Designing a new asymmetric encryption scheme based on a special group ring
*Bilel SelikhCorresponding authorselikh.bilel@ens-bousaada.dzbilel.selikh@univ-msila.dzDepartment of MathematicsLaboratoire de Mathématiques et Physique Appliquées, École Normale Supérieure de BousaâdaBousaâda, 28001, AlgeriaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Aug 2025
- Published Online:
- 02 Jun 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDMSC-2655
- Pages:
- 2873–2883
Abstract
In this work, we introduce a novel asymmetric encryption scheme founded on a non-commutative algebraic group defined over a specialized group ring, with the proposed system relying on the computational hardness of both the factorization with discrete logarithm problem (FDLP) and the conjugacy search problem (CSP), which holds a pivotal role in preserving the scheme’s security. Security analysis confirms that the scheme meets basic cryptographic standards, providing a good level of protection against known cryptographic attacks.
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References
[1] W. Diffie and M. Hellman, “New directions in cryptography,” IEEE Trans. Information Theory, vol. 22, no. 6, pp. 644-654 (1976).
[2] R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Communications of the ACM, vol. 21, no. 2, pp. 120-126 (1978).
[3] T. ElGamal, “A public key cryptosystem and a signature scheme based on discrete logarithms,” IEEE Trans. Information Theory, vol. 31, no. 4, pp. 469-472 (1985).
[4] R. Alvarez, F. Martinez, J. Vicent, and A. Zamora, “A new public key cryptosystem based on matrices,” in Proc. WSEAS Int. Conf. Information Security and Privacy, pp. 36-39 (2007).
[5] D. Kahrobaei, C. Koupparis, and V. Shpilrain, “Public key exchange using matrices over group rings,” Groups Complexity Cryptology, vol. 5, no. 1, pp. 97-115 (2013).
[6] S. Kanwal and R. Ali, “A cryptosystem with noncommutative platform groups,” Neural Computing and Applications, vol. 29, no. 11, pp. 1273-1278 (2018).
[7] C. M. Koupparis, Non-Commutative Cryptography: Diffie-Hellman and CCA-Secure Cryptosystems Using Matrices over Group Rings and Digital Signatures, Ph.D. dissertation, City University of New York, New York, NY, USA (2012).
[8] S. Makhlouf and K. Guenda, “A new public key cryptosystem based on group rings,” Advances in Mathematics: Scientific Journal, vol. 12, no. 2, pp. 357-366 (2023).
[9] A. J. Menezes and Y.H. Wu, “The discrete logarithm problem in GL(n,q),” Ars Combinatoria, vol. 47, pp. 23-32 (1997).
[10] G. Micheli, “Cryptanalysis of a non-commutative key exchange protocol,” Advances in Mathematics of Communications, vol. 9, no. 2, pp. 247-253 (2015).
[11] G. Mittal, S. Kumar, S. Narain, and S. Kumar, “Group ring based public key cryptosystems,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 25, no. 6, pp. 1683-1704 (2021).
[12] B. Selikh, On elliptic curves and application to cryptography, Ph.D. dissertation, Department of Mathematics, University of M’sila, Algeria (2022).
[13] B. Selikh, A. Chillali, D. Mihoubi, and N. Ghadbane, “A new public key cryptosystem based on the non-commutative ring R,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 27, no. 1, pp. 75-93 (2024).
[14] G. Mittal, S. Kumar, and S. Kumar, “Novel public-key cryptosystems based on NTRU and algebraic structure of group rings,” Journal of Information and Optimization Sciences, vol. 42, no. 7, pp. 1507-1521 (2021).
[15] Ö. Küsmüs and T. Hanoymak, “A novel public-key encryption scheme based on Bass cyclic units in integral group rings,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 25, no. 2, pp. 579-589 (2022).
[16] I. Gupta, A. Pandey, and M. K. Dubey, “A key exchange protocol using matrices over group rings,” Asian-European Journal of Mathematics, vol. 12, no. 5, pp. 1950075 (2019).
[17] M. Eftekhari, “A Diffie-Hellman key exchange protocol using matrices over non-commutative rings,” Groups Complexity Cryptology, vol. 4, no. 1, pp. 167-176 (2012).
[18] D. S. Passman, The Algebraic Structure of Group Rings. New York, NY, USA: Wiley (1977).
[19] A. Chatterjee and K. M. M. Aung, Fully Homomorphic Encryption in Real World Applications, Computer Architecture and Design Methodologies. Singapore: Springer Nature (2019).
[20] C. Boyd and A. Mathuria, Protocols for Authentication and Key Establishment, Information Security and Cryptography Series. Heidelberg, Germany: Springer-Verlag (2003).
