Group polynomials over rings

We consider polynomials over rings such that the polynomials represent Latin squares and define a group operation over the ring. We introduce the notion of a group polynomial, describe a number of properties of these polynomials and the groups generated. For the case of residue rings $\mathbb{Z}_{r^n}$, where $r$ is a prime number, we give a description of groups specified by polynomials and identify a class of group polynomials that can be used to construct controlled cryptographic transformations.