Relative Difference sets from Almost Perfect Nonlinear Functions

TL;DR

This paper investigates the relationship between Almost Perfect Nonlinear (APN) functions and relative difference sets, revealing that specific APN functions' image sets form relative difference sets and linking APN functions to bent functions.

Contribution

It establishes a novel connection between APN functions and relative difference sets, expanding understanding of their algebraic and combinatorial properties.

Findings

Image sets of certain 2-to-1 APN functions form relative difference sets

Connects APN functions with bent functions via Pott's result

Provides new insights into the structure of APN functions

Abstract

In this paper we explore a connection between certain Almost Perfect Nonlinear Functions (APN functions) and relative difference sets. In particular, we show that the image set of certain 2-to-1 APN functions is a relative difference set. Through a result of Pott this further provides a connection between APN functions and bent functions.