Coding theory for h(x)-Fibonacci polynomials

Yıl 2024, Cilt: 26 Sayı: 1, 226 - 236, 19.01.2024

Öznur Öztunç Kaymak

https://doi.org/10.25092/baunfbed.1347379

Öz

The amount of information transmitted over the internet network has dramatically increased with the prevailing of internet use. As a result of this increase, the algorithms used in data encryption methods have gained importance. In this paper, h(x)-Fibonacci coding/decoding method for h(x)-Fibonacci polynomials is introduced. The proposed method is fast because it is based on basic matrix operations, and it is suitable for cryptographic applications because it uses the ASCII character encoding system. For this reason, it differs from the classical algebraic methods in literature. Furthermore, the fact that h(x) is a polynomial improves the security of cryptography.

Anahtar Kelimeler

Coding/decoding algorithm, h(x)-Fibonacci polynomials, Fibonacci Numbers

h(x)-Fibonacci polinomları için kodlama teorisi

Öz

İnternet kullanımı gün geçtikçe yaygınlaştıkça, internet ağları üzerinden geçen bilgi miktarı da kayda değer ölçüde artmaktadır. Bu artışın sonucu olarak, bilgi şifreleme metotları da kullanılan algoritmalar da önem kazanmaktadır. Bu çalışmada, h(x)-Fibonacci polinomları için h(x)-Fibonacci şifreleme metodu tanıtılmıştır. Önerilen yöntem, temel matris işlemlerine dayandığından hızlıdır ve ASCII karakter kodlama sistemini kullandığından kriptografik uygulamalara uygundur. Bu nedenle literatürdeki klasik cebirsel yöntemlerden farklılık göstermektedir. Ayrıca h(x)'in bir polinom olması kriptografinin güvenliğini artırır.

Anahtar Kelimeler

Şifreleme/Deşifreleme Algoritması, h(x)-Fibonacci polinomları, Fibonacci Sayıları

Kaynakça

• Lee, G., Asci, M., Some properties of the (p,q)-Fibonacci and (p,q)-Lucas polynomials, J. Appl. Math, 264842, (2012).
• Simsek, Y. Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 117, 3, 130, (2023).
• Zhang, C., Khan, W. A. and Kızılateş, C., On (p,q)–Fibonacci and (p,q)–Lucas Polynomials Associated with Changhee Numbers and Their Properties, Symmetry, 15(4), 851, (2023).
• Prasad, B., Coding theory on Lucas p -numbers, Discrete Math. Algorithms Appl., 17,8, no.4, 17 pages, (2016).
• Basu, M. and Prasad, B., The generalized relations among the code elements for Fibonacci coding theory, Chaos Solitons Fractals, 41, 5, 2517-2525, (2009).
• Koshy T., Fibonacci and Lucas Numbers with Applications, Toronto, New York, NY, USA, (2001).
• Nalli, A., Haukkanen, P., On generalized Fibonacci and Lucas polynomials, Chaos Solitons Fractals, 42, 5, 3179–3186, (2009).
• Catarino, P., A note on h(x)–Fibonacci quaternion polynomials. Chaos Solitons Fractals, 77, 1–5, (2015).
• Birol, F., Koruoğlu, Ö., Linear groups related to Fibonacci polynomials, Adv. Studies: Euro-Tbilisi Math. J., 15, 4, 29-40, (2022).
• Wang, W., Wang, H., Generalized-Humbert Polynomials via Generalized Fibonacci Polynomials, Applied Mathematics and Computation, 307, 204–216, (2017).
• Kızılateş, C., Cekim. B, Tuglu N and Kim T., New families of three-variable polynomials coupled with well-known polynomials and numbers, Symmetry, 264, 11, 1-13, (2019).
• Taş, N., Uçar, S., Özgür, N. Y. and Kaymak, Ö. Ö., A new coding/decoding algorithm using Fibonacci numbers, Discrete Math. Algorithms Appl., 10, 2, 1850028, (2018).
• Stakhov, A. P., Fibonacci matrices, a generalization of the Cassini formula and a new coding theory, Chaos Solitons Fractals, 30, 1, 56-66, (2006).
• Uçar, S.; Tas, N.; Özgür, N.Y. A new Application to coding Theory via Fibonacci numbers, Math. Sci. Appl. Notes, 7, 62–70, (2019).
• Basu, M., Prasad, B., Coding theory on the m-extension of the Fibonacci p- numbers, Chaos, Solitons & Fractals, 42, 4, 2522-2530, (2009).
• Prasad, B., Coding theory on (h(x),g(y))-extension of the Fibonacci p-numbers polynomials, Universal Journal of Computational Mathematics, 2, 1, 6-10, (2014).
• Prasad, B., Basu, M., Coding theory on h(x) Fibonacci p-numbers polynomials, Discrete Mathematics, Algorithms and Applications, 4, 3, 1250030, (2012).
• Stakhov, A.P., The ‘‘golden’’ matrices and a new kind of cryptography, Chaos, Solitons & Fractals, 32,1138–1146, (2007).
• Akbiyik, M., Alo, J., On Third-Order Bronze Fibonacci Numbers, Mathematics, 9, 20:2606, (2021).
• Aydınyuz, S., Asçi, M., Error detection and correction for coding theory on k -order Gaussian Fibonacci matrices, Mathematical Biosciences and Engineering, 20(2): 1993–2010, (2022).
• Asci, M., Aydınyuz, S., k -order Gaussian Fibonacci polynomials and applications to the coding/decoding theory, Journal of Discrete Mathematical Sciences and Cryptography, 25, 5, 1399-1416, (2022).
• Basu, M., Das, M., Coding theory on generalized Fibonacci n-step polynomials, Journal of Information and Optimization Sciences, 38, 1, (2017).
• ASCII, https://en.wikipedia.org/wiki/ASCII, (14.08.2023).
• Newline, https://en.wikipedia.org/wiki/Newline, (19.07.2023).
• Asci, M., Aydinyuz S., k-Order Fibonacci Polynomials on AES-Like Cryptology, CMES-Computer Modeling in Engineering & Sciences, 131(1), 277–293, (2022).
• Shaik, A., ASCII Binary Self Generated Key Encryption, The International Journal of Computer Science & Applications (TIJCSA), 1(3), 112-115, (2012).