Euclidean algorithm and congruence relations

Abstract We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

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Congruence relations on almost distributive lattices-II

Asian-European Journal of Mathematics ◽

10.1142/s1793557121500054 ◽

2019 ◽

pp. 2150005

Author(s):

Gezahagne Mulat Addis

Keyword(s):

Distributive Lattice ◽

Congruence Relation ◽

Congruence Class ◽

Distributive Lattices ◽

Ideal Formula ◽

Congruence Relations

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.

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Congruence Relations on Residuated Semigroups (II)

Journal of the London Mathematical Society ◽

10.1112/jlms/s1-39.1.150 ◽

1964 ◽

Vol s1-39 (1) ◽

pp. 150-158 ◽

Cited By ~ 2

Author(s):

R. McFadden

Keyword(s):

Congruence Relations

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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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Current situation and development direction of sports industry in colleges and universities based on euclidean algorithm

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-219032 ◽

2021 ◽

pp. 1-7

Author(s):

Shunjiang Ma ◽

Gaicheng Liu ◽

Zhiwu Huang

Keyword(s):

Computer Technology ◽

Colleges And Universities ◽

Status Quo ◽

Euclidean Algorithm ◽

Test Results ◽

Sports Industry ◽

Development Direction ◽

The Status ◽

Traditional Sports ◽

And Function

With the development of sports in colleges and universities, the research on innovation reform of sports industry has been deepened. Therefore, based on the above situation, a study of the status quo and development direction of sports industry in colleges and universities based on the Euclid algorithm is proposed. In the research here, according to the traditional sports industry concept to sum up, and then according to the advantages of computer technology to deal with the relevant data. In order to realize good overlap between data, an application of Euclidean algorithm is proposed. In the test of Euclidean algorithm, the efficiency and function of the algorithm are tested comprehensively, and the test results show that the research is feasible.

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