Algorithm for Obtaining Complete Irreducible Polynomials over Given Galois Field for New Method of Digital Monitoring of Information Space
Shaltykova D. Massalimova A. Vitulyova Y. Suleimenov I.
November 2025Multidisciplinary Digital Publishing Institute (MDPI)
Computers
2025#14Issue 11
Irreducible polynomials are widely used in modern cryptography; however, algorithms for finding such polynomials remain quite complex and require significant computational resources. In this study, a new approach to finding irreducible equations over Galois fields (Formula presented.) is proposed. It is shown that such irreducible equations can be obtained by solving a system of linear equations over the base Galois field, generated by any element of the field (Formula presented.) that is distinct from the elements of the base field and from elements corresponding to lower-degree extensions. The connection of the proposed approach with algorithms based on the Frobenius automorphism is established. The case corresponding to the field (Formula presented.) and matrices over this field is examined in detail. It has been shown that the proposed method makes it possible to obtain complete sets of irreducible polynomials over a given Galois field. It has also been demonstrated that generating such sets is of particular interest for the development of new methods of digital monitoring of the information space, which are based on analogies with error-correcting coding techniques.
algebraic extensions , cubic equations , digital monitoring of the information space , Frobenius isomorphism , Galois fields , irreducible equations , primitive elements
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National Engineering Academy of the Republic of Kazakhstan, Almaty, 050060, Kazakhstan
Academy of Public Administration Under the President of the Republic of Kazakhstan, Astana, 010000, Kazakhstan
JSC “Institute of Digital Engineering and Technology”, Almaty, 050013, Kazakhstan
Department Smart Technologies in Engineering, International Engineering Technological University, Almaty, 050060, Kazakhstan
National Engineering Academy of the Republic of Kazakhstan
Academy of Public Administration Under the President of the Republic of Kazakhstan
JSC “Institute of Digital Engineering and Technology”
Department Smart Technologies in Engineering
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