Designing a Pseudo-Random Bit Generator Using Generalized Cascade Fractal Function

Yıl 2021, Cilt: 3 Sayı: 1, 11 - 19, 30.06.2021

Shafali Agarwal

https://doi.org/10.51537/chaos.835222

Cited By: 6

Öz

A cascade function is designed by combining two seed maps that resultantly has more parameters, high complexity, randomness, and more unpredictable behavior. In the paper, a cascade fractal function, i.e. cascade-PLMS is proposed by considering the phoenix and lambda fractal functions. The constructed cascade-PLMS exhibits the required fractal features such as fractional dimension, self-similar structure, and covering entire phase space by the data sequence in addition to the chaotic properties. Due to the chaotic behavior, the proposed function is utilized to generate a pseudo-random number sequence in both integer and binary format. This is the result of an extreme scalability feature of a fractal function that can be implemented on a large scale. A sequence generator is designed by performing the linear function operation to the real and imaginary part of a cascade-PLMS, cascade-PLJS separately, and the iteration number at which the cascade-PLJS converges to the fixed point. The performance analysis results show that the given method has a large keyspace, fast key generation speed, high key sensitivity, and strong randomness. Therefore, the scheme can be efficiently used further to design a secure cryptosystem with the ability to withstand various attacks.

Anahtar Kelimeler

Cascade phoenix lambda fractal, PRNG, Mandelbrot set, dynamic behavior, key security analysis.

Kaynakça

• Agarwal, S. (2020). A New Composite Fractal Function and Its Application in Image Encryption. Journal of Imaging, 6(7), 70. https://doi.org/10.3390/jimaging6070070
• Alvarez, G., & Li, S. (2006). SOME BASIC CRYPTOGRAPHIC REQUIREMENTS FOR CHAOS-BASED CRYPTOSYSTEMS. International Journal of Bifurcation and Chaos, 16(08), 2129–2151. https://doi.org/10.1142/S0218127406015970
• Artuğer, F., & Özkaynak, F. (2020). A Novel Method for Performance Improvement of Chaos-Based Substitution Boxes. Symmetry, 12(4), 571. https://doi.org/10.3390/sym12040571
• Ayubi, P., Setayeshi, S., & Rahmani, A. M. (2020). Deterministic chaos game: A new fractal based pseudo-random number generator and its cryptographic application. Journal of Information Security and Applications, 52, 102472. https://doi.org/10.1016/j.jisa.2020.102472
• Bai, S., Zhou, L., Yan, M., Ji, X., & Tao, X. (2020). Image Cryptosystem for Visually Meaningful Encryption Based on Fractal Graph Generating. IETE Technical Review, 0(0), 1–12. https://doi.org/10.1080/02564602.2020.1799875
• Bonilla, L. L., Alvaro, M., & Carretero, M. (2016). Chaos-based true random number generators. Journal of Mathematics in Industry, 7(1), 1. https://doi.org/10.1186/s13362-016-0026-4
• Chen, G., & Ueta, T. (1999). Yet another chaotic attractor. International Journal of Bifurcation and Chaos, 09(07), 1465–1466. https://doi.org/10.1142/S0218127499001024
• Çi̇men, M. E., Gari̇p, Z., Boyraz, Ö. F., Pehli̇van, İ., Yildiz, M. Z., & Boz, A. F. (2020). An Interface Design for Calculation of Fractal Dimension. Chaos Theory and Applications, 2(1), 3–9.
• Czyz, J. (1994). The hausdorff measures, hausdorff dimensions and fractals. In Paradoxes of Measures and Dimensions Originating in Felix Hausdorff’s Ideas (Vol. 1–0, pp. 219–413). WORLD SCIENTIFIC. https://doi.org/10.1142/9789814368193_0004
• Devaney, R., & Devaney, R. L. (2003). An Introduction to Chaotic Dynamical Systems, 2nd Edition (2 edition). CRC Press.
• Devaney, R., & Keen, L. (Eds.). (1989). Chaos and Fractals: The Mathematics Behind the Computer Graphics (Vol. 39). American Mathematical Society. https://doi.org/10.1090/psapm/039
• Hamza, R. (2017). A novel pseudo random sequence generator for image-cryptographic applications. Journal of Information Security and Applications, 35, 119–127. https://doi.org/10.1016/j.jisa.2017.06.005
• Hua, Z., Jin, F., Xu, B., & Huang, H. (2018). 2D Logistic-Sine-coupling map for image encryption. Signal Processing, 149, 148–161. https://doi.org/10.1016/j.sigpro.2018.03.010
• Jafari Barani, M., Ayubi, P., Yousefi Valandar, M., & Irani, B. Y. (2020). A new Pseudo random number generator based on generalized Newton complex map with dynamic key. Journal of Information Security and Applications, 53, 102509. https://doi.org/10.1016/j.jisa.2020.102509
• Khelaifi, F., & He, H. (2020). Perceptual image hashing based on structural fractal features of image coding and ring partition. Multimedia Tools and Applications, 79(27), 19025–19044. https://doi.org/10.1007/s11042-020-08619-w
• Lynnyk, V., Sakamoto, N., & Čelikovský, S. (2015). Pseudo random number generator based on the generalized Lorenz chaotic system. IFAC-PapersOnLine, 48(18), 257–261. https://doi.org/10.1016/j.ifacol.2015.11.046
• Mandelbrot, B. B. (1982). The Fractal Geometry of Nature (2nd prt. edition). Times Books.
• Motýl, I., & Jašek, R. (n.d.). Advanced user authentication process based on the principles of fractal geometry. Recent Advances in Signal Processing, 4.
• Moysis, L., Tutueva, A., Volos, C., Butusov, D., Munoz-Pacheco, J. M., & Nistazakis, H. (2020). A Two-Parameter Modified Logistic Map and Its Application to Random Bit Generation. Symmetry, 12(5), 829. https://doi.org/10.3390/sym12050829
• Moysi̇s, L., Tutueva, A., Volos, C. K., & Butusov, D. (2020). A Chaos Based Pseudo-Random Bit Generator Using Multiple Digits Comparison. Chaos Theory and Applications, 2(2), 58–68.
• Nayak, S. R., & Mishra, J. (2019). Analysis of Medical Images Using Fractal Geometry [Chapter]. Histopathological Image Analysis in Medical Decision Making; IGI Global. https://doi.org/10.4018/978-1-5225-6316-7.ch008
• Peitgen, H.-O., Jürgens, H., & Saupe, D. (1992). Chaos and Fractals: New Frontiers of Science. Springer-Verlag. https://doi.org/10.1007/978-1-4757-4740-9
• Sahari, M. L., & Boukemara, I. (2018). A pseudo-random numbers generator based on a novel 3D chaotic map with an application to color image encryption. Nonlinear Dynamics, 94(1), 723–744. https://doi.org/10.1007/s11071-018-4390-z
• Shannon, C. E. (1949). Communication Theory of Secrecy Systems*. Bell System Technical Journal, 28(4), 656–715. https://doi.org/10.1002/j.1538-7305.1949.tb00928.x
• Stallings, W. (2006). Cryptography and network security: Principles and practice (4th ed). Pearson/Prentice Hall.
• Takens, F. (1988). An introduction to chaotic dynamical systems. Acta Applicandae Mathematica, 13(1), 221–226. https://doi.org/10.1007/BF00047506
• Wang, L., & Cheng, H. (2019). Pseudo-random number generator based on logistic chaotic system. Entropy, 21(10), 960.
• Zhao, Y., Gao, C., Liu, J., & Dong, S. (2019). A self-perturbed pseudo-random sequence generator based on hyperchaos. Chaos, Solitons & Fractals: X, 4, 100023. https://doi.org/10.1016/j.csfx.2020.100023
• Zhou, Y., Hua, Z., Pun, C.-M., & Philip Chen, C. L. (2015). Cascade Chaotic System With Applications. IEEE Transactions on Cybernetics, 45(9), 2001–2012. https://doi.org/10.1109/TCYB.2014.2363168