Research Article
Year 2024,
Volume: 26 Issue: 1, 226 - 236, 19.01.2024
Abstract
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.
References
- 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).
Year 2024,
Volume: 26 Issue: 1, 226 - 236, 19.01.2024
Abstract
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.
References
- 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).
There are 26 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Applied Mathematics (Other) |
| Journal Section | Research Articles |
| Authors | |
| Early Pub Date | January 6, 2024 |
| Publication Date | January 19, 2024 |
| Submission Date | August 21, 2023 |
| Published in Issue | Year 2024 Volume: 26 Issue: 1 |