Designing multi-dimensional logistic map with fixed-point finite precision

  • Fahad A. Munir
  • Muhammad ZiaEmail author
  • Hasan Mahmood
Original paper


In cryptographic algorithms, random sequences of longer period and higher nonlinearity are always desirable in order to increase resistance against cryptanalysis. The use of chaotic maps is an attractive choice as they exhibit properties that are suitable for cryptography. In continuous phase space of the logistic map, proper control parameters and initial state result into aperiodic trajectories. However, when the phase space of the logistic map is quantized, the trajectories terminate in finite and stable periodic orbits due to quantization error. The dynamic degradation of the logistic map can be mitigated using nonlinear feedback and cascading multiple chaotic maps. We propose a logistic map-based, finite precision multi-dimensional logistic map, that incorporates nonlinear feedback and modulus operations to perturb the chaotic trajectories. We present complexity, average cycle length and randomness analysis to evaluate the proposed method. The simulation results and analysis reveal that the proposed MDLM approach achieves longer period and higher randomness.


Digital chaotic map Multi-dimensional logistic map Finite precision Chaotic maps Fixed-point processing 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Kolumban, G.: Theoretical noise performance of correlator-based chaotic communications schemes. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 47(12), 1692–1701 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Quyen, N.X., Yem, V.V., Hoang, T.M.: A chaos-based secure direct-sequence/spread-spectrum communication system. Abstr. Appl. Anal. 2013, 11 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Wang, S., Liu, W., Lu, H., Kuang, J., Hu, G.: Periodicity of chaotic trajectories in realizations of finite computer precisions and its implication in chaos communications. Int. J. Mod. Phys. B 18(17n19), 2617–2622 (2004)CrossRefGoogle Scholar
  4. 4.
    Wang, S., Kuang, J., Li, J., Luo, Y., Lu, H., Hu, G.: Chaos-based secure communications in a large community. Phys. Rev. E 66(6), 065202 (2002)CrossRefGoogle Scholar
  5. 5.
    Heidari-Bateni, G., McGillem, C.D.: A chaotic direct-sequence spread-spectrum communication system. IEEE Trans. Commun. 42(234), 1524–1527 (1994)CrossRefGoogle Scholar
  6. 6.
    Lau, F.C., Chi, K.T.: Chaos-Based Digital Communication Systems: Operating Principles, Analysis Methods and Performance Evaluation. Springer, Berlin (2013)Google Scholar
  7. 7.
    Zidan, M.A., Radawan, A.G., Salama, K.N.: Random number generation based on digital differential chaos. In: Proceedings of IEEE 54th International Midwest Symposium on Circuits and Systems (MWSCAS), pp. 1–4 (2011)Google Scholar
  8. 8.
    Matthews, R.: On the derivation of a chaotic encryption algorithm. Cryptologia 13(1), 29–42 (1989)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Vlad, A., Luca, A., Hodea, O., Tataru, R.: Generating chaotic secure sequences using tent map and a running-key approach. Proc. Rom. Acad. Ser. A 14(Special Issue), 295–302 (2013)MathSciNetGoogle Scholar
  10. 10.
    Kocarev, L.: Chaos-based cryptography: a brief overview. IEEE Circuits Syst. Mag. 1(3), 6–21 (2001)CrossRefGoogle Scholar
  11. 11.
    Li, C., Xie, T., Liu, Q., Cheng, G.: Cryptanalyzing image encryption using chaotic logistic map. Nonlinear Dyn. 78(2), 1545–1551 (2014)CrossRefGoogle Scholar
  12. 12.
    Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd edn. Perseus Books Group, New York City (2014)zbMATHGoogle Scholar
  13. 13.
    Khan, M., Shah, T., Mahmood, H., Gondal, M.A., Hussain, I.: A novel technique for the construction of strong S-boxes based on chaotic Lorenz systems. Nonlinear Dyn. 70(3), 2303–2311 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Deng, Y., Hu, H., Xiong, W., Xiong, N.N., Liu, L.: Analysis and design of digital chaotic systems with desirable performance via feedback control. IEEE Trans. Syst. Man Cybern. Syst. 45(8), 1187–1200 (2015)CrossRefGoogle Scholar
  15. 15.
    Persohn, K.J., Povinelli, R.J.: Analyzing logistic map pseudorandom number generators for periodicity induced by finite precision floating-point representation. Chaos Solitons Fractals 45(3), 238–245 (2012)CrossRefGoogle Scholar
  16. 16.
    Guo, J.: Analysis of chaotic systems. (2014). Accessed 22 Aug 2017
  17. 17.
    Beck, C.: Scaling behavior of random maps. Phys. Let. A 136(3), 121–125 (1989)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Li, C., Lin, D., Lü, J., Hao, F.: Cryptanalyzing an image encryption algorithm based on autoblocking and electrocardiography. IEEE Multimed 25(4), 46–56 (2018)CrossRefGoogle Scholar
  19. 19.
    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
  20. 20.
    Wheeler, D.D.: Problems with chaotic cryptosystems. Cryptologia 13(3), 243–250 (1989)CrossRefGoogle Scholar
  21. 21.
    Li, C.Y., Chen, J.S., Chang, T.Y.: A chaos-based pseudo random number generator using timing-based reseeding method. In: Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS), pp. 21–24 (2006)Google Scholar
  22. 22.
    Wang, Q., Yu, S., Li, C., Lü, J., Fang, X., Guyeux, C., Bahi, J.M.: Theortical design and FPGA-based implementation of higher-dimensional digital chaotic systems. IEEE Trans. Circuits Syst. I Regul. Pap. 63(3), 401–412 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Liu, L., Miao, S.: A universal method for improving the dynamical degradation of a digital chaotic system. Physica Scripta 90(8), 085205 (2015)CrossRefGoogle Scholar
  24. 24.
    Liu, Y., Luo, Y., Song, S., Cao, L., Liu, J., Harkin, J.: Counteracting dynamical degradation of digital chaotic chebyshev map via perturbation. Int. J. Bifurc. Chaos 27(3), 1750033 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    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
  26. 26.
    Garcia-Bosque, M., Pérez-Resa, A., Sánchez-Azqueta, C., Adlea, C., Celma, S.: Chaos-based bitwise dynamical pseudorandom number generator on FPGA. IEEE Trans. Instrum. Meas. 68(1), 291–293 (2018)CrossRefGoogle Scholar
  27. 27.
    Sprott, J.C.: Numerical Calculation of Largest Lyapunov Exponent. (2015). Accessed 11 Feb 2017
  28. 28.
    Rukhin, A., Soto, J., Nechvatal, J., Smid, M., Barker, E., Leigh, S., Levenson, M., Vangel, M., Banks, D., Heckert, A., Dray, J., Vo, S.: A statistical test suite for random and pseudorandom number generators for cryptographic applications. Tech. rep., DTIC Document (2001)Google Scholar
  29. 29.
    L’Ecuyer, P., Simard, R.: A C Library for Emperical Testing of Random Number Generators. ACM Trans. Math. Softw. (TOMS) 33(4), Art. no. 22 (2007)Google Scholar
  30. 30.
    Marsaglia, G.: DIEHARD: a battery of tests of randomness. (1996)
  31. 31.
    Li, C., Feng, B., Li, S., Kurths, J., Chen, G.: Dynamic analysis of digital chaotic maps via state-mapping networks. IEEE Trans. Circuits Syst. I Regul. Pap. 66(6), 2322–2335 (2019)CrossRefGoogle Scholar
  32. 32.
    Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis, 2nd edn. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  33. 33.
    Siu, S.W.K.: Lyapunov Exponent Toolbox. (1998). Accessed 15 Mar 2017
  34. 34.
    Papoulis, A., Pillai, S.U.: Probability, Random Variables and Stochastic Processes, 4th edn. Tata McGraw-Hill Education, New York City (2002)Google Scholar
  35. 35.
    Li, W., Songs, B., Ding, Q.: Discrete chaos circuit random characteristic analysis. In: Proceedings of IEEE 3rd International Conference on Robot, Vision and Signal Processing (RVSP), pp. 280–284 (2015)Google Scholar
  36. 36.
    Chapra, S.C.: Applied Numerical Methods with Matlab: For Engineers and Scientists, 3rd edn. Tata McGraw-Hill Education, New York City (2007)Google Scholar
  37. 37.
    Warner, S., Costenoble, S.R.: Finite Math and Applied Calculus, 6th edn. Cengage Learning, Boston (2013)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of ElectronicsQuaid-i-Azam UniversityIslamabadPakistan

Personalised recommendations