NUMERICAL METHODS FOR SOLUTION OF PROBLEM OF MATHEMATICAL STRONG BOX

Abstract

The algorithms necessary for constructing extensions ef residue fields are considered: a Rabin test for checking irreducibility of polynomials, construction of addition and multiplication tables by modulo of irreducible polynomial, calculation of opposite and inverse elements based on these tables. The features of the numbering of elements of finite fields and its influence on the efficiency ef performing basic operations on field elements are described. An algorithm for constructing a basis for a set of solutions of homogeneous systems of linear equations and an algorithm for building of common solution ef inhomogeneous systems of linear eqations over a finite field are proposed. All of the algorithms have polinomial estimations of time complexity and demonstrated by using tables for different systems ef equations and different parameters of these algorithms. The use of systems of linear equations in the problem ef mathematical safe is considered. An adaptation of the problem to the field is proposed. Various cases of the representation of a mathematical safe, the conditions for solution existence, algorithms for solving a problem in these cases, and their effectiveness in the field under consideration are described. Possible applications of systems of equations over the field in coding and cryptography are indicated.