Nonlinear functions and difference sets on group actions

TL;DR

This paper explores the properties, characterizations, and constructions of nonlinear functions and difference sets within finite group actions, extending existing theories and providing new insights into their structures.

Contribution

It introduces the concept of a $(G, H)$-related difference family and characterizes $G$-perfect nonlinear functions and difference sets using Fourier analysis, advancing the theoretical framework.

Findings

Characterization of $G$-perfect nonlinear functions via difference families.

Fourier transform characterization of $G$-difference sets when $G$ is abelian.

Existence and construction methods for $G$-perfect nonlinear and $G$-bent functions.

Abstract

Let $G$, $H$ be finite groups and let $X$ be a finite $G$-set. $G$-perfect nonlinear functions from $X$ to $H$ have been studied in several papers. They have more interesting properties than perfect nonlinear functions from $G$ itself to $H$. By introducing the concept of a $(G, H)$-related difference family of $X$, we obtain a characterization of $G$-perfect nonlinear functions on $X$. When $G$ is abelian, we characterize a $G$-difference set of $X$ by the Fourier transform on a normalized $G$-dual set $\widehat X$. We will also investigate the existence and constructions of $G$-perfect nonlinear functions and $G$-bent functions. Several known results in [2,6,10,17] are direct consequences of our results.