linearized PP. So it suffices to search for non-exceptional PPs of degree $8$ over $\mathbb{F}_{2^r}$, which exist only when $r\leqslant9$ 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 $\mathbb{F}_{2^r}$ (with $r>3$) exists if and only if $r\in\{4,5,6\}$, and such $f$ is explicitly listed up to linear transformations.