特征标、Weil和与$c$ -微分一致性及其在扰动Gold函数中的应用
Characters, Weil sums and $c$-differential uniformity with an application to the perturbed Gold function
摘要：
基于新定义的$c$ -微分均匀性的~ \cite{EFRST20}概念在EA或CCZ-等价~ \cite{SPRS20}下不是不变的这一观察，我们在~ \cite{SG20}中表明，对于某些~ $c$，添加一些适当的线性化单项式显著地增加了反函数的$c$ -微分均匀性。我们在这里继续调查。首先，通过分析所涉及的方程，我们找到了由一个单一的单项扰动的黄金函数的均匀性的界限，表现出我们以前观察到的反函数的差异。其次，为了处理通过任何线性化多项式的扰动的一般情况，我们使用有限域中的特征来表示有限域$\F_{p^n}$上的$(n,n)$ -函数的$c$ -微分分布表（DDT）中的所有条目，并且进一步地，我们使用该方法来找到被扰动的Gold函数的$c$ -DDT的所有条目的显式表达式（通过任意线性化多项式）。
英文摘要：
Building upon the observation that the newly defined~\cite{EFRST20} concept of $c$-differential uniformity is not invariant under EA or CCZ-equivalence~\cite{SPRS20}, we showed in~\cite{SG20} that adding some appropriate linearized monomials increases the $c$-differential uniformity of the inverse function, significantly, for some~$c$. We continue that investigation here. First, by analyzing the involved equations, we find bounds for the uniformity of the Gold function perturbed by a single monomial, exhibiting the discrepancy we previously observed on the inverse function. Secondly, to treat the general case of perturbations via any linearized polynomial, we use characters in the finite field to express all entries in the $c$-Differential Distribution Table (DDT) of an $(n,n)$-function on the finite field $\F_{p^n}$, and further, we use that method to find explicit expressions for all entries of the $c$-DDT of the perturbed Gold function (via an arbitrary linearized polynomial).
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