Recently, a new concept called multiplicative differential cryptanalysis and the corresponding $c$-differential uniformity were introduced by Ellingsen et al. (2020), and then some low differential uniformity functions were constructed. In this paper, we further study the constructions of perfect $c$-nonlinear (PcN) power functions. First, we give a conjecture on all power functions to be PcN over $\mathrm{GF}\left(2^m\right)$. Second, several classes of PcN power functions are obtained over finite fields of odd characteristic for $c=-1$ and our theorems generalize some results in Bartoli and Timpanella (2020), Hasan et al. (2021) and Zha and Hu (2021). Finally, the $c$-differential spectrum of a class of almost perfect $c$-nonlinear (APcN) power functions is determined.

最近，一个称为乘法差分密码分析的新概念和相应的$C$-Ellingsen 等人介绍了差异均匀性。（2020），然后构造了一些低微分均匀性函数。在本文中，我们进一步研究了完美的构造$C$-非线性（PcN）幂函数。首先，我们猜想所有的幂函数都是 PcN over$\mathrm{GF}\left(2^{米}\right)$. 其次，在奇特性的有限域上获得了几类 PcN 幂函数$C=-1$我们的定理概括了 Bartoli 和 Timpanella (2020)、Hasan 等人的一些结果。（2021 年）和查和胡（2021 年）。最后，$C$-几乎完美的一类微分谱$C$-确定非线性（APcN）幂函数。