Multipermutations on the Cartesian product of groups and their properties

Multipermutations are introduced by C.-P. Schnorr and S. Vaudenay as formalization of perfect diffusion in block ciphers. In this paper, we consider a group $X$ and a set $H$ of transformations on $X^2$ introduced by S. Vaudenay. Any bijective transformation from $H$ is a multipermutation. Multipermutations from $H$ are defined by orthomorphisms and complete mappings on $X$. For a set $W$ of distinct cosets of a normal subgroup $W_{0}$ in $X$, we provide multipermutations from $H$ such that they perfectly diffuse one of partitions $W^2$ or $X \times W$. As an example, we prove that Feistel-like involutions on $X$, which are components of the CS-cipher encryption function, perfectly diffuse $X \times W$ for any subgroup $W_{0}$.