Substitution Box on Maximal Cyclic Subgroup of Units of a Galois Ring

In this paper, we construct a new substitution box (S-box) structure based on the elements of the maximal cyclic subgroup of the multiplicative group of units in a finite Galois ring instead of Galois field. We analyze the potency of the proposed S-box by using the majority logic criterion. Moreover, we illustrate the utility of the projected S-box in watermarking.

Amalgamated Products and the Howson Property

We show that if A is a torsion-free word hyperbolic group which belongs to class (Q), that is all finitely generated subgroups of A are quasiconvex in A, then any maximal cyclic subgroup U of A is a Burns subgroup of A. This, in particular, implies that if B is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated) then A *UB, ⧼A, t | Ut = V⧽ are also Howson groups. Finitely generated free groups, fundamental groups of closed hyperbolic surfaces and some interesting 3-manifold groups are known to belong to class (Q) and our theorem applies to them. We also describe a large class of word hyperbolic groups which are not Howson.