Nilpotent linearized polynomials over finite fields and applications

Let q be a prime power and Fqn be the finite field with qn elements, where n>1. We introduce the class of the linearized polynomials L(X) over Fqn such thatL(t)(X):=L∘L∘⋯∘L︸ttimes(X)≡0(modXqn−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 Fqn 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.