On $-1$-differential uniformity of ternary APN power functions

TL;DR

This paper investigates the $-1$-differential uniformity of ternary APN power functions over GF(3^n), identifying functions with low uniformity and near-perfect nonlinearity, contributing to cryptographic function design.

Contribution

It introduces the study of $-1$-differential uniformity for ternary APN power functions and finds functions with low uniformity and near-perfect nonlinearity.

Findings

Identified ternary power functions with low $-1$-differential uniformity.

Discovered some functions are almost perfect $-1$-nonlinear.

Extended understanding of differential properties in finite fields.

Abstract

Very recently, a new concept called multiplicative differential and the corresponding $c$-differential uniformity were introduced by Ellingsen et al. A function $F(x)$ over finite field $\mathrm{GF}(p^n)$ to itself is called $c$-differential uniformity $\delta$, or equivalent, $F(x)$ is differentially $(c,\delta)$ uniform, when the maximum number of solutions $x\in\mathrm{GF}(p^n)$ of $F(x+a)-F(cx)=b$, $a,b,c\in\mathrm{GF}(p^n)$, $c\neq1$ if $a=0$, is equal to $\delta$. The objective of this paper is to study the $-1$-differential uniformity of ternary APN power functions $F(x)=x^d$ over $\mathrm{GF}(3^n)$. We obtain ternary power functions with low $-1$-differential uniformity, and some of them are almost perfect $-1$-nonlinear.