Investigations on $c$-Boomerang Uniformity and Perfect Nonlinearity

TL;DR

This paper explores the extension of $c$-differential concepts to boomerang uniformity, analyzing their implications for perfect nonlinearity and cryptographic S-boxes, building on prior definitions and investigations.

Contribution

It introduces the extension of $c$-differential uniformity to boomerang uniformity and examines its relationship with perfect nonlinearity in cryptographic functions.

Findings

Extended $c$-differential uniformity to boomerang uniformity.

Characterized perfect nonlinear functions under the new framework.

Analyzed implications for S-boxes in block ciphers.

Abstract

We defined in~\cite{EFRST20} a new multiplicative $c$-differential, and the corresponding $c$-differential uniformity and we characterized the known perfect nonlinear functions with respect to this new concept, as well as the inverse in any characteristic. The work was continued in~\cite{RS20}, investigating the $c$-differential uniformity for some further APN functions. Here, we extend the concept to the boomerang uniformity, introduced at Eurocrypt '18 by Cid et al.~\cite{Cid18}, to evaluate S-boxes of block ciphers, and investigate it in the context of perfect nonlinearity and related functions.