The Mathematical Safe Problem and Its Solution (Part 1)

Abstract

Introduction. The problem of the mathematical safe arises in the theory of computer games and cryptographic applications. The article considers the formulation of the mathematical safe problem and the approach to its solution using systems of linear equations in finite rings and fields. The purpose of the article is to formulate a mathematical model of the mathematical safe problem and its reduction to systems of linear equations in different domains; to consider solving the corresponding systems in finite rings and fields; to consider the principles of constructing extensions of residue fields and solving systems in the relevant areas. Results. The formulation of the mathematical safe problem is given and the way of its reduction to systems of linear equations is considered. Methods and algorithms for solving this type of systems are considered, where exist methods and algorithms for constructing the basis of a set of solutions of linear equations and derivative methods and algorithms for constructing the basis of a set of solutions of systems of linear equations for residue fields, ghost rings, finite rings and finite fields. Examples are given to illustrate their work. The principles of construction of extensions of residue fields by the module of an irreducible polynomial, and examples of operations tables for them are considered. The peculiarities of solving systems of linear equations in such fields are considered separately. All the above algorithms are accompanied by proofs and estimates of their time complexity. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in many variations of its formulation. The second part of the paper will consider the application of these methods and algorithms to solve the problem of mathematical safe in its various variations. Keywords: mathematical safe, finite rings, finite fields, method, algorithm, solution.