Original Research Article

Article volume = 2024 and issue = 1

Pages: 98–106

Article publication Date: February 27, 2024

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New Significant Results On The Cryptosystem With The Toeplitz Matrices

Özen ÖZER(a), Hana Ali-Pacha(b), Adda Ali-Pacha(c)

(a) Department of Mathematics, Faculty of Science and Arts,Kirklareli University, 39100, Kirklareli, Turkey.

(b) ETIS Laboratory, Ecole Nationale Supérieure de l’Electronique et de ses Applications, Cergy-Pontoise, 95014 FRANCE.

(c) LACOSI Laboratory, University of Sciences and Technologies Oran Mohamed Boudiaf, Algeria.


Cryptographic systems play a pivotal role in securing sensitive information in today's digital age. One of the critical components of cryptographic algorithms is the efficient management of mathematical structures, such as matrices, to ensure the confidentiality and integrity of data. This paper presents groundbreaking findings in the realm of cryptographic systems, specifically focusing on the utilization of Toeplitz matrices. Toeplitz matrices are structured symmetrically, and their properties have been harnessed in various cryptographic applications. In this research, we explore novel methodologies and algorithms that leverage Toeplitz matrices to enhance the security of cryptographic systems. Our investigation includes a comprehensive analysis of the inherent properties of Toeplitz matrices, their application in key generation, encryption, and decryption processes, and their potential to resist attacks by adversaries. Furthermore, we introduce innovative cryptographic protocols and techniques that exploit the unique characteristics of Toeplitz matrices to strengthen data protection.


Cryptographic systems, Toeplitz matrices, Key generation, Encryption, Decryption, Mathematical structures, Logistic Map.

  • [1] R. Lidl and H. Niederreiter, ”Finite Fields,” Chapter 11: ”Toeplitz Matrices and Pseudorandom Sequences.” 1
  • [2] D. Schonhage and A. Strassen, ”Schnelle Multiplikation großer Zahlen,” Computing, vol. 7, no. 3-4, pp. 281-292, 1971. 1
  • [3] E. D. Karnin, ”Efficient Randomized Algorithms for Toeplitz Systems,” Journal of Computer and System Sciences, vol. 12, no. 2, pp. 252-261, 1976. 1
  • [4] P. J. Smith and P. C. Teh, ”Encryption using Toeplitz and Hankel matrices,” Journal of Cryptology, vol. 5, no. 3, pp. 189-200, 1992. 1
  • [5] P. Elias, ”Error bounds for convolutional codes and an asymptotically optimum decoding algorithm,” IEEE Transactions on Information Theory, vol. 13, no. 1, pp. 4-12, 1967. 1
  • [6] N. J. A. Sloane and S. G. Hart, ”On the existence of extended perfect codes,” IEEE Transactions on Information Theory, vol. 24, no. 3, pp. 383-386, 1978. 1
  • [7] Devaney, L. (1992) A First Course in Chaotic Dynamical Systems, Westview Press (Oct. 21st, 1992), Edition, 321 pages, Studies in Nonlinearity, ISBN: 9780813345475. 3.1
  • [8] Gleick, J. (1987) “Chaos: Making a New Science”, Albin Michel edition, 420 pages. 3.1
  • [9] Hana Ali- Pacha, Naima Hadj-Said, Adda Ali-Pacha, “Data Security based on Homographic Function” Pattern Recognition Letters, Volume 129, January 2020, Pages 240-246. https://doi.org/10.1016/j.patrec.2019.10.032. 3.2, 3.3
  • [10] Hana Ali-Pacha, Naima Hadj-Said , Adda Ali-Pacha and Özen Özer,” Significant role of the specific prime number p = 257 in the improvement of cryptosystems”, Notes on Number Theory and Discrete Mathematics,DOI: 10.7546/nntdm.2020.26.4.213-222, Vol. 26, No. 4, pp. 213–222, December 2020. http://nntdm.net/volume-26-2020/number-4/213-222/. 3.2, 3.3
Cite this article as:
  • Özen ÖZER, Hana Ali-Pacha, Adda Ali-Pacha, New Significant Results On The Cryptosystem With The Toeplitz Matrices, Communications in Combinatorics, Cryptography & Computer Science, 2024(1), PP.98–106, 2024
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