Enhancing unimodal digital chaotic maps through hybridisation

  • Moatsum Alawida
  • Azman Samsudin
  • Je Sen TehEmail author
Original Paper


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.


Chaotic map Dynamical degradation Pseudorandom number generator Unimodal map Sine map Logistic map 



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.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Hamza, R.: A novel pseudo random sequence generator for image-cryptographic applications. J. Inf. Secur. Appl. 35, 119–127 (2017)Google Scholar
  2. 2.
    Ozturk, I., Kilic, R.: A novel method for producing pseudo random numbers from differential equation-based chaotic systems. Nonlinear Dyn. 80, 1147–1157 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Lambi, D., Nikoli, M.: Pseudo-random number generator based on discrete-space chaotic map. Nonlinear Dyn. 90, 223–232 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Wang, Y., Liu, Z., Ma, J., He, H.: A pseudorandom number generator based on piecewise logistic map. Nonlinear Dyn. 83, 2373–2391 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Liu, L., Miao, S.: Delay-introducing method to improve the dynamical degradation of a digital chaotic map. Inf. Sci. 396, 113 (2017)CrossRefGoogle Scholar
  7. 7.
    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)CrossRefzbMATHGoogle Scholar
  8. 8.
    Li, S., Chen, G., Mou, X.: On the dynamical degradation of digital piecewise linear chaotic maps. Int. J. Bifurc. Chaos 15(10), 31193151 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefGoogle Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hua, Z.Y., Zhou, Y.C.: One-dimensional nonlinear model for producing chaos. IEEE Trans. Circuits Syst. I Regul. Pap. 65(1), 235246 (2018)CrossRefGoogle Scholar
  14. 14.
    Teh, J.S., Tan, K., Alawida, M.: A chaos-based keyed hash function based on fixed point representation. Clust. Comput. (2018). Google Scholar
  15. 15.
    Teh, J.S., Samsudin, A., Akhavan, A.: Parallel chaotic hash function based on the shuffle-exchange. Nonlinear Dyn. 81, 10671079 (2015)CrossRefGoogle Scholar
  16. 16.
    Zhu, Z.W., Leung, H.: Identification of linear systems driven by chaotic signals using nonlinear prediction. IEEE Trans. Circuits Syst. 49(2), 170180 (2002)Google Scholar
  17. 17.
    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)Google Scholar
  18. 18.
    Ye, G., Huang, X.: An efficient symmetric image encryption algorithm based on an intertwining logistic map. Neurocomputing 251, 4553 (2017)CrossRefGoogle Scholar
  19. 19.
    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)CrossRefGoogle Scholar
  20. 20.
    Wheeler, D.D., Matthews, R.A.J.: Supercomputer investigations of a chaotic encryption algorithm. Cryptologia 15(2), 140152 (1991)CrossRefGoogle Scholar
  21. 21.
    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)CrossRefGoogle Scholar
  22. 22.
    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)CrossRefGoogle Scholar
  23. 23.
    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)CrossRefGoogle Scholar
  24. 24.
    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)CrossRefGoogle Scholar
  25. 25.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    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)CrossRefGoogle Scholar
  27. 27.
    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)CrossRefGoogle Scholar
  28. 28.
    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)Google Scholar
  29. 29.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    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)MathSciNetzbMATHGoogle Scholar
  31. 31.
    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)CrossRefGoogle Scholar
  32. 32.
    Li, C., Lin, D., Lü, J., Hao, F.: Cryptanalyzing an image encryption algorithm based on autoblocking and electrocardiography. IEEE MultiMed. (2019). Google Scholar
  33. 33.
    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). Google Scholar
  34. 34.
    Skokos, C.: The Lyapunov characteristic exponents and their computation. Lect. Notes Phys. 790, 63–135 (2010)CrossRefGoogle Scholar
  35. 35.
    Sprott, J.C.: Chaos and Time-Series Analysis, pp. 116–117. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  36. 36.
    Grassbergert, P., Procaccia, T.: Characterization of strange attractors. Phys. Rev. Lett. 50(5), 346–349 (1983)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Ozkaynak, F.: Brief review on application of nonlinear dynamics in image encryption. Nonlinear Dyn. 92, 305313 (2018)CrossRefGoogle Scholar
  38. 38.
    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)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Computer SciencesUniversity Sains Malaysia (USM)GelugorMalaysia

Personalised recommendations