On Some Permutation Binomials and Trinomials Over $\mathbb{F}_{2^n}$

TL;DR

This paper fully characterizes certain permutation binomials and trinomials over finite fields of characteristic two, providing new insights into their structure and conditions for permutation properties.

Contribution

It offers a complete characterization of permutation binomials and trinomials over _{2^n}, with the binomial case derived from the trinomial results, advancing understanding of permutation polynomials.

Findings

Characterization of permutation binomials of a specific form over _{2^n}

Characterization of permutation trinomials of a specific form over _{2^t}

The binomial characterization follows from the trinomial results

Abstract

In this work, we completely characterize (i) permutation binomials of the form $x^{{{2^n -1}\over {2^t-1}}+1}+ ax \in \mathbb{F}_{2^n}[x], n = 2^st, a \in \mathbb{F}_{2^{2t}}^{*}$, and (ii) permutation trinomials of the form $x^{2^s+1}+x^{2^{s-1}+1}+\alpha x \in \mathbb{F}_{2^t}[x]$, \end{enumerate} where $s,t$ are positive integers. The first result, which was our primary motivation, is a consequence of the second result. The second result may be of independent interest.