Nilpotent linearized polynomials over finite fields and applications

Let q be a prime power and F-qn be the finite field with q(n) elements, where n > 1. We introduce the class of the linearized polynomials L(X) over F-qn such that L(t) (X) := LoLo ... oL(X) (t times) equivalent to 0 (mod X-qn - X) for some t >= 2, called nilpotent linearized polynomials (NLP's). We discuss the existence and construction of NLP's and, as an application, we show how to obtain permutations of F-qn. from these polynomials. For some of those permutations, we can explicitly give the compositional inverse map and the cycle decomposition. This paper also contains a method for constructing involutions over binary fields with no fixed points, which are useful in block ciphers. (C) 2017 Elsevier Inc. All rights reserved.