Galois Fields

Simulation of Fast Transformation Procedures in Galois Fields to Estimate Efficiency of Perspective Signals Generation and Processing Devices

2021 Systems of Signals Generating and Processing in the Field of on Board Communications ◽

10.1109/ieeeconf51389.2021.9416089 ◽

2021 ◽

Author(s):

V. Y. Mikhaylov ◽

R. B. Mazepa

Keyword(s):

Galois Fields

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Normal Bases on Galois Ring Extensions

Symmetry ◽

10.3390/sym10120702 ◽

2018 ◽

Vol 10 (12) ◽

pp. 702

Author(s):

Aixian Zhang ◽

Keqin Feng

Keyword(s):

Signal Processing ◽

Finite Field ◽

Block Cipher ◽

Field Extension ◽

Galois Ring ◽

Normal Basis ◽

Galois Rings ◽

Ring Extension ◽

Galois Fields ◽

Normal Bases

Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension.

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Public key of cryptography over extended Galois fields

2010 20th International Crimean Conference "Microwave & Telecommunication Technology" ◽

10.1109/crmico.2010.5632471 ◽

2010 ◽

Author(s):

G. C. Bodyan ◽

D. G. Bodyan ◽

T. V. Shestakov

Keyword(s):

Public Key ◽

Galois Fields

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Generators for primary closures of Galois fields

Finite Fields and Their Applications ◽

10.1016/s1071-5797(02)00021-7 ◽

2003 ◽

Vol 9 (1) ◽

pp. 122-128 ◽

Cited By ~ 4

Author(s):

Dirk Hachenberger

Keyword(s):

Galois Fields

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Algebraic and Stochastic Coding Theory ◽

10.1201/b11707-12 ◽

2017 ◽

pp. 186-205

Keyword(s):

Galois Fields

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Permutation polynomials over Galois fields of characteristic

Bayero Journal of Pure and Applied Sciences ◽

10.4314/bajopas.v13i2.11 ◽

2021 ◽

Vol 13 (2) ◽

pp. 84-86

Author(s):

Z.L. Dahiru ◽

A.M. Lawan

Keyword(s):

Permutation Polynomial ◽

Permutation Polynomials ◽

Galois Fields ◽

Characteristic 2

In this paper, a class of permutation polynomial known as o-polynomial over Galois fields of characteristic 2 was studied. A necessary and sufficients condition for a monomial 𝑥2k to be an o-polynomial over F2t is given and two results obtained by Gupta and Sharma (2016) were deduced.

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DIFFERENTIATION OF POLYNOMIALS IN SEVERAL VARIABLES OVER GALOIS FIELDS OF FUZZY CARDINALITY AND APPLICATIONS TO REED-MULLER CODES

Vestnik of Don State Technical University ◽

10.23947/1992-5980-2018-18-3-339-348 ◽

2018 ◽

Vol 18 (3) ◽

pp. 339-348

Author(s):

V. M. Deundyak ◽

N. S. Mogilevskaya

Keyword(s):

Coding Theory ◽

Communication Systems ◽

Communication Channels ◽

Galois Field ◽

Second Order ◽

Galois Fields ◽

Practical Applications ◽

Several Variables ◽

Confidential Data ◽

Partial Distortion

Introduction. Polynomials in several variables over Galois fields provide the basis for the Reed-Muller coding theory, and are also used in a number of cryptographic problems. The properties of such polynomials specified over the derived Galois fields of fuzzy cardinality are studied. For the results obtained, two real-world applications are proposed: partitioning scheme and Reed-Muller code decoder.Materials and Methods. Using linear algebra, theory of Galois fields, and general theory of polynomials in several variables, we have obtained results related to the differentiation and integration of polynomials in several variables over Galois fields of fuzzy cardinality. An analog of the differentiation operator is constructed and studied for vectors.Research Results. On the basis of the obtained results on the differentiation and integration of polynomials, a new decoder for Reed-Muller codes of the second order is given, and a scheme for organizing the partitioned transfer of confidential data is proposed. This is a communication system in which the source data on the sender is divided into several parts and, independently of one another, transmitted through different communication channels, and then, on the receiver, the initial data is restored of the parts retrieved. The proposed scheme feature is that it enables to protect data, both from the nonlegitimate access, and from unintentional errors; herewith, one and the same mathematical apparatus is used in both cases. The developed decoder for the second-order Reed-Muller codes prescribed over the derived odd Galois field may have a constraint to the recoverable error level; however, its use is advisable for a number of the communication channels.Discussion and Conclusions. The proposed practical applications of the results obtained are useful for the organization of reliable communication systems. In future, it is planned to study the restoration process of the original polynomial by its derivatives, in case of their partial distortion, and the development of appropriate applications.

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