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

Abstract

In this work, we completely characterize (i) permutation binomials of the form $x^{\frac{2^n-1}{2^t-1}+1}+ax\in {\mathbb{F}}_{2^n}[x],n=2^{s}t,a\in {\mathbb{F}}_{2^{2t}}^{\ast }$, 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.

16 pages