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

Authors:
A. S. Kuz'Min;V. T. Markov;A. A. Nechaev;V. A. Shishkin;A. B. Shishkov
Affiliations:
Moscow State Institute of Radio Engineering, Electronics and Automation (Technical University), Moscow, Russia;Lomonosov Moscow State University, Moscow, Russia;Lomonosov Moscow State University, Moscow, Russia;Moscow State Institute of Radio Engineering, Electronics and Automation (Technical University), Moscow, Russia;Moscow State Institute of Radio Engineering, Electronics and Automation (Technical University), Moscow, Russia
Venue:
Problems of Information Transmission
Year:
2008

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Abstract

We study the parameters of bent and hyper-bent (HB) functions in n variables over a field $$ P = \mathbb{F}_q $$ with q = 2驴 elements, 驴 1. Any such function is identified with a function F: Q 驴 P, where $$ P . The latter has a reduced trace representation F = tr P Q (驴), where 驴(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 驴 = 1 to the case 驴 1 is obtained if instead of the nonlinearity degree of a function one considers its binary nonlinearity index (in the case 驴 = 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.