Finding functions, particularly permutations, with good differential properties has received a lot of attention due to their possible applications. For instance, in combinatorial design theory, a correspondence of perfect $c$-nonlinear functions and difference sets in some quasigroups was recently shown [1]. Additionally, in a recent manuscript by Pal and Stanica [20], a very interesting connection between the $c$-differential uniformity and boomerang uniformity when $c=-1$ was pointed out, showing that that they are the same for an odd APN permutations. This makes the construction of functions with low $c$-differential uniformity an intriguing problem. We investigate the $c$-differential uniformity of some classes of permutation polynomials. As a result, we add four more classes of permutation polynomials to the family of functions that only contains a few (non-trivial) perfect $c$-nonlinear functions over finite fields of even characteristic. Moreover, we include a class of permutation polynomials with low $c$-differential uniformity over the field of characteristic~$3$. As a byproduct, our proofs shows the permutation property of these classes. To solve the involved equations over finite fields, we use various techniques, in particular, we find explicitly many Walsh transform coefficients and Weil sums that may be of an independent interest.

翻译：寻找具有良好差分性质的函数（尤其是置换函数）因其潜在应用而受到广泛关注。例如，在组合设计理论中，近期研究[1]展示了完全$c$-非线性函数与某些拟群中差集之间的对应关系。此外，Pal和Stanica在最新手稿[20]中指出，当$c=-1$时，$c$-差分一致性与回旋镖一致性之间存在非常有趣的联系，证明对于奇APN置换，两者是相同的。这使得构造低$c$-差分一致性的函数成为一个引人关注的问题。本文研究了几类置换多项式的$c$-差分一致性。由此，我们向仅包含少数（非平凡）完全$c$-非线性函数的偶特征有限域函数族中新增了四类置换多项式。此外，我们还在特征为$3$的有限域上发现了一类具有低$c$-差分一致性的置换多项式。作为副产品，我们的证明揭示了这些类别的置换性质。为求解有限域上的相关方程，我们采用了多种技术，特别是显式计算了许多可能具有独立研究价值的Walsh变换系数和Weil和。