A conjecture on permutation trinomials over finite fields of characteristic two

TL;DR

This paper proves a recent conjecture about permutation trinomials over finite fields of characteristic two by analyzing quadratic factors of an 11th degree polynomial, settling a specific case where n=2m and gcd(m,5)=1.

Contribution

It provides a proof for a conjecture on permutation trinomials over finite fields of characteristic two, specifically for the case where n=2m and gcd(m,5)=1.

Findings

Conjecture on permutation trinomials is confirmed for specified parameters.

Analysis of quadratic factors is key to the proof.

Results contribute to understanding permutation polynomials over finite fields.

Abstract

In this paper, by analyzing the quadratic factors of an $11$-th degree polynomial over the finite field $\ftwon$, a conjecture on permutation trinomials over $\ftwon[x]$ proposed very recently by Deng and Zheng is settled, where $n=2m$ and $m$ is a positive integer with $\gcd(m,5)=1$.