Geodesic
On a class of power piecewise affine permutations on nonabelian groups of order $2^m$ with cyclic subgroups of index~$2$
Prikladnaya Diskretnaya Matematika. Supplement, no. 12 (2019), pp. 27-29

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It is known that four nonabelian groups of order $2^m$, where $m \ge 4$, 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 $\mathbb{Z}_{2^m}$, 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.
Keywords: nonabelian group, dihedral group, generalized quaternion group, orthomorphism.
Mots-clés : bijection criterion 
@article{PDMA_2019_12_a6,
     author = {B. A. Pogorelov and M. A. Pudovkina},
     title = {On a class of power piecewise affine permutations on nonabelian groups of order $2^m$ with cyclic subgroups of index~$2$},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {27--29},
     publisher = {mathdoc},
     number = {12},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2019_12_a6/}
}
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B. A. Pogorelov; M. A. Pudovkina. On a class of power piecewise affine permutations on nonabelian groups of order $2^m$ with cyclic subgroups of index~$2$. Prikladnaya Diskretnaya Matematika. Supplement, no. 12 (2019), pp. 27-29. http://geodesic.mathdoc.fr/item/PDMA_2019_12_a6/