Solution to a problem of S. Payne

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