Permutation trinomials over $\mathbb{F}_{2^m}$: a corrected version

TL;DR

This paper revises and completes the classification of permutation trinomials over finite fields of characteristic two, confirming a conjecture, discovering new classes, and analyzing their equivalences.

Contribution

It corrects and extends previous classifications of permutation trinomials over _{2^m}, proving a conjecture and identifying new permutation polynomials.

Findings

Confirmed all permutation trinomials over _{2^m} in Zieve's work.

Proved a conjecture by Gupta and Sharma.

Discovered new permutation trinomials and their equivalences.

Abstract

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over $\mathbb{F}_{2^m}$ in Zieve's paper. We prove a conjecture proposed by Gupta and Sharma and obtain some new permutation trinomials over $\mathbb{F}_{2^m}$. Finally, we show that some classes of permutation trinomials with parameters are QM equivalent to some known permutation trinomials.