The Mathematical Safe Problem and Its Solution (Part 1)

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.

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The Mathematical Safe Problem and Its Solution (Part 2)

Cybernetics and Computer Technologies ◽

10.34229/2707-451x.21.1.2 ◽

2021 ◽

pp. 16-28

Author(s):

Sergii Kryvyi ◽

Hryhorii Hoherchak

Keyword(s):

Commutative Ring ◽

Finite Fields ◽

Computer Games ◽

Diophantine Equations ◽

Linear Equations ◽

Finite Rings ◽

Prime Field ◽

Linear Diophantine Equations ◽

Systems Of Linear Equations ◽

Mathematical Safe

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.

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Cryptanalysis of Loiss Stream Cipher-Revisited

Journal of Applied Mathematics ◽

10.1155/2014/457275 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7

Author(s):

Lin Ding ◽

Chenhui Jin ◽

Jie Guan ◽

Qiuyan Wang

Keyword(s):

Time Complexity ◽

Stream Cipher ◽

Linear Equations ◽

Data Complexity ◽

Secret Key ◽

Key Recovery ◽

Systems Of Linear Equations ◽

Key Recovery Attack ◽

Better Than

Loiss is a novel byte-oriented stream cipher proposed in 2011. In this paper, based on solving systems of linear equations, we propose an improved Guess and Determine attack on Loiss with a time complexity of 2231and a data complexity of 268, which reduces the time complexity of the Guess and Determine attack proposed by the designers by a factor of 216. Furthermore, a related key chosenIVattack on a scaled-down version of Loiss is presented. The attack recovers the 128-bit secret key of the scaled-down Loiss with a time complexity of 280, requiring 264chosenIVs. The related key attack is minimal in the sense that it only requires one related key. The result shows that our key recovery attack on the scaled-down Loiss is much better than an exhaustive key search in the related key setting.

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A removal lemma for linear configurations in subsets of the circle

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091512000399 ◽

2013 ◽

Vol 56 (3) ◽

pp. 657-666 ◽

Cited By ~ 2

Author(s):

Pablo Candela ◽

Olof Sisask

Keyword(s):

Finite Fields ◽

Linear Equations ◽

Circle Group ◽

Systems Of Linear Equations ◽

Removal Lemma

AbstractWe obtain a removal lemma for systems of linear equations over the circle group, using a similar result for finite fields due to Král′, Serra and Vena, and we discuss some applications.

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An algorithmic feature of a phase transition of random systems of linear equations over finite fields

2012 IEEE Fifth International Conference on Advanced Computational Intelligence (ICACI) ◽

10.1109/icaci.2012.6463298 ◽

2012 ◽

Author(s):

Jing Shen ◽

Yun Fan

Keyword(s):

Phase Transition ◽

Finite Fields ◽

Random Systems ◽

Linear Equations ◽

Systems Of Linear Equations

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Small FPGA implementations for solving systems of linear equations in finite fields

2015 6th IEEE International Conference on Software Engineering and Service Science (ICSESS) ◽

10.1109/icsess.2015.7339120 ◽

2015 ◽

Author(s):

Haibo Yi ◽

Weijian Li

Keyword(s):

Finite Fields ◽

Linear Equations ◽

Systems Of Linear Equations

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On Coset Coverings of Solutions of Homogeneous Cubic Equations over Finite Fields

The Electronic Journal of Combinatorics ◽

10.37236/1566 ◽

2001 ◽

Vol 8 (1) ◽

Author(s):

Ara Aleksanyan ◽

Mihran Papikian

Keyword(s):

Finite Field ◽

Finite Fields ◽

Linear Equations ◽

Initial Equation ◽

Arbitrary Size ◽

Cubic Equation ◽

Systems Of Linear Equations

Given a cubic equation $x_1y_1z_1+x_2y_2z_2+\cdots +x_ny_nz_n=b$ over a finite field, it is necessary to determine the minimal number of systems of linear equations over the same field such that the union of their solutions exactly coincides with the set of solutions of the initial equation. The problem is solved for arbitrary size of the field. A covering with almost minimum complexity is constructed.

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