On ARX-like ciphers based on different codings of 2-groups with a cyclic subgroup of index 2

A large number of block ciphers are based on easily and efficiently implemented group operations on 2-groups such as the additive group of the residue ring Z2m , the additive group of the vector space Vm (2) over GF(2) and their combination. ARX-like ciphers use the operations of cyclic shifts and additions in Z2m, Vm(2). For developing techniques of building and analysing new symmetric-key block ciphers, we study group properties of m-bit ARX-like ciphers based on regular groups generated by (0,1,. . . ,2^m- 1) and different codings of permutation representations of nonabelian 2-groups with a cyclic subgroup of index 2. There are exactly four isomorphism classes of the nonabelian 2-groups such as the dihedral group D2m , the generalized quaternion group Q2m , the quasidihedral group SD2m and the modular maximal-cyclic group M2m . For such groups, we get imprimitivity criterions and give conditions on codings in order that the group of the ARX-like cipher should be equal to the symmetric group S2m . We also provide examples of three natural codings and their group properties.

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