This paper makes the first bridge between the classical differential/boomerang uniformity and the newly introduced $c$-differential uniformity. We show that the boomerang uniformity of an odd APN function is given by the maximum of the entries (except for the first row/column) of the function's $(-1)$-Difference Distribution Table. In fact, the boomerang uniformity of an odd permutation APN function equals its $(-1)$-differential uniformity. We next apply this result to easily compute the boomerang uniformity of several odd APN functions. In the second part we give two classes of differentially low-uniform functions obtained by modifying the inverse function. The first class of permutations (CCZ-inequivalent to the inverse) over a finite field $\mathbb{F}_{p^n}$ ($p$, an odd prime) is obtained from the composition of the inverse function with an order-$3$ cycle permutation, with differential uniformity $3$ if $p=3$ and $n$ is odd; $5$ if $p=13$ and $n$ is even; and $4$ otherwise. The second class is a family of binomials and we show that their differential uniformity equals~$4$. We next complete the open case of $p=3$ in the investigation started by G\" olo\u glu and McGuire (2014), for $p\geq 5$, and continued by K\"olsch (2021), for $p=2$, $n\geq 5$, on the characterization of $L_1(X^{p^n-2})+L_2(X)$ (with linearized $L_1,L_2$) being a permutation polynomial. Finally, we extend to odd characteristic a result of Charpin and Kyureghyan (2010) providing an upper bound for the differential uniformity of the function and its switched version via a trace function.

翻译：摘要：本文首次构建了经典微分/回旋镖一致性与新引入的$c$-微分一致性之间的桥梁。我们证明奇APN函数的回旋镖一致性可由其$(-1)$-差分分布表中（首行/列除外）的最大条目给出。事实上，奇置换APN函数的回旋镖一致性等于其$(-1)$-微分一致性。随后应用该结果便捷计算了若干奇APN函数的回旋镖一致性。第二部分给出两类通过修改逆函数所得的低微分一致函数。第一类置换（与逆函数CCZ不等价）定义于有限域$\mathbb{F}_{p^n}$（$p$为奇素数），由逆函数与三阶循环置换复合得到，其微分一致性在$p=3$且$n$为奇数时为$3$；$p=13$且$n$为偶数时为$5$；其他情形为$4$。第二类为二元多项式族，其微分一致性等于$4$。继而我们完善了Göloğlu与McGuire（2014）对$p\geq 5$情形、以及Kölsch（2021）对$p=2$且$n\geq 5$情形所启动的关于$L_1(X^{p^n-2})+L_2(X)$（其中$L_1,L_2$为线性化多项式）为置换多项式特征的未解决情形（$p=3$）。最后我们将Charpin与Kyureghyan（2010）的结论推广至奇特征，该结论通过迹函数给出了函数与其切换版本的微分一致性的上界。