On Some Permutation Binomials of the Form $x^{\frac{2^n-1}{k}+1} +ax$ over $\mathbb{F}_{2^n}$ : Existence and Count

Based on a criterion of permutation polynomials of the form $x^rf(x^{\frac{q-1}{m}})$ by Wan and Lidl (1991) and some very elementary techniques we show existence of permutation binomials of the following forms 1 $x(x^{\frac{2^n-1}{3}}+a) \in \mathbb{F}_{2^n}[x]$, for n>4 2 $x^{\frac{2^{2n}-1}{2^{n}-1} + 1}+ax = x^{2^n+2} + ax \in \mathbb{F}_{2^{2n}}[x]$, for n≥3. In (i), we extend a result of Carlitz (1962) for even characteristic. Moreover we present the count of such permutation binomials when a is in a certain subfield of $\mathbb{F}_{2^n}$. In (ii), we reprove, using much simpler technique, a recent result of Charpin and Kyureghyan (2008) and give the number of permutation binomials of this form. Finally, we discuss some cryptographic relevance of these results.