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Building upon the observation that the newly defined~\cite{EFRST20} conceptof $c$-differential uniformity is not invariant under EA orCCZ-equivalence~\cite{SPRS20}, we showed in~\cite{SG20} that adding someappropriate linearized monomials increases the $c$-differential uniformity ofthe inverse function, significantly, for some~$c$. We continue thatinvestigation here. First, by analyzing the involved equations, we find boundsfor 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 linearizedpolynomial, 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 finitefield $\F_{p^n}$, and further, we use that method to find explicit expressionsfor all entries of the $c$-DDT of the perturbed Gold function (via an arbitrarylinearized polynomial).

Characters, Weil sums and $c$-differential uniformity with an application to the perturbed Gold function