On Cryptographic Applications of Matrices Acting on Finite Commutative Groups and Rings

In this paper, we investigate matrices acting on finite commutative groups and rings. In fact, we study modules on ring of matrices over ZN and also modules over the ring (F2 , ⨁,∧); these new algebraic constructions are a generalization of some of the constructions which were previously presented by the authors of this paper. We present new linearized and nonlinear MDS diffusion layers, based on this mathematical investigation. Also, we study some types of nonlinear number generators over Z2n and we present a lower bound on the period of these new nonlinear number generators. As a consequence, we present nonlinear recurrent sequences over Z2n with periods which are multiples of the period of the corresponding sigma-LFSR’s.