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

TL;DR

This paper investigates how perturbations of the Gold function affect its $c$-differential uniformity, providing bounds and explicit formulas using character sums, with implications for cryptographic function analysis.

Contribution

It extends previous work by analyzing the impact of linearized polynomial perturbations on the $c$-differential uniformity of Gold functions, offering explicit formulas and bounds.

Findings

Bounds for the $c$-differential uniformity of perturbed Gold functions.

Explicit expressions for the $c$-DDT entries of perturbed Gold functions.

Discrepancies observed in uniformity behavior compared to inverse functions.

Abstract

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).