The structure of a class of automata is analyzed. These automata are analogs of symmetric chaotic dynamical systems over a finite ring, namely, the Guckenheimer-Holmes cycle and free-running systems. Problems of parametric identification and identification of initial states are solved, and a set of fixed points of automaton mappings is characterized.

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A general scheme is investigated that is destined for obtaining estimates based on the cardinality of subsets of a fixed set of automata over some finite commutative-associative ring with unit element. A scheme is proposed to solve parametric systems of polynomial equations based on classes of associated elements of the ring. Some general characteristics of automata over the ring are established.

Some general properties of families of hash functions defined by strongly connected automata without output function over a finite ring are analyzed. The probabilities of randomly choosing a sequence for which a hash function assumes a given value and also two different sequences of the same length for which the values of a hash function coincide. The computational security of the investigated hash-functions is characterized.

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