A Generalization of APN Functions for Odd Characteristic

TL;DR

This paper introduces a modified definition of APN functions tailored for finite fields of odd characteristic, exploring their properties to extend cryptographic and geometric applications beyond characteristic two.

Contribution

It proposes a new framework for APN functions in odd characteristic fields and analyzes their properties, filling a gap in existing cryptographic research.

Findings

Modified APN definition for odd characteristic fields

Identification of key properties of the new APN functions

Potential applications in cryptography and finite geometry

Abstract

Almost perfect nonlinear (APN) functions on finite fields of characteristic two have been studied by many researchers. Such functions have useful properties and applications in cryptography, finite geometries and so on. However APN functions on finite fields of odd characteristic do not satisfy desired properties. In this paper, we modify the definition of APN function in the case of odd characteristic, and study its properties.