## 2021, issue 1, p. 16-28

Received 06.12.2020; Revised 30.01.2021; Accepted 25.03.2021

Published 30.03.2021; First Online 03.04.2021

https://doi.org/10.34229/2707-451X.21.1.2

MSC 12F05, 68W05

The Mathematical Safe Problem and Its Solution (Part 2)

Sergii Kryvyi 1 * ,   Hryhorii Hoherchak 1 1 Taras Shevchenko National University of Kyiv, Ukraine

Introduction. The problem of mathematical safe arises in the theory of computer games and cryptographic applications. The article considers numerous variations of the mathematical safe problem and examples of its solution using systems of linear Diophantine equations in finite rings and fields.

The purpose of the article. To present methods for solving the problem of a mathematical safe for its various variations, which are related both to the domain over which the problem is considered and to the structure of systems of linear equations over these domains. To consider the problem of a mathematical safe (in matrix and graph forms) in different variations over different finite domains and to demonstrate the work of methods for solving this problem and their efficiency (systems over finite simple fields, finite fields, ghost rings and finite associative-commutative rings).

Results.  Examples of solving the problem of a mathematical safe, the conditions for the existence of solutions in different areas, over which this problem is considered. The choice of the appropriate area over which the problem of the mathematical safe is considered, and the appropriate algorithm for solving it depends on the number of positions of the latches of the safe. All these algorithms are accompanied by estimates of their time complexity, which were considered in the first part of this paper.

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 a large number of variations of its formulation (over finite prime field, finite field, primary associative-commutative ring and finite associative-commutative ring with unit).

Keywords: mathematical safe, finite rings, finite fields, method, algorithm.

Cite as: Kryvyi S., Hoherchak H. The Mathematical Safe Problem and Its Solution (Part 2). Cybernetics and Computer Technologies. 2021. 1. P. 16–28. (in Ukrainian) https://doi.org/10.34229/2707-451X.21.1.2

References

1.     Kryvyi S., Hoherchak H. The Mathematical Safe Problem and Its Solution (Part 1). Cybernetics and Computer Technologies. 2020. 4. P. 24–35. (in Ukrainian) https://doi.org/10.34229/2707-451X.20.4.2

2.     Kryvyi S.L. Numerical methods of solution of mathematical safe problem. Cybernetics and Systems Analysis. 2019. 55 (5). P. 714–730. https://doi.org/10.1007/s10559-019-00182-z

3.     Кostrikin А.I. Introduction in algebra. М.: Nauka, 1977. 495 p. (in Russian)

ISSN 2707-451X (Online)

ISSN 2707-4501 (Print)