Abstract
Despite sharing many similar properties with cryptography, digitizing chaotic maps for the purpose of developing chaos-based cryptosystems leads to dynamical degradation, causing many security issues. This paper introduces a hybrid chaotic system that enhances the dynamical behaviour of these maps to overcome this problem. The proposed system uses cascade and combination methods as a nonlinear chaotification function. To depict the capability of the proposed system, we apply it to classical chaotic maps and analyse them using theoretical analysis, conventional, fractal and randomness evaluations. Results show that the enhanced maps have a larger chaotic range, low correlation, uniform data distribution and better chaotic properties. As a proof of concept, simple pseudorandom number generators are then designed based on a classical map and its enhanced variant. Security comparisons between the two generators indicate that the generator based on the enhanced map has better statistical properties as compared to its classical counterpart. This finding showcases the capability of the proposed system in improving the performance of chaos-based algorithms.
Similar content being viewed by others
References
Hamza, R.: A novel pseudo random sequence generator for image-cryptographic applications. J. Inf. Secur. Appl. 35, 119–127 (2017)
Ozturk, I., Kilic, R.: A novel method for producing pseudo random numbers from differential equation-based chaotic systems. Nonlinear Dyn. 80, 1147–1157 (2015)
Lambi, D., Nikoli, M.: Pseudo-random number generator based on discrete-space chaotic map. Nonlinear Dyn. 90, 223–232 (2017)
Murillo-Escobar, M.A., Cruz-Hernndez, C., Cardoza-Avendao, L., Mndez-Ramrez, R.: A novel pseudorandom number generator based on pseudorandomly enhanced logistic map. Nonlinear Dyn. 87, 407–425 (2017)
Wang, Y., Liu, Z., Ma, J., He, H.: A pseudorandom number generator based on piecewise logistic map. Nonlinear Dyn. 83, 2373–2391 (2016)
Liu, L., Miao, S.: Delay-introducing method to improve the dynamical degradation of a digital chaotic map. Inf. Sci. 396, 113 (2017)
Deng, Y., Hu, H., Xiong, N., Xiong, W., Liu, L.: A general hybrid model for chaos robust synchronization and degradation reduction. Inf. Sci. 305, 146164 (2015)
Li, S., Chen, G., Mou, X.: On the dynamical degradation of digital piecewise linear chaotic maps. Int. J. Bifurc. Chaos 15(10), 31193151 (2005)
Wang, Q., Yu, S., Li, C., Lü, J., Fang, X., Guyeux, C., Bahi, J.M.: Theoretical design and FPGA-based implementation of higher-dimensional digital chaotic systems. IEEE Trans. Circuits Syst. I Regul. Pap. 63(3), 401412 (2016)
Hua, Z.Y., Zhou, Y.C., Pun, C.M., Chen, C.L.P.: 2D Sine logistic modulation map for image encryption. Inf. Sci. 297, 8094 (2015)
Zhou, Y.C., Hua, Z.Y., Pun, C.M., Chen, C.L.P.: Cascade chaotic system with applications. IEEE Trans. Cybern. 45(9), 20012012 (2015)
Liu, L., Liu, B., Hu, H., Miao, S.: Reducing the dynamical degradation by bi-coupling digital chaotic maps. Int. J. Bifurc. Chaos 28(5), 1850059 (2018)
Hua, Z.Y., Zhou, Y.C.: One-dimensional nonlinear model for producing chaos. IEEE Trans. Circuits Syst. I Regul. Pap. 65(1), 235246 (2018)
Teh, J.S., Tan, K., Alawida, M.: A chaos-based keyed hash function based on fixed point representation. Clust. Comput. (2018). https://doi.org/10.1007/s10586-018-2870-z
Teh, J.S., Samsudin, A., Akhavan, A.: Parallel chaotic hash function based on the shuffle-exchange. Nonlinear Dyn. 81, 10671079 (2015)
Zhu, Z.W., Leung, H.: Identification of linear systems driven by chaotic signals using nonlinear prediction. IEEE Trans. Circuits Syst. 49(2), 170180 (2002)
Wu, X.G., Hu, H.P., Zhang, B.L.: Parameter estimation only from the symbolic sequences generated by chaos system. Chaos Solitons Fractals 22(2), 359366 (2004)
Ye, G., Huang, X.: An efficient symmetric image encryption algorithm based on an intertwining logistic map. Neurocomputing 251, 4553 (2017)
Garcia-Bosque, M., Prez-Resa, A., Snchez-Azqueta, C., Aldea, C., Celma, S.: Chaos-based bitwise dynamical pseudorandom number generator on FPGA. IEEE Trans. Instrum. Meas. 68(1), 291–293 (2019)
Wheeler, D.D., Matthews, R.A.J.: Supercomputer investigations of a chaotic encryption algorithm. Cryptologia 15(2), 140152 (1991)
Hua, Z.Y., Zhou, B.H., Zhou, Y.C.: Sine chaotification model for enhancing chaos and its hardware implementation. IEEE Trans. Ind. Electron. 66(2), 1273–1284 (2019)
Nagaraj, N., Shastry, M.C., Vaidya, P.G.: Increasing average period lengths by switching of robust chaos maps in finite precision. Eur. Phys. J. Spec. Top. 165, 7383 (2008)
Wu, Y., Zhou, Y.C., Bao, L.: Discrete wheel-switching chaotic system and applications. IEEE Trans. Circuits Syst. I Regul. Pap. 61(12), 34693477 (2014)
Hu, H.P., Xu, Y., Zhu, Z.G.: A method of improving the properties of digital chaotic system. Chaos Solitons Fractals 38(2), 439446 (2008)
Liu, L.F., Lin, J., Miao, S.X., Liu, B.C.: A double perturbation method for reducing dynamical degradation of the digital baker map. Int. J. Bifurc. Chaos 27(7), 1750103 (2017)
Tao, S., Ruli, W., Yixun, Y.X.: Perturbance-based algorithm to expand cycle length of chaotic key stream. Electron. Lett. 34(9), 873–874 (1998)
Lv-Chen, C., Yu-Ling, L., Sen-Hui, Q., Jun-Xiu, L.: A perturbation method to the tent map based on Lyapunov exponent and its application. Chin. Phys. B 24(10), 100501 (2015)
Hua, Z.Y., Zhou, Y.C., Pun, C.M., Chen, C.L.P.: A new 1D parameter-control chaotic framework. In: Proceedings of the SPIE 9030, Mobile Devices and Multimedia: Enabling Technologies, Algorithms, and Applications 2014, 90300M. 110 (2014)
Liu, Y.Q., Luo, Y.L., Song, S.X., Cao, L.C., Liu, J.X., Harkin, J.: Counteracting dynamical degradation of digital chaotic Chebyshev map via perturbation. Int. J. Bifurc. Chaos 27(3), 1750033 (2017)
Zhang, T.F., Li, S.L., Ge, R.J., Yuan, M., Ma, Y.D.: A novel 1D hybrid chaotic map-based image compression and encryption using compressed sensing and Fibonacci–Lucas transform. Math. Probl. Eng. 2016, 7683687 (2016)
Li, C., Lin, D., Feng, B., Lü, J., Hao, F.: Cryptanalysis of a chaotic image encryption algorithm based on information entropy. IEEE Access 6, 75834–75842 (2018)
Li, C., Lin, D., Lü, J., Hao, F.: Cryptanalyzing an image encryption algorithm based on autoblocking and electrocardiography. IEEE MultiMed. (2019). https://doi.org/10.1109/MMUL.2018.2873472
Li, C.G., Feng, B.B., Li, S.J., Kurths, J., Chen, G.R.: Dynamic analysis of digital chaotic maps via state-mapping networks. IEEE Trans. Circuits Syst. I Regul. Pap. (2018). https://doi.org/10.1109/TCSI.2018.2888688
Skokos, C.: The Lyapunov characteristic exponents and their computation. Lect. Notes Phys. 790, 63–135 (2010)
Sprott, J.C.: Chaos and Time-Series Analysis, pp. 116–117. Oxford University Press, Oxford (2003)
Grassbergert, P., Procaccia, T.: Characterization of strange attractors. Phys. Rev. Lett. 50(5), 346–349 (1983)
Ozkaynak, F.: Brief review on application of nonlinear dynamics in image encryption. Nonlinear Dyn. 92, 305313 (2018)
Rukhin, A., Soto, J., Nechvatal, J., Smid, M., Barker, E.: A statistical test suite for random and pseudorandom number generators for cryptographic applications. Technical report, pp. 800822. NIST Special Publication (2001)
Acknowledgements
This work has been partially supported by Universiti Sains Malaysia under Grant No. 304/PKOMP/6316280 and the National Natural Science Foundation of China under Grant No. 61702212.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Alawida, M., Samsudin, A. & Teh, J.S. Enhancing unimodal digital chaotic maps through hybridisation. Nonlinear Dyn 96, 601–613 (2019). https://doi.org/10.1007/s11071-019-04809-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-019-04809-w