Families of semi-automata defined by a recurrence relation in a finite quasigroup are investigated. Initially, these families are defined in an abstract finite quasigroup, and their structure is studied. It is shown that from a probabilistic point of view these semi-automata are the best mathematical models for computationally secure families of iterated hash functions. Then families of semi-automata in T-quasigroups determined by a finite Abelian group are defined, and their structure is studied. Representation of these semi-automata by the parallel composition of the ones defined in T-quasigroups determined by cyclic groups of prime power order is considered. This decomposition results in speed up the functioning and reducing space complexity of a semi-automaton. In addition, families of semi-automata in the Abelian group of an elliptic curve over a finite field are investigated.