Skobelev V. Elaboration and investigation of methods of analysis of automata models defined on algebraic structures.

The thesis is devoted to elaboration of mathematical tools intended for de-scriptive, algorithmic and metric analysis of families of automata presented via systems of recurrent relations on algebraic structures over a finite ring under supposition of their potential application for resolving problems of information protection. There are elaborated mathematical tools intended for investigation of systems of equations and disequalities over a finite ring, namely: the method for analysis of statistical equivalence of multy-variable mappings of a finite set into itself, the scheme for presentation of the set of solutions of algebraic equations with parameters in implicit form, the scheme of a solver for checking satisfya-bility of formulae of linear arithmetic. The problem of design of imitational model for a family of automata presented via systems of recurrent relations over a finite ring is resolved, and the class of asymptotically exact imitational models is extracted. It is investigated computational security of families of hash-functions, realized by strongly-connected automata without output function presented via systems of recurrent relations over a finite ring. It is investigated families of automata over a finite associative-commutative ring, which transition and output functions are algebraic sum of a function of the state of automaton and a function of an input symbol, under condition that the value of every component of transition function is some element of fixed ideals of the ring. It is investigated families of automata presented over a finite ring via systems of recurrent relations over varieties with algebra, as well as over parameterized varieties with some extracted set of trajectories. Families of automata presented via systems of recurrent relations on elliptic curve over a finite field are investigated. The results obtained in this thesis can be used in development of software intended for re-solving problems of transformation and protection of information.