Low $c$-differential and $c$-boomerang uniformity of the swapped inverse function

TL;DR

This paper analyzes the $c$-differential and $c$-boomerang uniformity of the $(0,1)$-swapped inverse function in characteristic 2, establishing upper bounds of 4 and 5 respectively, which are tight for dimensions at least 4.

Contribution

It characterizes the $c$-differential and $c$-boomerang uniformity of the swapped inverse function in characteristic 2, providing tight bounds for all $c eq 1$.

Findings

$c$-differential uniformity is at most 4 for all $c eq 1$

$c$-boomerang uniformity is at most 5 for all $c eq 1$

Bounds are attained for $n \\geq 4$.

Abstract

Modifying the binary inverse function in a variety of ways, like swapping two output points has been known to produce a $4$-differential uniform permutation function. Recently, in \cite{Li19} it was shown that this swapped version of the inverse function has boomerang uniformity exactly $10$, if $n\equiv 0\pmod 6$, $8$, if $n\equiv 3\pmod 6$, and 6, if $n\not\equiv 0\pmod 3$. Based upon the $c$-differential notion we defined in \cite{EFRST20} and $c$-boomerang uniformity from \cite{S20}, in this paper we characterize the $c$-differential and $c$-boomerang uniformity for the $(0,1)$-swapped inverse function in characteristic~$2$: we show that for all~$c\neq 1$, the $c$-differential uniformity is upper bounded by~$4$ and the $c$-boomerang uniformity by~$5$ with both bounds being attained for~$n\geq 4$.