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 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.

Every Boolean function of n variables is identified with a function F : Q → P, where Q = GF(2n), P = GF(2). A. Youssef and G. Gong showed that for n = 2λ there exist functions F which have equally bad approximations not only by linear functions (that is, by functions tr (μx), where μ ∈ Q* and tr: Q → P is the trace function), but also by proper monomial functions (functions tr(μxδ), where (δ, 2n − 1) = 1). Such functions F were called hyper-bent functions (HB functions, HBF), and for any n = 2λ a non-empty class of HBF having the property F(0) = 0 was constructed. This class consists of the functions F(x) = such that the equation F(x) = 1 has exactly (2λ − 1)2λ−1 solutions in Q. In the present paper, we give some essential restrictions on the parameters of an arbitrary HBF showing that the class of HBF is far less than that of bent functions. In particular, we show that any HBF is a bent function having the degree of nonlinearity λ, and for some n (for instance, if λ > 2 and 2λ − 1 is prime, or λ ∈ {4,9,25,27}) the class of HBF is exhausted by the functions F(x) = described by A. Youssef and G. Gong. For n = 4, in addition to 10 HBF listed above there exist 18 more HBF with property F(0) = 0. The question of whether there exist other hyper-bent functions for n > 4 remains open.

AbstractThe minimal polynomials of some decimations from geometric progressions and linear recurring sequences of maximal period over a finite field of characteristic 2 are studied. The step of these decimations increases exponentially.