Multipermutations and perfect diffusion of partitions

Multipermutations are introduced by C.-P. Schnorr and S. Vaudenay as formalization of perfect diffusion in block ciphers. In this paper, we consider an abelian 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 on $X$. The set $H$ is nonempty iff there exists an orthomorphism on $X$. We consider a set $W$ of distinct cosets of $W_{0}$ in $X$. We describe multipermutations from $H$ such that they perfectly diffuse one of partitions $W^2$ or $X \times W$. As an example, we prove that $8$-bit and $16$-bit transformations of CS-cipher perfectly diffuse such partitions.