Paper 2024/1247

A Note on the Quasigroup Lai-Massey Structures

Abstract

In our paper, we explore the consequences of replacing the commutative group operation used in Lai-Massey structures with a quasigroup operation. We introduce four quasigroup versions of the Lai-Massey structure, and prove that for quasigroups isotopic with a group $\mathbb{G}$, the complexity of launching a differential attack against these variants of the Lai-Massey structure is equivalent to attacking an alternative structure based on $\mathbb{G}$. Then we provide the conditions needed for correct decryption, and further refine the resulting structure. The emerging structure is both intriguing and novel, and we hope that it will form the basis for future secure block ciphers based on non-commutative groups. In the case of commutative groups, we show that the resulting structure reduces to the classical Lai-Massey structure.