Solution to a problem of S. Payne

A problem posed by S. Payne calls for determination of all linearized polynomials $f(x)\in\mathbb{F} _{2^n}[x]$ such that $f(x)$and $f(x)/x$ are permutations of $\mathbb{F} _{2^n}$ and $\mathbb{F} _{2^n}^*$respectively. We show that such polynomials are exactly of the form $f(x)=ax^{2^k}$ with $a\in\mathbb{F} _{2^n}^*$ and $(k,n)=1$. In fact, we solve a $q$-ary version of Payne's problem.