论文标题
低$ c $差异均匀性的有限场上的功率功能
Power Functions over Finite Fields with Low $c$-Differential Uniformity
论文作者
论文摘要
最近，在[c-差异，乘法统一性和（几乎）完美的c-nonelinearity，ieee trans中，Ellingsen \ textit {et al}引入了一个新概念（以及相应的$ c $差异均匀性）。通知。理论，2020年]，是由实践差分密码分析动机。与经典的完美非线性功能不同，即使对于特征两个，也有完美的$ c $ - nlinlinear函数。本文的目的是研究功率函数$ f（x）= x^d $在有限的字段上，低$ c $差异均匀性。某些功率功能被证明是完美的$ c $ - nonlinear或几乎完美的$ c $ - nonlinear。值得注意的是，我们完全确定$ c $ - 差异均匀的均匀性，几乎是完美的非线性函数。我们还为Bartoli和Timpanella在2019年提出的最新猜想提供了肯定的解决方案，该猜想与出色的准平面功率函数有关。
Very recently, a new concept called multiplicative differential (and the corresponding $c$-differential uniformity) was introduced by Ellingsen \textit{et al} in [C-differentials, multiplicative uniformity and (almost) perfect c-nonlinearity, IEEE Trans. Inform. Theory, 2020] which is motivated from practical differential cryptanalysis. Unlike classical perfect nonlinear functions, there are perfect $c$-nonlinear functions even for characteristic two. The objective of this paper is to study power function $F(x)=x^d$ over finite fields with low $c$-differential uniformity. Some power functions are shown to be perfect $c$-nonlinear or almost perfect $c$-nonlinear. Notably, we completely determine the $c$-differential uniformity of almost perfect nonlinear functions with the well-known Gold exponent. We also give an affirmative solution to a recent conjecture proposed by Bartoli and Timpanella in 2019 related to an exceptional quasi-planar power function.
