Decomposing Generalized Bent and Hyperbent Functions

Abstract

In this paper, we introduce generalized hyperbent functions from $ {\mathbb F}_{2^{n}}$ to $ {\mathbb Z}_{2^{k}}$ , and investigate decompositions of generalized (hyper)bent functions. We show that generalized (hyper)bent functions $f$ from $ {\mathbb F}_{2^{n}}$ to $ {\mathbb Z}_{2^{k}}$ consist of components which are generalized (hyper)bent functions from $ {\mathbb F}_{2^{n}}$ to $ {\mathbb Z}_{2^{k^\prime }}$ for some $k^\prime . For even $n$ , most notably we show that the g-hyperbentness of $f$ is equivalent to the hyperbentness of the components of $f$ with some conditions on the Walsh–Hadamard coefficients. For odd $n$ , we show that the Boolean functions associated to a generalized bent function form an affine space of semibent functions. This complements a recent result for even $n$ , where the associated Boolean functions are bent.