交换反函数的低$c$ -差分和$c$ -回飞棒均匀性
Low $c$-differential and $c$-boomerang uniformity of the swapped inverse function
摘要：
已知以各种方式修改二进制反函数，如交换两个输出点，以产生$4$ -差分均匀置换函数。最近，在\cite{Li19}中，证明了这种交换版本的反函数具有回飞镖均匀性，如果是$n\equiv 0\pmod 6$，则为$10$，如果是$n\equiv 3\pmod 6$，则为$8$，如果是$n\not\equiv 0\pmod 3$，则为6。基于我们在\cite{EFRST20}中定义的$c$ -微分概念和\cite{S20}中定义的$c$ -回飞镖一致性，本文刻画了特征~ $2$中$(0,1)$ -交换的反函数的$c$ -微分和$c$ -回飞镖一致性：我们证明了对于所有的~ $c\neq 1$，$c$ -微分均匀性的上界为~ $4$，$c$ -回飞棒均匀性的上界为~ $5$，两个界都可由~ $n\geq 4$得到。
英文摘要：
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~$n\geq 4$.
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