Proof of a Conjecture on Permutation Polynomials over Finite Fields

TL;DR

This paper proves a conjecture regarding a specific permutation polynomial over finite fields and extends the result to a broader class of such polynomials.

Contribution

The paper confirms a conjecture about a particular permutation polynomial and provides a generalization to a wider family of permutation polynomials over finite fields.

Findings

Confirmed the conjecture that a specific polynomial is a permutation polynomial.

Extended the result to a more general class of permutation polynomials.

Provided a proof technique applicable to similar problems.

Abstract

Let $k$ be a positive integer and $S_{2k}={\tt x}+{\tt x}^4+...+{\tt x}^{4^{2k-1}}\in\Bbb F_2[{\tt x}]$. It was recently conjectured that ${\tt x}+S_{2k}^{4^{2k}}+S_{2k}^{4^k+3}$ is a permutation polynomial of $\Bbb F_{4^{3k}}$. In this note, the conjecture is confirmed and a generalization is obtained.