Fourier Transforms and Bent Functions on Finite Abelian Group-Acted Sets

TL;DR

This paper generalizes Fourier analysis and bent functions from finite abelian groups to sets acted upon by such groups, introducing a new dual set and characterizing bent functions in this broader context.

Contribution

It introduces the $G$-dual set and Fourier transforms on $X$, unifies the treatment of bent functions on groups and $G$-acted sets, and relates bentness to distance from $G$-linear functions.

Findings

Characterization of bent functions via Fourier transforms on $\, ilde{X}$

Unified framework for bent functions on groups and $G$-acted sets

Bentness determined by distance from $G$-linear functions

Abstract

Let $G$ be a finite abelian group acting faithfully on a finite set $X$. As a natural generalization of the perfect nonlinearity of Boolean functions, the $G$-bentness and $G$-perfect nonlinearity of functions on $X$ are studied by Poinsot et al. [6,7] via Fourier transforms of functions on $G$. In this paper we introduce the so-called $G$-dual set $\widehat X$ of $X$, which plays the role similar to the dual group $\widehat G$ of $G$, and the Fourier transforms of functions on $X$, a generalization of the Fourier transforms of functions on finite abelian groups. Then we characterize the bent functions on $X$ in terms of their own Fourier transforms on $\widehat X$. Bent (perfect nonlinear) functions on finite abelian groups and $G$-bent ($G$-perfect nonlinear) functions on $X$ are treated in a uniform way in this paper, and many known results in [4,2,6,7] are obtained as direct consequences. Furthermore, we will prove that the bentness of a function on $X$ can be determined by its distance from the set of $G$-linear functions. In order to explain the main results clearly, examples are also presented.