Random Number Generator and Secure Communication Applications Based on Infinitely Many Coexisting Chaotic Attractors

This paper aims to investigate a 3D chaotic system for applications on the secure communication system and random number generation. There existsa sinusoidal nonlinearity in the system making it uncommon of its type. Infinitely many chaotic attractors indicate multi-stability of the system, which isdesired; for instance, the same system can be implemented for a multiple channel secure communication and switching between the channels can beachieved just by changing initial conditions. A brief mathematical analysis of the system is performed, and the circuit of the system is designed usingactive circuit elements. A synchronized system for secure communication is mathematically analyzed on MATLAB and simulated on PSPICE OrCAD.Synchronization of the system with the proposed circuit structure shows that this dynamic system can be used for chaotic communication. In addition,as an application of cryptography, a NIST* statistical test is performed on 10 bitstreams generated by the system. The bitstream produced has successfullypassed all tests giving results in the length of the generated bit

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1. E. N. Lorenz, “Hvad er matematik?”, American Association for the Advancement of Science, 1972. 2. E. N. Lorenz, “Deterministic Nonperiodic Flow,” J Atmospheric Sci, vol. 20, pp. 130-141, 1963. [Crossref]

3. J. E. Skinner, M. Molnar, T. Vybiral, M. Mitra, “Application of Chaos Theory to Biology and Medicine”, Integr Physiol Behav Sci, vol. 27, no. 1, pp. 39-53, 1992. [Crossref]

4. D. Levy, “Chaos Theory and Strategy: Theory Application, and managerial Implication,” Strategic Manag J, vol. 15, pp. 167-178, 1994. [Crossref]

5. J. Trygestad, “Chaos in the Classroom: An Application of Chaos Theory,” Paper presented at the Annual Meeting of the American Educational Research Association, Chicago, 1997.

6. V. Akmansoy, S. Kartal, “Chaos Theory and its Application to Education: Mehmet Akif Ersoy University Case,” Educational Sciences: Theory & Practice, vol. 14, no. 2, pp. 510-518, 2014. [Crossref]

7. S. Ayers, “The Application of Chaos Theory to Psychology”, Theory & Psychology, vol. 7, no. 3, pp. 373-398, 1997. [Crossref]

8. D. Stapleton, J. B. Hanna, R. R. Jonathan, “Enhancing supply chain solutions with the application of chaos theory”, Supply Chain Management: An International J, vol. 11, no. 2, pp. 108-114, 2006. [Crossref]

9. A. Adewumi, J. Kagamba, A. Alochukwu, “Application of Chaos Theory in the Prediction of Motorised Traffic Flows on Urban Networks”, Mathematical Problems in Engineerin, vol. 2016, p. 15, 2016. [Crossref]

10. C. Frazier, K. M. Kockelman, “Chaos Theory and Transportation Systems”, J Transportation Res Board, vol. 1897, p. 9-17, 2004. [Crossref]

11. L. M. Pecora, T. L. Carroll, “Synchronization in Chaotic Systems,” Physical Rev Lett, vol. 64, no. 8, 1990. [Crossref]

12. E. Ott, C. Grebogi, J. A. Yorke, “Controlling Chaos”, Physical Rev Lett, vol. 64, no. 11, pp. 1196-1199, 1990. [Crossref]

13. K. Pyragas, “Continuous control of chaos by self-controlling feedback”, Physics Letters A, vol. 170, no. 6, pp. 421-428, 1992. [Crossref]

14. S. Troy, C. Grebogi, E. Ott, J. A. Yorke, “Using Small Perturbations to Control Chaos”, Nature, vol. 363, pp. 411-417, 1993. [Crossref]

15. S. Hayes, C. Greboi, E. Ott, A. Mark, “Experimental Control of Chaos for Communication”, Physical Rev Lett, vol. 73, no. 13, pp. 1781- 1784, 1994. [Crossref]

16. J. Lu, G. Chen, D. Cheng, “A New Chaotic System and Beyond: the Generalized Lorenz-like System”, Int J Bifurcation Chaos, vol. 14, no. 5, pp. 1507-1537, 2004. [Crossref]

17. V. Sundarapandian, I. Pehlivan, “Analysis, control, synchronization, and circuit design of a novel chaotic”, Mathematical and Computer Modelling, vol. 55, pp. 1904-1915, 2012. [Crossref]

18. Q. Lai, A. Akgul, C. Li, G. Xu, Ü. Çavusoglu, “A New Chaotic System with Multiple Attractors”, Entropy, vol. 2018, p. 15, 2017.

19. Y. B. Huang, S. H. Wang, Y. Wang, H. Li, “A New Four-Dimensional Chaotic System and Its Application in Speech Encryption”, In Information Communication Technologies Conference, 2020. [Crossref]

20. C. Zhu, Y. Liu, Y. Guo, “Theoretic and Numerical Study of a New Chaotic System”, Intelligent Information Management, vol. 2, pp. 104-109, 2010. [Crossref]

21. S. Vaidyanathan, A. Sambas, M. Mamat, W. Mada Sanjaya, “A new three-dimensional chaotic system with a hidden attractor, circuit design and application in wireless mobile robot”, Arch Control Sci, vol. 27(LXIII), no. 4, pp. 551-554, 2017. [Crossref]

22. Q. Lai, P. D. K. Kuate, F. Liu, H. H. C. Iu, “An Extremely Simple Chaotic System with infinitely many coexisting attractors”, IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 67, no.6, pp. 1129-1133, 2019. [Crossref]

23. M. Sharafi, F. Fotouhi-Ghazvini, M. Shirali, M. Ghassemian, “A Low Power Cryptography Solution Based on Chaos Theory in Wireless Sensor Nodes”, IEEE Access, vol. 7, pp. 8737-8753, 2019. [Crossref]

24. B. Triandi, E. Ekadiansyah, R. Puspasari, L. T. Iwan, F. Rahmad, “Improve Security Algorithm Cryptography Vigenere Cipher Using Chaos Functions”, in 6th International Conference on Cyber and IT Service Management (CITSM), Parapat, Indonesia, 2018. [Crossref]

25. Y. Liu, Z. Jiang, X. Xu, F. Zhang, J. Xu, “Optical image encryption algorithm based on hyper-chaos and public-key cryptography”, Optics & Laser Technology, vol. 127, no. 106171, 2020. [Crossref]

26. M. Lawnik, “Generalized logistic map and its application in chaos based cryptography,” in Journal of Physics: Conference Series, Volume 936, 6th International Conference on Mathematical Modelling in Physical Sciences (IC-MSQUARE 2017), Pafos, Cyprus, 2017. [Crossref]

27. P. Antonik, M. Gulina, J. Pauwels, S. Massar, “Using a reservoir computer to learn chaotic attractors, with applications to chaos synchronization and cryptography”, Phys Rev, vol. 98, no. 1, pp. 012215-012224, 2018. [Crossref]

28. R. Wang, P. Du, W. Zhong, H. Han, H. Sun, “Analyses and Encryption Implementation of a New Chaotic System”, Complexity, vol. 2020, 2020. [Crossref]

29. M. Irfan, A. Asim, M. A. K, M. Ehatisham-ul-Haq, S. N. M, A. Saboor, A. Waqar, “Pseudorandom Number Generator (PRNG) Design Using Hyper-Chaotic Modified Robust Logistic Map(HC-MRLM)”, Electronics, vol. 9, no. 104, 2020. [Crossref]

30. K. Demir, S. Ergün, “An Analysis of Deterministic Chaos as an Entropy”, Entropy, vol. 20, no. 957, 2018. [Crossref]

31. K. Demir, S. Ergün, “Security analysis of a random number generator based on a chaotic hyperjerk system”, EPL, vol. 129, 2020. [Crossref]

32. A. V. Tutueva, E. G. Nepomuceno, A. I. Karimov, V. S. Andreev, D. N. Butusov, “Adaptive chaotic maps and their application to pseudo-random”, Chaos, Solitons and Fractals, vol. 133, no. 109615, 2020. [Crossref]

33. P. Ayubi, S. Setayeshi, A. M. Rahman, “Deterministic chaos game: A new fractal based pseudo-random”, J Information Security App, vol. 52, no. 102472, 2020. [Crossref]

34. J. C. Sprott, W. J. Thio, “A Chaotic Circuit for Producing Gaussian”, Int J Bifurcation Chaos, vol. 30, no. 8, 2020. [Crossref]

35. L. M. Pecora, T. L. Carroll, “Synchronization of chaotic systems”, Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 25, no. 097611, 2015. [Crossref]

36. H. I. Deniz, Z. G. Cam Taskiran, H. Sedef, “Chaotic Lorenz Synchronization Circuit Design for Secure Communication,” in 2018 6th International Conference on Control Engineering & Information Technology (CEIT), Istanbul, 2018. [Crossref]

37. K. M. Cuomo, A. V. Oppenheim, S. H. Strogatz, “Synchronization of Lorenz-based chaotic circuits with applications to communications”, IEEE Transactionson Circuits and Systems II: Analog and Digital Signal Processing, vol. 40, no. 10, p. 626-633, 1993. [Crossref]

38. A. Sambas, W. S. Mada Sanjaya, M. Mamat and O. Tacha, “Design and numerical simulation of unidirectional chaotic synchronization and its application in secure communication system,” J Engineering Science and Technology Review, vol. 6, no. 4, p. 66-73, 2013.

39. A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, “A statistical test suite for random and pseudorandom number generators for cryptographic applications”, Tech Rep, Booz-Allen and Hamilton Inc Mclean Va, 2001. [Crossref]

40. A. N. Pisarchik, U. Feudel, “Control of multistability”, Physics Reports, vol. 540, no. 4, pp. 167-218, 2014. [Crossref]

41. R. C. Hilborn, Amanda, L. Cross, Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers, New York: Oxford University Press, 2004.

42. A. Wolf, J. B. Swıft, H. L. Swinney, J. A. Vastano, “Determining lyapunov exponents from a time series”, Physica, vol. 16, no. D, pp. 285-317, 1985. [Crossref]

43. A. Bedri Özer, E. Akın, “Tools for Detecting Chaos”, Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 9, no. 1, pp. 60-66, 2005.

44. Analog Devices, AD633 Low Cost Analog Multiplier, rev. K. 2015.

45. Analog Devices, AD844 60 MHz 2000 V/us Monolithic Op Amp, rev. F. 2009.

46. E. Tlelo-Cuautle, J. D. Díaz-Muñoz, A. M. González-Zapata, R. Li, W. D. León-Salas, F. V. Fernández, O. Guillén-Fernández, I. Cruz-Vega, “Chaotic image encryption using hopfield and hindmarsh-rose neurons implemented on FPGA”, Sensors, vol. 20, no. 5, pp. 1326, 2020. [Crossref]

47. R. Trejo-Guerra, E. Tlelo-Cuautle, J. M. Jimenez-Fuentes, C. Sánchez-López, J. M. Munoz-Pacheco, G. Espinosa-Flores-Verdad, J. M. Rocha-Perez, “Integrated circuit generating 3-and 5-scroll attractors”, Communications in Nonlinear Science and Numerical Simulation, vol. 17, no.11, pp. 4328-4335, 2012. [Crossref]

48. R. Trejo-Guerra, E. Tlelo-Cuautle, C. Cruz-Hernández, C. Sanchez-Lopez, “Chaotic communication system using Chua’s oscillators realized with CCII+ s”, Int J Bifurcation Chaos, vol. 19, no.12, pp. 4217-4226, 2009. [Crossref]

49. B. Muthuswamy, “Implementing memristor based chaotic circuits”, Int J Bifurcation Chaos, vol. 20, no. 05, pp. 1335-1350, 2010. [Crossref]