Lyapunov Exponent Enhancement in Chaotic Maps with Uniform Distribution Modulo One Transformation

Abstract

Most of the chaotic maps are not suitable for chaos-based cryptosystems due to their narrow chaotic parameter range and lacking of strong unpredictability. This work presents a nonlinear transformation approach for Lyapunov exponent enhancement and robust chaotification in discrete-time chaotic systems for generating highly independent and uniformly distributed random chaotic sequences. The outcome of the new chaotic systems can directly be used in random number and random bit generators without any post-processing algorithms for various information technology applications. The proposed Lyapunov exponent enhancement based chaotic maps are analyzed with Lyapunov exponents, bifurcation diagrams, entropy, correlation and some other statistical tests. The results show that excellent random features can be accomplished even with one-dimensional chaotic maps with the proposed approach.

Keywords

• chaos
• Lyapunov exponent
• Random numbers
• Cryptography
• image encryption

References

1. Ablay, G., 2016 Chaotic map construction from common nonlinearities and microcontroller implementations. International Journal of Bifurcation and Chaos 26: 1650121.
2. Asgari-Chenaghlu, M., M.-A. Balafar, and M.-R. Feizi-Derakhshi, 2019 A novel image encryption algorithm based on polynomial combination of chaotic maps and dynamic function generation. Signal Processing 157: 1–13.
3. Awrejcewicz, J., A. V. Krysko, N. P. Erofeev, V. Dobriyan, M. A. Barulina, et al., 2018 Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems. Entropy 20: 175.
4. Banerjee, S., L. Rondoni, and M. Mitra, editors, 2012 Applications of Chaos and Nonlinear Dynamics in Science and Engineering - Vol. 2. Springer-Verlag, Berlin Heidelberg.
5. Bassham, L. E., A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, et al., 2010 A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications. Technical report, National Institute of Standards & Technology, Gaithersburg, MD, United States.
6. Benamara, O., F. Merazka, and K. Betina, 2016 An improvement of a cryptanalysis algorithm. Information Processing Letters 116: 192–196.
7. Dekking, F. M., C. Kraaikamp, H. P. Lopuhaä, and L. E. Meester, 2005 A Modern Introduction to Probability and Statistics: Understanding Why and How. Springer-Verlag, London.
8. Dorfman, J. R., 1999 An Introduction to Chaos in Nonequilibrium Statistical Mechanics. Cambridge Lecture Notes in Physics, Cambridge University Press, Cambridge.

9. El-Hameed, H. A. A., N. Ramadan, W. El-Shafai, A. A. M. Khalaf, H. E. H. Ahmed, et al., 2021 Cancelable biometric security system based on advanced chaotic maps. The Visual Computer .
10. Falniowski, F., 2014 On the Connections of Generalized Entropies With Shannon and Kolmogorov–Sinai Entropies. Entropy 16.
11. Farajallah, M., S. El Assad, and O. Deforges, 2016 Fast and secure chaos-based cryptosystem for images. International Journal of Bifurcation and Chaos 26: 1650021(1–21).
12. Garasym, O., I. Taralova, and R. Lozi, 2016 New Nonlinear CPRNG Based on Tent and Logistic Maps. In Complex Systems and Networks: Dynamics, Controls and Applications, edited by J. Lü, X. Yu, G. Chen, and W. Yu, Understanding Complex Systems, pp. 131–161, Springer, Berlin, Heidelberg.
13. Hamza, R., 2017 A novel pseudo random sequence generator for image-cryptographic applications. Journal of Information Security and Applications 35: 119–127.
14. Hu, G. and B. Li, 2021 Coupling chaotic system based on unit transform and its applications in image encryption. Signal Processing 178: 107790.
15. Hua, Z., Y. Zhang, and Y. Zhou, 2020 Two-Dimensional Modular Chaotification System for Improving Chaos Complexity. IEEE Transactions on Signal Processing 68: 1937–1949.
16. Hua, Z., B. Zhou, and Y. Zhou, 2019a Sine Chaotification Model for Enhancing Chaos and Its Hardware Implementation. IEEE Transactions on Industrial Electronics 66: 1273–1284.
17. Hua, Z., Y. Zhou, and H. Huang, 2019b Cosine-transform-based chaotic system for image encryption. Information Sciences 480: 403–419.
18. Jafari Barani, M., P. Ayubi, M. Yousefi Valandar, and B. Y. Irani, 2020 A new Pseudo random number generator based on generalized Newton complex map with dynamic key. Journal of Information Security and Applications 53: 102509.
19. James, F., 2006 Statistical Methods In Experimental Physics. World Scientific, Hackensack, NJ, second edition.
20. Karmeshu and N. R. Pal, 2003 Uncertainty, Entropy and Maximum Entropy Principle — An Overview. In Entropy Measures, Maximum Entropy Principle and Emerging Applications, edited by Karmeshu, Studies in Fuzziness and Soft Computing, pp. 1–53, Springer, Berlin, Heidelberg.
21. Khan, J. S. and S. K. Kayhan, 2021 Chaos and compressive sensing based novel image encryption scheme. Journal of Information Security and Applications 58: 102711.
22. Lan, R., J. He, S. Wang, T. Gu, and X. Luo, 2018 Integrated chaotic systems for image encryption. Signal Processing 147: 133–145.
23. Liu, L., S. Miao, M. Cheng, and X. Gao, 2016 A pseudorandom bit generator based on new multi-delayed Chebyshev map. Information Processing Letters 116: 674–681.
24. Luo, Y., S. Zhang, J. Liu, and L. Cao, 2020 Cryptanalysis of a Chaotic Block Cryptographic System Against Template Attacks. International Journal of Bifurcation and Chaos 30: 2050223.
25. Murillo-Escobar, M. A., C. Cruz-Hernández, L. Cardoza-Avendaño, and R. Méndez-Ramírez, 2017 A novel pseudorandom number generator based on pseudorandomly enhanced logistic map. Nonlinear Dynamics 87: 407–425.
26. Pak, C. and L. Huang, 2017 A new color image encryption using combination of the 1D chaotic map. Signal Processing 138: 129–137.
27. Parvaz, R. and M. Zarebnia, 2018 A combination chaotic system and application in color image encryption. Optics & Laser Technology 101: 30–41.
28. Pikovsky, A. and A. Politi, 2016 Lyapunov Exponents: A Tool to Explore Complex Dynamics. Cambridge University Press, Cambridge.
29. Pulido-Luna, J. R., J. A. López-Rentería, N. R. Cazarez-Castro, and E. Campos, 2021 A two-directional grid multiscroll hidden attractor based on piecewise linear system and its application in pseudo-random bit generator. Integration 81: 34–42.
30. Ruelle, D., 1997 Chaos, predictability, and idealization in physics. Complexity 3: 26–28.
31. Stallings, W., 2006 Cryptography and Network Security: Principles and Practice. Prentice Hall.
32. Strogatz, S. H., 2015 Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press, Boulder, CO, second edition.
33. Talhaoui, M. Z., X. Wang, and M. A. Midoun, 2021 A new onedimensional cosine polynomial chaotic map and its use in image encryption. The Visual Computer 37: 541–551.
34. Vallejo, J. C. and M. A. F. Sanjuán, 2019 Predictability of Chaotic Dynamics : A Finite-time Lyapunov Exponents Approach. Springer Series in Synergetics, Springer International Publishing, Switzerland, second edition.
35. Wang, X. and P. Liu, 2021 Image encryption based on roulette cascaded chaotic system and alienated image library. The Visual Computer.
36. Xiang, H. and L. Liu, 2020 An improved digital logistic map and its application in image encryption. Multimedia Tools and Applications 79: 30329–30355.
37. Zahmoul, R., R. Ejbali, and M. Zaied, 2017 Image encryption based on new Beta chaotic maps. Optics and Lasers in Engineering 96: 39–49.
38. Zhou, Y., L. Bao, and C. L. P. Chen, 2014 A new 1D chaotic system for image encryption. Signal Processing 97: 172–182.