While the classical differential uniformity (c = 1) is invariant under the CCZ-equivalence, the newly defined (Ellingsen et al., IEEE Trans. Inf. Theory 66(9), 5781–5789, 2020) concept of c-differential uniformity (cDU), as was observed in Hasan et al. (2020), is not invariant under EA or CCZ-equivalence, for c≠ 1. In this paper, we find an intriguing behavior of the inverse function, namely, that adding some appropriate linearized monomials increases the c-differential uniformity significantly, for some c. For example, adding the linearized monomial \(x^{2^{d}}\) to \(x^{2^{n}-2}\), where d is the largest nontrivial divisor of n, increases the mentioned c-differential uniformity from 2 or 3 (for c≠ 0,1) to ≥ 2^d + 2, which in the case of the inverse function (as used in the AES) on \({\mathbb {F}}_{2^{8}}\) is a significant value of 18. We consider the case of perturbations via more general linearized polynomials and give bounds for the cDU based upon character sums. We further provide some computational results on other known Sboxes.

虽然经典微分均匀性（c = 1）在CCZ等价性下是不变的，但新定义的（c Eldersen et al。，IEEE Trans。Inf。Theory 66（9），5781-5789，2020）概念是c-微分均匀性（cDU），正如在Hasan等人（2005年）中所观察到的 （2020）在EA或CCZ等价性下对于c ≠1并非不变。在本文中，我们发现了反函数的一种有趣行为，即，添加一些合适的线性单项式显着提高了c-微分均匀性，对于一些c。例如，将线性化的单项式\（x ^ {2 ^ {d}} \）添加到\（x ^ {2 ^ {n} -2} \），其中d是n的最大非平凡因数，将上述c-微分均匀性从2或3（对于c ≠0,1）增加到≥2 ^d + 2，对于反函数（如AES中使用的）\（{\ mathbb {F}} _ {2 ^ {8}} \）的有效值为18。我们考虑通过更通用的线性多项式进行扰动的情况，并基于字符和为cDU赋予边界。我们还将在其他已知的Sbox上提供一些计算结果。