On some permutation binomials and trinomials over $$mathbb {F}_{2^n}$$

In this work, we completely characterize (1) 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 (2) permutation trinomials of the form (x^{2^s+1}+x^{2^{s-1}+1}+alpha x in mathbb {F}_{2^t}[x]), 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.