SavedAdd your journal

Journals / Gazi University Journal of Science / 2026 / Volume: 38 Issue: 3

On Encryption by Stirling Polynomial Matrices

Hatice Kayhan(PAMUKKALE ÜNİVERSİTESİ)0009-0004-6165-7337
Serpil Halıcı(Pamukkale Üniversitesi)0000-0002-8071-0437
Pages
—
Publication Date
2025-09-01
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

In our study, the examination and verification of this algorithm are given with an application.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this article, by examining Stirling numbers, a new matrix type containing these numbers is defined and this matrix is called Stirling matrix.
📊
Key Findings & Data:Based on the fact that the use of matrices is very common and convenient in the field of encryption, we used the newly defined Stirling matrix to perform AES type encryption.
💡
Conclusion & Impact:In our study, the examination and verification of this algorithm are given with an application.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Stirling polynomials · Galois field · Coding- decoding

Abstract

In this article, by examining Stirling numbers, a new matrix type containing these numbers is defined and this matrix is called Stirling matrix. Based on the fact that the use of matrices is very common and convenient in the field of encryption, we used the newly defined Stirling matrix to perform AES type encryption. Inspired by the properties of integer sequences, a new recurrence relation that gives Stirling polynomials is defined. The Stirling polynomials used in our study are associated with k-order generalized matrices. LU decomposition of this generalized matrix is performed and examined. The L matrix obtained with the help of this decomposition is used in AES-like encryption methods. In our study, the examination and verification of this algorithm are given with an application.

🔖

Cite This Article

Kayhan, H., Halıcı, S., *, H. K. (2025). On Encryption by Stirling Polynomial Matrices. Gazi University Journal of Science, Volume: 38 Issue: 3. https://doi.org/10.35378/gujs.1602943

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (25)

  1. Graham, R. L., Knuth, D. E., Patashnik, O., “Concrete Math: A Foundation for Computer Science”, 2nd ed., Addison-Wesley, Massachusetts, 257-267, (1994).
  2. Stirling, J., Methodus Differentialis sive Tractatus de Summatione et Interpolatione Serierum Infinitarum, G. Strahan, London, 8-11, (1730).
  3. Pearson, K., “Historical note on the origin of the normal curve of errors,” Biometrika, 16(3/4): 402-404, (1924). DOI: https://doi.org/10.2307/2331714
  4. Dominici, D., Variations on a theme by James Stirling, arXiv preprint math/0603007, (2006).
  5. Abel, N. H., “Mémoire Sur Les Équations Algeébriques Où on Démontre L'impssibilité de la Résolution de L'equation Générale Du Cinquième Dégré” Christiania, Groendahl, University of Olso, (1824).
  6. Mongelli, P., “Total positivity properties of Jacobi–Stirling numbers,” Advances in Applied Mathematics, 48(2): 354-364, (2012). DOI: https://doi.org/10.1016/j.aam.2011.06.008
  7. Aziza, H. A., “Combinatoric sums and their applications in probability and statistics,” Akdeniz University (Master's thesis), Antalya, (2016).
  8. Cheon, G. S., Kim, J. S., “Stirling matrix via Pascal matrix,” Linear Algebra and its Applications, 329(1/3): 49-59, (2001). DOI: https://doi.org/10.1016/S0024-3795(01)00234-8
  9. Lee, G. Y., Kim, J. S., Cho, S. H., “Some combinatorial identities via Fibonacci numbers,” Discrete applied mathematics, 130(3): 527-534, (2003). DOI: https://doi.org/10.1016/S0166-218X(03)00331-7
  10. He, T. X., “Use Impulse Response Sequences in the Construction of Number Sequence Identities,” arXiv preprint arXiv:1303.7466, (2013).
  11. Boyadzhiev, K. N., “Close encounters with the Stirling numbers of the second kind,” Mathematics Magazine, 85(4): 252-266, (2012). DOI: https://doi.org/10.4169/math.mag.85.4.252
  12. Butzer, P. L., Markett, C., Schmidt, M., “Stirling numbers, central factorial numbers, and representations of the Riemann zeta function,” Results in mathematics, 19(3): 257-274, (1991). DOI: https://doi.org/10.1007/BF03323285
  13. Dere, R., “Generating Functions and Applications of Some Special Polynomials on Umbral Algebra,” Akdeniz University (Master's thesis), Antalya, (2011).
  14. Yıldız, Y., “Stirling Polynomial Families,” Akdeniz University (Master's thesis), Antalya, (2022).
  15. Nesin, A., “Stirling Sayıları,” Sayma-Kombinasyon Hesapları (6. Baskı), Nesin Yayıncılık, İstanbul, 155-166, (2019).
  16. Hetyei, G., “The Stirling Polynomial of a Simplicial Complex,” Discrete & Computational Geometry, 35(3): 437-455, (2006). DOI: https://doi.org/10.1007/s00454-005-1190-2
  17. Massey, J. L., Costello, D. J., Justesen, J., “Polynomial weights and code constructions,” IEEE Transactions on Information Theory, 19(1): 101-110, (1973). DOI: 10.1109/TIT.1973.1054936
  18. Başar, Ü., “Merkezi Faktöriyel Sayılarının Üreteç Fonksiyonları ve Uygulamaları,” Akdeniz University (Master's thesis), Antalya, (2016).
  19. Halıcı, S., Koca, N., “On Excellent Safe Primary Numbers and Encryption,” In Conference Proceedings of Science and Technology, 3(2): 247-251, (2020).
  20. Klima, R. E., Sigmon, N. P., “Cryptology: Classical and modern with maplets,” 2nd ed., New York: Chapman and Hall/CRC, (2012).
  21. Avaroglu, E., Koyuncu, I., Ozer, A. B., M. Turk, “Hybrid pseudo-random number generator for cryptographic systems,” Nonlinear Dynamics, 82(1/2): 239–248, (2015). DOI: https://doi.org/10.1007/s11071-015-2152-8
  22. Kadıoğlu, E., “Cisim Genişlemeleri ve Galois Grupları,” Atatürk University (Master's thesis), Erzurum, (1986).
  23. Paar, C., Pelzl, J., “Understanding cryptography: A textbook for students and practitioners,” London: Springer Science, Business Media, Berlin, Heidelberg, (2009).
  24. Pehlivan, M.K., Duru, N., Sakallı, M. T., “Sonlu Cisimler Teorisine Dayalı Gri Seviye Görüntü Şifreleme,” Bilecik Şeyh Edebali University Journal of Science, 3(2): 10-17, (2016).
  25. Diskaya, O., Avaroglu, E., Menken, H., “The classical AES-like cryptology via the Fibonacci polynomial matrix,” Turkish Journal of Engineering, 4(3): 123–128, (2020). DOI: 10.31127/tuje.646926