Skip to main content
SpringerLink
Log in

Improving chaos-based pseudo-random generators in finite-precision arithmetic

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

One of the widely-used ways in chaos-based cryptography to generate pseudo-random sequences is to use the least significant bits or digits of finite-precision numbers defined by the chaotic orbits. In this study, we show that the results obtained using such an approach are very prone to rounding errors and discretization effects. Thus, it appears that the generated sequences are close to random even when parameters correspond to non-chaotic oscillations. In this study, we confirm that the actual source of pseudo-random properties of bits in a binary representation of numbers can not be chaos, but computer simulation. We propose a technique for determining the maximum number of bits that can be used as the output of a pseudo-random sequence generator including chaos-based algorithms. The considered approach involves evaluating the difference of the binary representation of two points obtained by different numerical methods of the same order of accuracy. Experimental results show that such estimation can significantly increase the performance of the existing chaos-based generators. The obtained results can be used to reconsider and improve chaos-based cryptographic algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Data availability statement

Not applicable.

References

  1. Irani, B.Y., Ayubi, P., Jabalkandi, F.A., Valandar, M.Y., Barani, M.J.: Digital image scrambling based on a new one-dimensional coupled sine map. Nonlinear Dyn. 97(4), 2693–2721 (2019)

    Article  Google Scholar 

  2. Lambić, D., Nikolić, M.: Pseudo-random number generator based on discrete-space chaotic map. Nonlinear Dyn. 90(1), 223–232 (2017)

    Article  MathSciNet  Google Scholar 

  3. Karimov, T., Nepomuceno, E.G., Druzhina, O., Karimov, A., Butusov, D.: Chaotic oscillators as inductive sensors: theory and practice. Sensors 19(19), 4314 (2019)

    Article  Google Scholar 

  4. Parastesh, F., Jafari, S., Azarnoush, H., Hatef, B., Namazi, H., Dudkowski, D.: Chimera in a network of memristor-based Hopfield neural network. Eur. Phys. J. Spec. Top. 228(10), 2023–2033 (2019)

    Article  Google Scholar 

  5. Moysis, L., Petavratzis, E., Volos, C., Nistazakis, H., Stouboulos, I.: A chaotic path planning generator based on logistic map and modulo tactics. Rob. Auton. Syst. 124, 103377 (2020)

    Article  Google Scholar 

  6. Datcu, O., Macovei, C., Hobincu, R.: Chaos based cryptographic pseudo-random number generator template with dynamic state change. Appl. Sci. 10(2), 451 (2020)

    Article  Google Scholar 

  7. Kuiate, G.F., Rajagopal, K., Kingni, S.T., Tamba, V.K., Jafari, S.: Autonomous Van der Pol-Duffing snap oscillator: analysis, synchronization and applications to real-time image encryption. Int. J. Dyn. Control 6(3), 1008–1022 (2018)

    Article  MathSciNet  Google Scholar 

  8. Dmitriev, A.S., Mokhseni, T.I., Teran, K.S.: Differentially coherent information transmission based on chaotic radio pulses. J. Commun. Technol. Electron. 63(10), 1183–1190 (2018)

    Article  Google Scholar 

  9. Öztürk, I., Kılıç, R.: A novel method for producing pseudo random numbers from differential equation-based chaotic systems. Nonlinear Dyn. 80(3), 1147–1157 (2015)

    Article  MathSciNet  Google Scholar 

  10. Ye, G., Jiao, K., Huishan, W., Pan, C., Huang, X.: An asymmetric image encryption algorithm based on a fractional-order chaotic system and the rsa public-key cryptosystem. Int. J. Bifurc. Chaos 30(15), 2050233 (2020)

    Article  MathSciNet  Google Scholar 

  11. Ye, G., Pan, C., Huang, X., Mei, Q.: An efficient pixel-level chaotic image encryption algorithm. Nonlinear Dyn. 94(1), 745–756 (2018)

    Article  Google Scholar 

  12. Hasimoto-Beltrán, R., Mota-García, E.: Real-time secure multimedia communication system based on chaos theory. In: Pacific-Rim Conference on Multimedia, pp. 441–445. Springer (2007)

  13. Ye, H.-S., Zhou, N.-R., Gong, L.-H.: Multi-image compression-encryption scheme based on quaternion discrete fractional Hartley transform and improved pixel adaptive diffusion. Signal Processing 175, 107652 (2020)

    Article  Google Scholar 

  14. Kanso, A., Smaoui, N.: Irregularly decimated chaotic map (s) for binary digits generations. Int. J. Bifurc. Chaos 19(04), 1169–1183 (2009)

    Article  MathSciNet  Google Scholar 

  15. Palacios-Luengas, L., Pichardo-Méndez, J.L., Díaz-Méndez, J.A., Rodríguez-Santos, F., Vázquez-Medina, R.: PRNG based on skew tent map. Arab. J. Sci. Eng. 44(4), 3817–3830 (2019)

    Article  Google Scholar 

  16. François, M., Defour, D., Negre, C.: A fast chaos-based pseudo-random bit generator using binary64 floating-point arithmetic. Informatica 38, 115–124 (2014)

    MathSciNet  Google Scholar 

  17. Flores-Vergara, A., García-Guerrero, E.E., Inzunza-González, E., López-Bonilla, O.R., Rodríguez-Orozco, E., Cárdenas-Valdez, J.R., Tlelo-Cuautle, E.: Implementing a chaotic cryptosystem in a 64-bit embedded system by using multiple-precision arithmetic. Nonlinear Dyn. 96(1), 497–516 (2019)

    Article  Google Scholar 

  18. Lui, O.Y., Yuen, C.H., Wong, K.W.: A pseudo-random number generator employing multiple Renyi maps. Int. J. Mod. Phys. C 24(11), 1350079 (2013)

    Article  Google Scholar 

  19. Alawida, M., Samsudin, A., Teh, J.: Enhanced digital chaotic maps based on bit reversal with applications in random bit generators. Inf. Sci. 512, 1155–1169 (2020)

    Article  Google Scholar 

  20. Institute of Electrical and Electronics Engineers. 754-2008-IEEE standard for floating-point arithmetic. IEEE (2008)

  21. Lambić, D.: Security analysis and improvement of the pseudo-random number generator based on quantum chaotic map. Nonlinear Dyn. 94(2), 1117–1126 (2018)

    Article  MathSciNet  Google Scholar 

  22. Kwok, H.S., Tang, W.K.: A fast image encryption system based on chaotic maps with finite precision representation. Chaos Solitons Fractals 32(4), 1518–1529 (2007)

    Article  MathSciNet  Google Scholar 

  23. Akhshani, A., Akhavan, A., Mobaraki, A., Lim, S.C., Hassan, Z.: Pseudo random number generator based on quantum chaotic map. Commun. Nonlinear Sci. Numer. Simul. 19(1), 101–111 (2014)

    Article  Google Scholar 

  24. Liu, L., Miao, S., Cheng, M., Gao, X.: A pseudorandom bit generator based on new multi-delayed Chebyshev map. Inf. Process. Lett. 116(11), 674–681 (2016)

    Article  Google Scholar 

  25. García-Martínez, M., Ontañón-García, L., Campos-Cantón, E., Čelikovskỳ, S.: Hyperchaotic encryption based on multi-scroll piecewise linear systems. Appl. Math. Comput. 270, 413–424 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Wu, X., Wang, D., Kurths, J., Kan, H.: A novel lossless color image encryption scheme using 2D DWT and 6D hyperchaotic system. Inf. Sci. 349, 137–153 (2016)

    Article  Google Scholar 

  27. Cuyt, A., Verdonk, B., Becuwe, S., Kuterna, P.: A remarkable example of catastrophic cancellation unraveled. Computing 66(3), 309–320 (2001)

    Article  MathSciNet  Google Scholar 

  28. Butusov, D., Karimov, A., Tutueva, A., Kaplun, D., Nepomuceno, E.G.: The effects of Padé numerical integration in simulation of conservative chaotic systems. Entropy 21(4), 362 (2019)

    Article  Google Scholar 

  29. Karimov, T.I., Butusov, D.N., Pesterev, D.O., Predtechenskii, D.V., Tedoradze, R.S.: Quasi-chaotic mode detection and prevention in digital chaos generators. In: 2018 IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering (EIConRus), pp. 303–307. IEEE (2018)

  30. Li, S., Chen, G., Mou, X.: On the dynamical degradation of digital piecewise linear chaotic maps. Int. J. Bifurc. Chaos 15(10), 3119–3151 (2005)

    Article  MathSciNet  Google Scholar 

  31. 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)

    Article  Google Scholar 

  32. Teh, J.S., Alawida, M., Ho, J.J.: Unkeyed hash function based on chaotic sponge construction and fixed-point arithmetic. Nonlinear Dyn. 100, 713–729 (2020)

    Article  Google Scholar 

  33. Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration illustrated by the Störmer–Verlet method. Acta Numer. 12, 399–450 (2003)

    Article  MathSciNet  Google Scholar 

  34. 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, Booz-Allen and Hamilton Inc. McLean (2001)

  35. Alcin, M.: The Runge Kutta-4 based 4D hyperchaotic system design for secure communication applications. Chaos Theory Appl. 2(1), 23–30 (2020)

    Google Scholar 

  36. Goldberg, D.: What every computer scientist should know about floating-point arithmetic. ACM Comput. Surv. (CSUR) 23(1), 5–48 (1991)

    Article  Google Scholar 

  37. Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)

    Article  Google Scholar 

  38. Ojoniyi, O.S., Njah, A.N.: A 5D hyperchaotic Sprott B system with coexisting hidden attractors. Chaos Solitons Fractals 87, 172–181 (2016)

    Article  MathSciNet  Google Scholar 

  39. Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4(3), 506–510 (1953)

    Article  MathSciNet  Google Scholar 

  40. Su, Z., Zhang, G., Jiang, J.: Multimedia security: a survey of chaos-based encryption technology. In: Multimedia—A Multidisciplinary Approach to Complex Issues, pp. 99–124. InTech (2012)

  41. Liu, B., Xiang, H., Liu, L.: Reducing the dynamical degradation of digital chaotic maps with time-delay linear feedback and parameter perturbation. Math. Probl. Eng. (2020). https://doi.org/10.1155/2020/4926937

  42. Fan, C., Ding, Q.: Effects of limited computational precision on the discrete chaotic sequences and the design of related solutions. Complexity (2019). https://doi.org/10.1155/2019/3510985

  43. Fan, C., Ding, Q.: Analysing the dynamics of digital chaotic maps via a new period search algorithm. Nonlinear Dyn. 97(1), 831–841 (2019)

    Article  Google Scholar 

  44. Adiyaman, Y., Emiroglu, S., Ucar, M.K., Yildiz, M.: Dynamical analysis, electronic circuit design and control application of a different chaotic system. Chaos Theory Appl. 2(1), 8–14 (2020)

    Google Scholar 

Download references

Acknowledgements

We thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions.

Funding

This study was supported by the Grant of the Russian Science Foundation (RSF), Project 20-79-10334.

Author information

Authors and Affiliations

  1. Department of Computer-Aided Design, Saint-Petersburg Electrotechnical University “LETI”, 5, Professora Popova st., Saint Petersburg, Russia, 197376

    Aleksandra V. Tutueva

  2. Youth Research Institute, Saint-Petersburg Electrotechnical University “LETI”, 5, Professora Popova st., Saint Petersburg, Russia, 197376

    Timur I. Karimov & Denis N. Butusov

  3. Laboratory of Nonlinear Systems - Circuits and Complexity (LaNSCom), Physics Department, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Lazaros Moysis & Christos Volos

  4. Control and Modelling Group (GCOM), Department of Electrical Engineering, Federal University of São João del-Rei, São João del-Rei, MG, 36307-352, Brazil

    Erivelton G. Nepomuceno

Authors
  1. Aleksandra V. Tutueva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Timur I. Karimov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lazaros Moysis
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Erivelton G. Nepomuceno
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Christos Volos
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Denis N. Butusov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Denis N. Butusov.

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.

Appendices

Appendix A: The results of NIST statistical testing with \(\alpha = 0.005\) for sequences obtained from the Rossler system

See Tables 11 and 12.

Table 11 chaotic mode (\(c=15\))
Full size table
Table 12 harmonic mode (\(c=5\))
Full size table

Appendix B: The results of NIST statistical testing with \(\alpha = 0.005\) for sequences obtained from the hyperchaotic Sprott B system

See Tables 13 and 14.

Table 13 chaotic mode (\(a=0.01\))
Full size table
Table 14 harmonic mode (\(a=-0.9\))
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tutueva, A.V., Karimov, T.I., Moysis, L. et al. Improving chaos-based pseudo-random generators in finite-precision arithmetic. Nonlinear Dyn 104, 727–737 (2021). https://doi.org/10.1007/s11071-021-06246-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06246-0

Keywords

Search

Navigation