Solution to a problem of S. Payne
Xiang‐dong Hou · Proceedings of the American Mathematical Society · 2003
A problem posed by S. Payne calls for determination of all linearized polynomials f ( x ) ∈ F 2 n [ x ] f(x)\in \mathbb {F}_{2^n}[x] such that f ( x ) f(x) and f ( x ) / x f(x)/x are permutations of F 2 n \mathbb {F}_{2^n} and F 2 n ∗ \mathbb {F}_{2^n}^* respectively. We show that such polynomials are exactly of the form f ( x ) = a x 2 k f(x)=ax^{2^k} with a ∈ F 2 n ∗ a\in \mathbb {F}_{2^n}^* and ( k , n ) = 1 (k,n)=1 . In fact, we solve a q