Skip to main content
SpringerLink
Log in

Chaotic S-box: six-dimensional fractional Lorenz–Duffing chaotic system and O-shaped path scrambling

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

Abstract

This paper is concerned with designing a chaotic encryption system to generate the nonlinear component, substitution box (S-box), of a block cipher system. Many existing S-boxes generation methods employ a single or complicate chaotic systems to yield S-boxes. All of these chaotic systems are integral and have promoted the development of the theoretical research of chaotic S-boxes. However, it is difficult to implement the integral chaotic S-box generation systems that are appropriate for practical engineering applications. In this paper, a six-dimensional fractional Lorenz–Duffing chaotic system and O-shaped path scrambling algorithm (FLDSOP) is developed to yield an S-box with good dynamic characteristics. First, FLDSOP leverages a six-dimensional fractional Lorenz–Duffing chaotic system to construct a preliminary S-box. Second, it designs an O-Shaped path scrambling scheme to disturb the order of elements in the obtain S-box. Experimental results have shown that the chaotic S-box produced by the proposed FLDSOP algorithm can effectively resist to multiple types of cryptanalysis attacks.

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

References

  1. Wang, X., Wang, Q.: A novel image encryption algorithm based on dynamic S-boxes constructed by chaos. Nonlinear Dyn. 75(3), 567–576 (2014)

    Article  Google Scholar 

  2. Zhang, X., Mao, Y.: An efficient chaotic image encryption based on alternate circular S-boxes. Nonlinear Dyn. 78(1), 359–369 (2014)

    Article  Google Scholar 

  3. Sam, I.S., Devaraj, P., Bhuvaneswaran, R.S.: An intertwining chaotic maps based image encryption scheme. Nonlinear Dyn. 69(4), 1995–2007 (2012)

    Article  MathSciNet  Google Scholar 

  4. Chen, Y.C., Zou, X.C., Liu, Z.L., et al.: Energy-efficient and security-optimized AES hardware design for ubiquitous computing. J. Syst. Eng. Electron. 19(4), 652–658 (2008)

    Article  Google Scholar 

  5. Lambi’c, D.: A novel method of S-box design based on chaotic map and composition method. Chaos Solitons Fractals 58, 16–21 (2014)

    Article  Google Scholar 

  6. Zhu, C.: A novel image encryption scheme based on improved hyperchaotic sequences. Opt. Commun. 285(1), 29–37 (2012)

    Article  Google Scholar 

  7. Jakimoski, G., Kocarev, L.: Chaos and cryptography: block encryption ciphers based on chaotic maps. IEEE Trans. Circuits Syst. 48(2), 163–169 (2001)

    Article  MathSciNet  Google Scholar 

  8. Tang, G., Liao, X., Chen, Y.: A novel method for designing S-boxes based on chaotic maps. Chaos Solitons Fractals 23(2), 413–419 (2005)

    Article  Google Scholar 

  9. Tang, G., Liao, X.: A method for designing dynamical S-boxes based on discretized chaotic map. Chaos Solitons Fractals 23(5), 1901–1909 (2005)

    Article  MathSciNet  Google Scholar 

  10. Behnia, S., Akhshani, A., Mahmodi, H., et al.: A novel algorithm for image encryption based on mixture of chaotic maps. Chaos Solitons Fractals 35(2), 408–419 (2008)

    Article  MathSciNet  Google Scholar 

  11. Khan, M., Shah, T., Mahmood, H., et al.: An efficient method for the construction of block cipher with multi-chaotic systems. Nonlinear Dyn. 71(3), 489–492 (2013)

    Article  MathSciNet  Google Scholar 

  12. Khan, J., Ahmad, J., Hwang, S.O.: An efficient image encryption scheme based on: Henon map, skew tent map and S-Box. In: IEEE 6th International Conference on Modeling, Simulation, and Applied Optimization (ICMSAO), pp. 1–6 (2015)

  13. Ahmad, M., Haleem, H., Khan, P.M.: A new chaotic substitution box design for block ciphers. In: IEEE International Conference on Signal Processing and Integrated Networks (SPIN), pp. 255–258 (2014)

  14. Hussain, I., Shah, T., Mahmood, H., et al.: A projective general linear group based algorithm for the construction of substitution box for block ciphers. Neural Comput. Appl. 22(6), 1085–1093 (2013)

    Article  Google Scholar 

  15. Shao, S.Y., Min, F.H., Ma, M.L., et al.: Non-inductive modular circuit of dislocated synchronization of fractional-order Chua’s system and its application. Acta Phys. Sin. 62(13), 130504 (2013)

    Google Scholar 

  16. Özkaynak, F., Çelik, V., Özer, A.B.: A new S-box construction method based on the fractional-order chaotic Chen system. Signal Image Video Process. (2016). https://doi.org/10.1007/s11760-016-1007-1

    Article  Google Scholar 

  17. Khan, M., Shah, T.: An efficient construction of substitution box with fractional chaotic system. Signal Image Video Process. 9(6), 1335–1338 (2015)

    Article  MathSciNet  Google Scholar 

  18. Tian, Y., Lu, Z., Gao, X.: Simulation of a six-dimensional fractional-order Lorenz-duffing system. Mod. Electron. Tech. 40(12), 22–27 (2017). (in Chinese)

    Google Scholar 

  19. Zhang, F.: A new six-dimensional chaotic algorithms and its application in image encryption. Microelectron. Comput. 30(10), 62–65 (2013). (in Chinese)

    Google Scholar 

  20. Charef, A., Sun, H.H., Tsao, Y.Y., et al.: Fractal system as represented by singularity function. IEEE Trans. Autom. Control 37(9), 1465–1470 (1992)

    Article  MathSciNet  Google Scholar 

  21. Detombe, J., Tavares, S.E.: Constructing large cryptographically strong S-boxes. In: Advances in Cryptology: Proceedings of AUSCRYPT’92. LNCS, pp. 165–181 (1993)

    Chapter  Google Scholar 

  22. Adams, C., Tavares, S.: The structured design of cryptographically good S-boxes. J. Cryptol. 3(1), 27–41 (1990)

    Article  MathSciNet  Google Scholar 

  23. Webster, A.F., Tavares, S.: On the design of S-boxes. In: Advances in Cryptology: Proceedings of CRYPTO’85. LNCS, pp. 523–534 (1986)

  24. Adams, C., Tavares, S.: Good S-boxes are easy to find. In: Advances in Cryptology: Proceedings of CRYPTO’89. LNCS, vol. 89, pp. 612–615 (1989)

  25. Biham, E., Shamir, A.: Differential cryptanalysis of DES like cryptosystems. J. Cryptol. 4(1), 3–72 (1991)

    Article  MathSciNet  Google Scholar 

  26. Matsui, M.: Linear cryptanalysis method of DES cipher. In: Advances in Cryptology, Proceedings of CRYPTO’93. LNCS, vol.765, pp. 386–397 (1994)

  27. Chen, G., Yong, C., Xiaofeng, L.: An extended method for obtaining S-boxes based on three-dimensional chaotic Baker maps. Chaos Solitons Fractals 31(3), 571–579 (2007)

    Article  MathSciNet  Google Scholar 

  28. Ozkaynak, F., Ozer, A.B.: A method for designing strong S-Boxes based on chaotic Lorenz system. Phys. Lett. A 374, 3733–3738 (2010)

    Article  Google Scholar 

  29. Khan, M., Shah, T., Mahmood, H., et al.: A novel technique for the construction of strong S-boxes based on chaotic Lorenz systems. Nonlinear Dyn. 70(3), 2303–2311 (2012)

    Article  MathSciNet  Google Scholar 

  30. Hussain, I., Tariq, S., Muhammad, A.G.: A novel approach for designing substitution-boxes based on nonlinear chaotic algorithm. Nonlinear Dyn. 70(3), 1791–1794 (2012)

    Article  MathSciNet  Google Scholar 

  31. Khan, M., Shah, T., Gondal, M.A.: An efficient technique for the construction of substitution box with chaotic partial differential equation. Nonlinear Dyn. 73(3), 1795–1801 (2013)

    Article  MathSciNet  Google Scholar 

  32. Khan, M., Shah, T.: A construction of novel chaos base nonlinear component of block cipher. Nonlinear Dyn. 76(1), 377–382 (2014)

    Article  MathSciNet  Google Scholar 

  33. Anees, A., Ahmed, Z.: A technique for designing substitution box based on Van der Pol Oscillator. Wirel. Pers. Commun. 82(3), 1497–1503 (2015)

    Article  Google Scholar 

  34. Khan, M., Shah, T., Batool, S.I.: Construction of S-box based on chaotic Boolean functions and its application in image encryption. Neural Comput. Appl. 27(3), 677–685 (2016)

    Article  Google Scholar 

  35. Khan, M., Asghar, Z.: A novel construction of substitution box for image encryption applications with Gingerbreadman chaotic map and \(\text{ S }_{8}\) permutation. Neural Comput. Appl. (2016). https://doi.org/10.1007/s00521-016-2511-5

    Article  Google Scholar 

  36. Çavuşoğlu, Ü., Zengin, A., Pehlivan, I., Kaçar, S.: A novel approach for strong S-box generation algorithm design based on chaotic scaled Zhongtang system. Nonlinear Dyn. 87(2), 1081–1094 (2017)

    Article  Google Scholar 

  37. Khan, M.: A novel image encryption scheme based on multiple chaotic S-boxes. Nonlinear Dyn. 82(1–2), 527–533 (2015)

    Article  MathSciNet  Google Scholar 

  38. Belazi, A., El-Latif, A.A.: A simple yet efficient S-box method based on chaotic sine map. Optik 130, 1438–1444 (2017)

    Article  Google Scholar 

  39. Farah, T., Rhouma, R., Belghith, S.: A novel method for designing S-box based on chaotic map and teaching-learning-based optimization. Nonlinear Dyn. (2016). https://doi.org/10.1007/s11071-016-3295-y

    Article  Google Scholar 

  40. Chen, G.: A novel heuristic method for obtaining S-boxes. Chaos Solitons Fractals 36(4), 1028–1036 (2008)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research is supported by the National Natural Science Foundation of China (Nos. 60975042, 51472066).

Author information

Authors and Affiliations

  1. College of Information and Communication Engineering, Harbin Engineering University, Harbin, 150001, China

    Tian Ye & Lu Zhimao

  2. Key Laboratory of Photonic and Electronic Bandgap Materials, Ministry of Education, School of Physics and Electronic Engineering, Harbin Normal University, Harbin, 150025, China

    Tian Ye

Authors
  1. Tian Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lu Zhimao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tian Ye.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ye, T., Zhimao, L. Chaotic S-box: six-dimensional fractional Lorenz–Duffing chaotic system and O-shaped path scrambling. Nonlinear Dyn 94, 2115–2126 (2018). https://doi.org/10.1007/s11071-018-4478-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-018-4478-5

Keywords

Search

Navigation