Randomness Analysis With Runge Kutta Methods

Öz

Chaotic systems are widely used in encryption because of their sensitivity to initial conditions and parameters, high ergodicity, mixing properties, and highly complex structures. Various analyzes are available to understand whether a system is chaotic. The most used analyzes are time series analysis, phase portraits, Lyapunov exponents, and bifurcation diagrams. Modeling of chaotic systems is also possible with numerical analysis methods. These methods are; Houses, Heun, 4th and 5th degree Runge Kutta methods are the most common differential solution methods.

Anahtar Kelimeler

• Random Number Generators
• Runge Kutta
• Chaotic Maps
• Floating Point Numbers

Runge Kutta Yöntemleriyle Rasgelelik Analizi

Öz

Kaotik sistemler, başlangıç koşullarına ve parametrelere olan duyarlılıkları, yüksek ergodikliği, karıştırma özellikleri ve oldukça karmaşık yapıları nedeniyle şifrelemede yaygın olarak kullanılmaktadır. Bir sistemin kaotik olup olmadığını anlamak için çeşitli analizler mevcuttur. En çok kullanılan analizler zaman serisi analizi, faz portreleri, Lyapunov üsleri ve çatallanma diyagramlarıdır. Kaotik sistemlerin modellenmesi sayısal analiz yöntemleri ile de mümkündür. Bu yöntemler; Houses, Heun, 4. ve 5. derece Runge Kutta yöntemleri en yaygın diferansiyel çözüm yöntemleridir.

Anahtar Kelimeler

• Rasgele Sayı Üreteçleri
• Runge Kutta
• Kaotik Haritalar
• Kayan Noktalı Sayılar

Kaynakça

1. [1] E. M. Esin, “Knutt / Durstenfeld Shuffle Algoritmasının Resim Şifreleme Amacıyla Kullanılması,” Politek. Derg., vol. 12, no. 3, pp. 151–155, 2009, doi: 10.2339/2009.12.3.
2. [2] A. Akgül, M. Z. Yıldız, Ö. F. Boyraz, E. Güleryüz, S. Kaçar, and B. Gürevin, “Microcomputer-based encryption of vein images with a non-linear novel system,” J. Fac. Eng. Archit. Gazi Univ., vol. 35, no. 3, pp. 1369–1385, 2020, doi: 10.17341/GaziMfd.558379.
3. [3] H. Li, L. Deng, and Z. Gu, “A Robust Image Encryption Algorithm Based on a 32-bit Chaotic System,” IEEE Access, vol. 8, pp. 30127–30151, 2020, doi: 10.1109/ACCESS.2020.2972296.
4. [4] S. Chen, X. X. Zhong, and Z. Z. Wu, “Chaos block cipher for wireless sensor network,” Sci. China, Ser. F Inf. Sci., vol. 51, no. 8, pp. 1055–1063, 2008, doi: 10.1007/s11432-008-0102-5.
5. [5] X. Tong, Z. Wang, Y. Liu, M. Zhang, and L. Xu, “A novel compound chaotic block cipher for wireless sensor networks,” Commun. Nonlinear Sci. Numer. Simul., vol. 22, pp. 120–133, 2015.
6. [6] Z. Liu et al., “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun., vol. 284, no. 1, pp. 123–128, 2011, doi: 10.1016/j.optcom.2010.09.013.
7. [7] Y. Zhang, “The unified image encryption algorithm based on chaos and cubic S-Box,” Inf. Sci. (Ny)., vol. 450, pp. 361–377, 2018, doi: 10.1016/j.ins.2018.03.055.
8. [8] R. U. Ginting and R. Y. Dillak, “Digital color image encryption using RC4 stream cipher and chaotic logistic map,” 2013 Int. Conf. Inf. Technol. Electr. Eng., pp. 101–105, 2013.

9. [9] A. Jolfaei and A. Mirghadri, “Image Encryption Using Chaos and Block Cipher,” Comput. Inf. Sci., vol. 4, no. 1, pp. 172–185, 2010, doi: 10.5539/cis.v4n1p172.
10. [10] M. Khan, T. Shah, and S. I. Batool, “Construction of S-box based on chaotic Boolean functions and its application in image encryption,” Neural Comput. Appl., vol. 27, no. 3, pp. 677–685, 2016, doi: 10.1007/s00521-015-1887-y.
11. [11] Y. Liu, X. Tong, and J. Ma, “Image encryption algorithm based on hyper-chaotic system and dynamic S-box,” Multimed. Tools Appl., vol. 75, no. 13, pp. 7739–7759, 2016, doi: 10.1007/s11042-015-2691-5.
12. [12] I. Cicek, A. E. Pusane, and G. Dundar, “A novel design method for discrete time chaos based true random number generators,” Integr. VLSI J., vol. 47, no. 1, pp. 38–47, 2014, doi: 10.1016/j.vlsi.2013.06.003.
13. [13] K. I. Farhana Sheikh Leonel Sousa, Ed., “Circuits and Systems for Security and Privacy” .
14. [14] A. Akgul and I. Pehlivan, “A New Three-Dimensional Chaotic System Without Equilibrium Points, Its Dynamical Analyses and Electronic Circuit Application,” Teh. Vjesn., vol. 23, pp. 209–214, 2016, doi: 10.17559/TV-20141212125942.
15. [15] N. Munir, M. Khan, T. Shah, A. S. Alanazi, and I. Hussain, “Cryptanalysis of nonlinear confusion component based encryption algorithm,” Integration, vol. 79, no. February, pp. 41–47, 2021, doi: 10.1016/j.vlsi.2021.03.004.
16. [16] H. G. Mohamed, D. H. Elkamchouchi, and K. H. Moussa, “A novel color image encryption algorithm based on hyperchaotic maps and mitochondrial DNA sequences,” Entropy, vol. 22, no. 2, pp. 7279–7297, 2020, doi: 10.3390/e22020158.