On $-1$-differential uniformity of ternary APN power functions
(2101.10543)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 $\mathrm{GF}(pn)$ to itself is called $c$-differential uniformity $\delta$, or equivalent, $F(x)$ is differentially $(c,\delta)$ uniform, when the maximum number of solutions $x\in\mathrm{GF}(pn)$ of $F(x+a)-F(cx)=b$, $a,b,c\in\mathrm{GF}(pn)$, $c\neq1$ if $a=0$, is equal to $\delta$. The objective of this paper is to study the $-1$-differential uniformity of ternary APN power functions $F(x)=xd$ over $\mathrm{GF}(3n)$. We obtain ternary power functions with low $-1$-differential uniformity, and some of them are almost perfect $-1$-nonlinear.
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