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

TL;DR

This paper investigates how the inverse function's $c$-differential uniformity changes when adding linearized monomials, revealing that such modifications can significantly increase its differential uniformity, especially in cryptographic contexts.

Contribution

It uncovers the behavior of the inverse function's $c$-differential uniformity under linearized monomial additions, highlighting non-invariance and potential cryptographic implications.

Findings

Adding linearized monomials increases $c$-differential uniformity.

For the inverse function, the uniformity can rise from 2 or 3 to over 18.

The behavior is exemplified with AES's inverse function on $_{2^8}$.

Abstract

While the classical differential uniformity ($c=1$) is invariant under the CCZ-equivalence, the newly defined \cite{EFRST20} concept of $c$-differential uniformity, in general is not invariant under EA or CCZ-equivalence, as was observed in \cite{SPRS20}. 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 $x^{p^d}$, where $d$ is the largest nontrivial divisor of $n$, increases the mentioned $c$-differential uniformity from~$2$ or $3$ (for $c\neq 0$) to $\geq p^{d}+2$, which in the case of AES' inverse function on $\F_{2^8}$ is a significant value of~$18$.