A Unified Method for Selecting Parameters and Primitive Elements in 2 × 2 Matrix Fields for Cryptographic Protocols
Baikenov A. Faure E. Shcherba A. Khaliavka V. Tynymbayev S. Abramkina O.
August 2025Multidisciplinary Digital Publishing Institute (MDPI)
Symmetry
2025#17Issue 8
This paper introduces a novel method for selecting parameters of finite fields formed by 2 × 2 matrices over a finite field of integers modulo a prime p. The method aims to simultaneously determine both the field parameters and primitive elements, thereby optimizing the construction of cryptographic algorithms. The proposed approach leverages the properties of quadratic residues and non-residues, simplifying the process of finding matrix field parameters while maintaining computational efficiency. The method is particularly effective when the prime number p is either a Mersenne prime or (p + 1)/2 is also a prime. This study demonstrates that the resulting matrix fields can be practically computed, offering a high degree of flexibility for cryptographic protocols such as key agreement and secure data transmission. Compared to previous methods, the new method reduces the parameter search space and provides a structured way to identify primitive elements without the need for a separate search procedure. The findings have significant implications for the development of efficient cryptographic systems using matrix-based finite fields.
cryptography , Galois field , key agreement protocol , matrix-based finite field , primitive elements , quadratic residues
Text of the article Перейти на текст статьи
Department of Telecommunications and Space Engineering, Almaty University of Power Engineering and Telecommunications Named After Gumarbek Daukeyev, Almaty, 050013, Kazakhstan
Research & Development Systems LLC, Karasai District, Pervomaysky Rural District, Bekbolat Village, 040911, Kazakhstan
Department of Information Security and Computer Engineering, Cherkasy State Technological University, Cherkasy, 18006, Ukraine
Department of Scientific and Technical Expertise, State Scientific and Research Institute of Cybersecurity Technologies and Information Protection, Kyiv, 03142, Ukraine
Department of Forensic Investigation and Duplication of Vehicle Identification Numbers, Cherkasy Research Forensic Centre of the Ministry of Internal Affairs of Ukraine, Cherkasy, 18023, Ukraine
Department of Computer Engineering, International University of Information Technology, Almaty, 050013, Kazakhstan
Department of Telecommunications and Space Engineering
Research & Development Systems LLC
Department of Information Security and Computer Engineering
Department of Scientific and Technical Expertise
Department of Forensic Investigation and Duplication of Vehicle Identification Numbers
Department of Computer Engineering
10 лет помогаем публиковать статьи Международный издатель
Книга Публикация научной статьи Волощук 2026 Book Publication of a scientific article 2026