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).

中文翻译：

---

字符、Weil 和和 $c$-微分均匀性以及对扰动 Gold 函数的应用

基于观察到新定义的~\cite{EFRST20} 概念 $c$-微分均匀性在 EA 或 CCZ 等价下不是不变的~\cite{SPRS20}，我们在~\cite{SG20} 中展示了添加一些适当的线性化单项式显着增加了反函数的 $c$-微分均匀性，对于某些~$c$。我们在这里继续调查。首先，通过分析所涉及的方程，我们找到了受单个单项式扰动的 Gold 函数均匀性的界限，展示了我们之前在反函数上观察到的差异。其次，为了通过任何线性化多项式处理扰动的一般情况，我们使用有限域中的字符来表达 $(n,n)$-函数的 $c$-微分分布表 (DDT) 中的所有条目有限域 $\F_{p^n}$，进而，