On a class of power piecewise affine permutations on nonabelian groups of order 2m with cyclic subgroups of index 2

It is known that four nonabelian groups of order 2^m, where m ^ 4, have cyclic subgroups of index 2. Examples are well-known dihedral groups and generalized quaternion groups. Any nonabelian group G of order 2^m with cyclic subgroups of index 2 can be considered similar to the additive abelian group of the residue ring Z₂m, which is used as a key-addition group of ciphers. In this paper, we define two classes of transformations on G, which are called power piecewise affine. For each class we prove a bijection criterion. Using these criteria, we can fully classify orthomorphisms or their variations among described classes of power piecewise affine permutations.

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