Construction methods for generalized bent functions

Abstract: Generalized bent (gbent) functions is a class of functions $f: \mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$, where $q \geq 2$ is a positive integer, that generalizes a concept of classical bent functions through their co-domain extension. A lot of research has recently been devoted towards derivation of the necessary and sufficient conditions when $f$ is represented as a collection of Boolean functions. Nevertheless, apart from the necessary conditions that these component functions are bent when $n$ is even (respectively semi-bent when $n$ is odd), no general construction method has been proposed yet for $n$ odd case. In this article, based on the use of the well-known Maiorana-McFarland (MM) class of functions, we give an explicit construction method of gbent functions, for any even $q >2$ when $n$ is even and for any $q$ of the form $q=2^r$ (for $r>1$) when $n$ is odd. Thus, a long-term open problem of providing a general construction method of gbent functions, for odd $n$, has been solved. The method for odd $n$ employs a large class of disjoint spectra semi-bent functions with certain additional properties which may be useful in other cryptographic applications.