Decomposing generalized bent and hyperbent functions

TL;DR

This paper introduces generalized hyperbent functions from finite fields to cyclic groups, explores their decompositions, and reveals structural properties relating to bent and semibent functions for different n.

Contribution

It defines generalized hyperbent functions, analyzes their decompositions, and characterizes associated Boolean functions for odd and even n, extending existing results.

Findings

Generalized hyperbent functions decompose into components with smaller codomain groups.

For odd n, associated Boolean functions form an affine space of semibent functions.

For even n, associated Boolean functions are bent.

Abstract

In this paper we introduce generalized hyperbent functions from $F_{2^n}$ to $Z_{2^k}$, and investigate decompositions of generalized (hyper)bent functions. We show that generalized (hyper)bent functions from $F_{2^n}$ to $Z_{2^k}$ consist of components which are generalized (hyper)bent functions from $F_{2^n}$ to $Z_{2^{k^\prime}}$ for some $k^\prime < k$. 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.