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

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 $\hat{I}´$, or equivalent, $F(x)$ is differentially $(c,\hat{I}´)$ uniform, when the maximum number of solutions $x\in \mathrm{GF}(p^n)$ of $F(x+a)-F(cx)=b$, $a,b,c\in \mathrm{GF}(p^n)$, $c\ne 1$ if $a=0$, is equal to $\hat{I}´$. 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.