Variations of orthomorphisms and pseudo-Hadamard transformations on nonabelian groups

An orthomorphism of a group $(X, \cdot )$ is a permutation $g:X \to X$ such that the mapping $x \mapsto {x^{ - 1}}g(x)$ is also a permutation. In the field of symmetric-key cryptography, orthomorphisms of Abelian groups have been used in the Lai–Massey scheme, the FOX family of block ciphers, the quasi-Feistel network, block ciphers in Davies–Meyer mode, and authentication codes. In this paper, we study orthomorphisms, complete mappings and their variations of nonabelian key-addition groups. In the SAFER block cipher, a linear transformation, called the pseudo-Hadamard transformation, has been used to provide the diffusion that a good cipher requires. We describe ten variations of the pseudo-Hadamard transformations on nonabelian groups, which are defined by a permutation $g:X \to X$. We have proved that our ten variations are permutations iff $g$ is an orthomorphism or its variation.