Bent and hyper-bent functions over a field of 2 l elements

We study the parameters of bent and hyper-bent (HB) functions in n variables over a field $$ P = \mathbb{F}_q $$ with q = 2? elements, ? > 1. Any such function is identified with a function F: Q ? P, where $$ P 1 is obtained if instead of the nonlinearity degree of a function one considers its binary nonlinearity index (in the case ? = 1 these parameters coincide). We construct a class of HB functions that generalize binary HB functions found in [1]; we indicate a set of parameters q and n for which there are no other HB functions. We introduce the notion of the period of a function and establish a relation between periods of (hyper-)bent functions and their frequency characteristics.