Variations of orthomorphisms and pseudo-hadamard transformations on nonabelian groups

. An orthomorphism of a group (X, ·) is a permutation g : X M X such that the mapping x M x^-1g(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 M X. We have proved that our ten variations are permutations iff g is an orthomorphism or its variation.

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