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Primitive Elements in the Finite Field of Square Matrices of Order 2 for Cryptographic Applications

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Information Technology for Education, Science, and Technics (ITEST 2024)

Part of the book series: Lecture Notes on Data Engineering and Communications Technologies ((LNDECT,volume 222))

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Abstract

This paper is based on the previous studies that allowed to define a family of square matrices of order 2 over the field of integers modulo prime. This family forms a finite field with the usual operations of matrix multiplication and addition. This paper aims at developing and applying an approach to determine primitive elements of such a finite field of square matrices. The relevance of this topic is explained by the fact that finding primitive elements is an integral component for applying a finite field in cryptographic transformation tasks, in particular, the Diffie-Hellman key agreement protocol. This study provides an answer to the conditions under which a matrix is a primitive element of a finite field, as well as the method for finding all primitive elements of a matrix field. It has been determined the number of different primitive elements of the field. Examples of searching for primitive elements for the finite field various parameters are provided. An example of using a finite field of square matrices of order 2 in the Diffie-Hellman key agreement protocol is demonstrated.

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Acknowledgements

This research was funded by the Ministry of Education and Science of Ukraine under grant 0123U100270.

Author information

Authors and Affiliations

  1. Cherkasy State Technological University, Shevchenko Blvd. 460, Cherkasy, 18006, Ukraine

    Anatoly Shcherba, Emil Faure & Viktor Khaliavka

  2. State Scientific and Research Institute of Cybersecurity Technologies and Information Protection, M. Zaliznyaka Street 3, Kyiv, 03142, Ukraine

    Emil Faure

  3. University of Vaasa, Wolffintie 32, 65200, Vaasa, Finland

    Tero Vartiainen

  4. Cherkasy Research Forensic Center of the Ministry of Internal Affairs of Ukraine, Lesia Ukrainka Street 21, Cherkasy, 18023, Ukraine

    Viktor Khaliavka

Authors
  1. Anatoly Shcherba
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  2. Emil Faure
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  3. Tero Vartiainen
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  4. Viktor Khaliavka
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Corresponding author

Correspondence to Emil Faure .

Editor information

Editors and Affiliations

  1. Department of Information Security and Computer Engineering, Cherkasy State Technological University, Cherkasy, Ukraine

    Emil Faure

  2. Department of Computer Science and Systems Analysis, Cherkasy State Technological University, Cherkasy, Ukraine

    Yurii Tryus

  3. Department of Computing Sciences, University of Vaasa, Vaasa, Finland

    Tero Vartiainen

  4. Department of Computer Science and Systems Analysis, Cherkasy State Technological University, Cherkasy, Ukraine

    Olena Danchenko

  5. Department of Instrumentation, Mechatronics and Computerized Technologies, Cherkasy State Technological University, Cherkasy, Ukraine

    Maksym Bondarenko

  6. Department of Instrumentation, Mechatronics and Computerized Technologies, Cherkasy State Technological University, Cherkasy, Ukraine

    Constantine Bazilo

  7. Department of Software of Automated System, Cherkasy State Technological University, Cherkasy, Ukraine

    Grygoriy Zaspa

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Shcherba, A., Faure, E., Vartiainen, T., Khaliavka, V. (2024). Primitive Elements in the Finite Field of Square Matrices of Order 2 for Cryptographic Applications. In: Faure, E., et al. Information Technology for Education, Science, and Technics. ITEST 2024. Lecture Notes on Data Engineering and Communications Technologies, vol 222. Springer, Cham. https://doi.org/10.1007/978-3-031-71804-5_17

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  • DOI: https://doi.org/10.1007/978-3-031-71804-5_17

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