Separable Extensions of Finite Fields and Finite Rings

Public-key cryptosystem based on invariants of diagonalizable groups

Groups – Complexity – Cryptology ◽

10.1515/gcc-2017-0003 ◽

2017 ◽

Vol 9 (1) ◽

Author(s):

František Marko ◽

Alexandr N. Zubkov ◽

Martin Juráš

Keyword(s):

Linear Algebra ◽

Finite Fields ◽

Number Fields ◽

Euclidean Algorithm ◽

Public Key ◽

Public Key Cryptosystem ◽

Finite Rings

AbstractWe develop a public-key cryptosystem based on invariants of diagonalizable groups and investigate properties of such a cryptosystem first over finite fields, then over number fields and finally over finite rings. We consider the security of these cryptosystem and show that it is necessary to restrict the set of parameters of the system to prevent various attacks (including linear algebra attacks and attacks based on the Euclidean algorithm).

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Integral Cayley Graphs Generated by Distance Sets in Vector Spaces over Finite Fields

The Electronic Journal of Combinatorics ◽

10.37236/2730 ◽

2013 ◽

Vol 20 (1) ◽

Author(s):

Ngoc Dai Nguyen ◽

Minh Hai Nguyen ◽

Duy Hieu Do ◽

Anh Vinh Le

Keyword(s):

Finite Fields ◽

Euclidean Distance ◽

Cayley Graphs ◽

Vector Spaces ◽

Sufficient Condition ◽

Finite Rings ◽

Distance Sets

Si Li and the fourth listed author (2008) considered unitary graphs attached to the vector spaces over finite rings using an analogue of the Euclidean distance. These graphs are shown to be integral when the cardinality of the ring is odd or the dimension is even. In this paper, we show that the statement also holds for the remaining case: the cardinality of the ring is even and the dimension is odd, by showing a sufficient condition for Cayley graphs generated by distance sets in vector spaces over finite fields to be integral.

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The Mathematical Safe Problem and Its Solution (Part 1)

Cybernetics and Computer Technologies ◽

10.34229/2707-451x.20.4.2 ◽

2020 ◽

pp. 15-38

Author(s):

S. Kryvyi ◽

H. Hoherchak

Keyword(s):

Mathematical Model ◽

Finite Fields ◽

Computer Games ◽

Time Complexity ◽

Linear Equations ◽

Irreducible Polynomial ◽

Finite Rings ◽

Systems Of Linear Equations ◽

Residue Fields ◽

Mathematical Safe

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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Dedekind-finite fields

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700008492 ◽

1978 ◽

Vol 19 (1) ◽

pp. 117-124 ◽

Cited By ~ 1

Author(s):

J.L. Hickman

Keyword(s):

Set Theory ◽

Finite Fields ◽

Nonnegative Integer ◽

Axiom Of Choice ◽

General Knowledge ◽

Embedding Theorems ◽

Finite Sets ◽

Finite Rings ◽

The Axiom Of Choice ◽

Positive Integers

Let p be a prime and let (mk)k<ω be a strictly increasing sequence of positive integers such that m0 = 1 and mk divides mk+1. A field F is said to be of type (p, (mk)k<ω) if it is the union of an increasing sequence (Fk)k<ω of fields such that Fk has pmk elements. A set X is called “finite” if it has n elements for some nonnegative integer n, and “Dedekind-finite” if every injection f: X → X is a bijection. If the Axiom of Choice is rejected, then it is relatively consistent to assume the existence of medial (that is, infinite, Dedekind-finite) sets. In this paper it is shown that given any type (p, (mk)k<ω) as above, it is relatively consistent with the usual axioms of set theory (minus Choice) to assume the existence of a medial field of type (p, (mk)k<ω). Conversely, it is shown that any medial field must be of type (p, (mk)k<ω) for some (p, (mk)k<ω) as above. The paper concludes with a few observations on Dedekind-finite rings. In the first part of the paper, a general knowledge of Fraenkel-Mostowski set theory and of the Jech-Sochor Embedding Theorems is assumed.

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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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