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

We study the parameters of bent and hyper-bent (HB) functions in n variables over a field $$ P = \mathbb{F}_q $$ with q = 2ℓ elements, ℓ > 1. Any such function is identified with a function F: Q → P, where $$ P < Q = \mathbb{F}_{qn} $$ . The latter has a reduced trace representation F = tr (Φ), where Φ(x) is a uniquely defined polynomial of a special type. It is shown that the most accurate generalization of results on parameters of bent functions from the case ℓ = 1 to the case ℓ > 1 is obtained if instead of the nonlinearity degree of a function one considers its binary nonlinearity index (in the case ℓ = 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.