Full Characterization of Generalized Bent Functions as (Semi)-Bent Spaces, Their Dual, and the Gray Image

Abstract

A natural generalization of bent functions is a class of functions from ${\mathbb F}_{2}^{n}$ to ${\mathbb Z}_{2^{k}}$ which is known as generalized bent (gbent) functions. The construction and characterization of gbent functions are commonly described in terms of the Walsh transforms of the associated Boolean functions. Using similar approach, we first determine the dual of a gbent function when $n$ is even. Then, depending on the parity of $n$ , it is shown that the Gray image of a gbent function is $(k-1)$ or $(k-2)$ plateaued, which generalizes previous results for $k$ = 2,3, and 4. We then completely characterize gbent functions as algebraic objects. More precisely, again depending on the parity of $n$ , a gbent function is a $(k-1)$ -dimensional affine space of bent functions or semi-bent functions with certain interesting additional properties, which we completely describe. Finally, we also consider a subclass of functions from ${\mathbb F}_{2}^{n}$ to ${\mathbb Z}_{2^{k}}$ , called ${\mathbb Z}_{q}$ -bent functions (which are necessarily gbent), which essentially gives rise to relative difference sets similarly to standard bent functions. Two examples of this class of functions are provided and it is demonstrated that many gbent functions are not ${\mathbb Z}_{q}$ -bent.