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

We study the parameters of bent and hyper-bent (HB) functions in n variables over a field $ P = \mathbb{F}_q We study the parameters of bent and hyper-bent (HB) functions in n variables over a field with q = 2^ℓ elements, ℓ > 1. Any such function is identified with a function F: Q → P, where . The latter has a reduced trace representation F = tr _P ^Q (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. Original Russian Text ? A.S. Kuz’min, V.T. Markov, A.A. Nechaev, V.A. Shishkin, A.B. Shishkov, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 1, pp. 15–37. Supported in part by the Russian Foundation for Basic Research, project nos. 05-01-01018 and 05-01-01048, and the President of the Russian Federation Council for State Support of Leading Scientific Schools, project nos. NSh-8564.2006.10 and NSh-5666.2006.1. A part of the results were obtained in the course of research in the Cryptography Academy of the Russian Federation.