Paper 2021/1676

Cryptographic Symmetric Structures Based on Quasigroups

Abstract

In our paper we study the effect of changing the commutative group operation used in Feistel and Lai-Massey symmetric structures into a quasigroup operation. We prove that if the quasigroup operation is isotopic with a group $\mathbb{G}$, the complexity of mounting a differential attack against our generalization of the Feistel structure is the same as attacking the unkeyed version of the general Feistel iteration based on $\mathbb{G}$. Also, when $\mathbb{G}$ is non-commutative we show that both versions of the Feistel structure are equivalent from a differential point of view. For the Lai-Massey structure we introduce four non-commutative versions, we argue for the necessity of working over a group and we provide some necessary conditions for the differential equivalency of the four notions.