The c-differential behavior of the inverse function under the EA-equivalence

Abstract

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.