Methods of construction and properties of logariphmic signatures


  • E.V. Kotukh
  • O.V. Severinov
  • A.V. Vlasov
  • L.S. Kozina
  • A.O. Tenytska
  • E.O. Zarudna



post-quantum cryptography, logarithmic signatures, coverings, non-abelian groups


Development and promising areas of research in the construction of practical models of quantum computers contributes to the search and development of effective cryptographic primitives. Along with the growth of the practical possibilities of using quantum computing, the threat to classical encryption and electronic signature schemes using classical mathematical problems as a basis, being overcome by the computational capabilities of quantum computers. This fact motivates the study of fundamental theorems concerning the mathematical and computational aspects of candidate post-quantum cryptosystems. Development of a new quantum-resistant asymmetric cryptosystem is one of the urgent problems. The use of logarithmic signatures and coverings of finite groups a promising direction in the development of asymmetric cryptosystems. The current state of this area and the work of recent years suggest that the problem of factorizing an element of a finite group in the theory of constructing cryptosystems based on non-Abelian groups using logarithmic signatures is computationally complex; it potentially provides the necessary level of cryptographic protection against attacks using the capabilities of quantum calculations. The paper presents logarithmic signatures as a special type of factorization in finite groups; it also considers their properties and construction methods.


Gonzáles Vasco M. I. On minimal length factorizations of _nite groups / M. I. Gonzáles Vasco, M. Rotteler, R. Steinwandt // Experimental Mathematics. 2003. Vol. 12 (1). P. 112.

Singhi N. Minimal logarithmic signatures for _nite groups of Lie type / N. Singhi, N. Singhi, S. Magliveras // Designs, Codes and Cryptography. 2010. Vol. 55 (2). P. 243260.

Magliveras S. New approaches to designing public key cryptosystems using one-way functions and trap-doors in nite groups / S. Magliveras, D. Stinson, T. van Trung // Journal of Cryptology. 2002. Vol. 15. P. 285297.

Lempken W. A public key cryptosystem based on non-abelian _nite groups / W. Lempken, T. van Trung, S.S. Magliveras, W. Wei // Journal of Cryptology. 2009. Vol. 22 (1). P. 6274.

Goldreich O. Foundations of Cryptography: Basic Tools / O. Goldreich // Cambridge University Press. 2001.

Nuss A. On group based public key cryptography [Electronic resource] : Phd thesis. Access mode :

Blackburn S. R. Cryptanalysis of the MST 3 public key cryptosystem / S. R. Blackburn, C. Cid, C. Mullan // Journal of Mathematical Cryptology. 2009. Vol. 3 (4). P. 321338.

Bohli J. Weak keys in MST / J. Bohli, M. I. Gonzáles Vasco, C. J. M. Martínez, R. Steinwandt // Designs, Codes and Cryptography. 2005. Vol. 37 (3). P. 509524.

Caranti A. The round functions of cryptosystem PGM generate the symmetric group / A. Caranti, F. D. Volta // Designs, Codes and Cryptography. 2006. Vol. 38 (1). P. 147155.

Magliveras S. Algebraic Properties of Cryptosystem PGM / S. Magliveras, N. D. Memon // Journal of Cryptology. 1992. Vol. 5 (3). P. 167183.

Khalimov G. MST3 Cryptosystem Based on a Generalized Suzuki 2-Groups [Electronic resource] / G. Khalimov, Y. Kotukh, S. Khalimova. Access mode :

Khalimov G., Kotukh Y., Khalimova S. MST3 cryptosystem based on the automorphism group of the hermitian function field' // IEEE International Scientific-Practical Conference: Problems of Infocommunications Science and Technology, PIC S and T 2019 – Proceedings, 2019, pр. 865 – 868.

Khalimov G., Kotukh Y., Khalimova S. Encryption scheme based on the automorphism group of the Ree function field // 2020 7th International Conference on Internet of Things: Systems, Management and Security, IOTSMS 2020, 2020, 9340192.



How to Cite

Kotukh, E. ., Severinov, O. ., Vlasov, A. ., Kozina, L. ., Tenytska, A. ., & Zarudna , E. . (2021). Methods of construction and properties of logariphmic signatures . Radiotekhnika, 2(205), 94–99.