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

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}}^{*}$,and(ii)permutationtrinomialsoftheform$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.