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

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 $n\equiv 0(\mathrm{mod}6)$, $8$, if $n\equiv 3(\mathrm{mod}6)$, and 6, if $n\not\equiv 0(\mathrm{mod}3)$. 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~$c\ne 1$, the $c$-differential uniformity is upper bounded by~$4$ and the $c$-boomerang uniformity by~$5$ with both bounds being attained for~$n\ge 4$.

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