Maximal cyclic subgroups of the groups of units of Galois rings: a computational approach

Abstract

We present a computational approach based on algorithmic techniques to obtain maximal cyclic subgroups of the groups of units in Galois rings. The objective of this work was to provide an automated methodology to obtain such maximal cyclic subgroups for minimizing the human effort in such calculations. Necessity of getting a stock of maximal cyclic subgroups is due to their novel role in the formation of cyclic codes over finite commutative rings and prophesied shifting of S-box construction through binary field extensions GF(2(h)), 1 <= h <= 8 to these maximal cyclic subgroups of the groups of units of finite Galois rings GR(2(k), h).