Slobodian S. Theorems on a normal limit distribution of the number of false solutions of a system of nonlinear random equations in the field GF(2).

The thesis is devoted to the further development of the theory of systems of nonlinear random equations over the field GF(2). Distribution of the number of solutions of a system of nonlinear random equations in the field GF(2) is investigated in the thesis. The normal limit distribution of the normalized number of solutions of a system of nonlinear random equations in the field GF(2) at various restrictions on distributions of coefficients of a system and the orders of their nonlinearity, quantity of nonzero components of a true solution is obtained. Theorems on a normal limit distribution of the normalized number of solutions of such system under conditions of increase of the number of nonzero components of a true solution of this system with increase of the number of its unknowns; at presence of a linear part in the system with positive probability; the increase of the number of zero components of a true solution of this system with increase of the number of its unknowns are proved. The generalization of the lemma on a metric modification of a method of moments and the explicit expression for r-factorial moment of the number of solutions of a system are used for proof of results of the thesis. Obtained results are both of theoretical and practical interest, in particular, for problems of information encoding at transfer by communication channels and guarding against the unauthorized access.