Nonlinear Dynamics

, Volume 88, Issue 4, pp 2473–2489 | Cite as

A novel secure image transmission scheme based on synchronization of fractional-order discrete-time hyperchaotic systems

  • Sarah Kassim
  • Hamid Hamiche
  • Saïd Djennoune
  • Maâmar Bettayeb
Original Paper


In this paper, a secure image transmission scheme based on synchronization of fractional-order discrete-time hyperchaotic systems is proposed. In this scheme, a fractional-order modified-Hénon map is considered as a transmitter, the system parameters and fractional orders are considered as secret keys. As a receiver, a step-by-step delayed observer is used, and based on this one, an exact synchronization is established. To make the transmission scheme secure, an encryption function is used to cipher the original information using a key stream obtained from the chaotic map sequences. Moreover, to further enhance the scheme security, the ciphered information is inserted by inclusion method in the chaotic map dynamics. The first contribution of this paper is to propose new results on the observability and the observability matching condition of nonlinear discrete-time fractional-order systems. To the best of our knowledge, these features have not been addressed in the literature. In the second contribution, the design of delayed discrete observer, based on fractional-order discrete-time hyperchaotic system, is proposed. The feasibility of this realization is demonstrated. Finally, different analysis are introduced to test the proposed scheme security. Simulation results are presented to highlight the performances of our method. These results show that, our scheme can resist different kinds of attacks and it exhibits good performance.


Private communication Chaotic synchronization Fractional-order systems Image encryption Modified-Hénon map Step-by-step observer Robustness 


  1. 1.
    Stallings, W.: Cryptography and Network Security, 5th edn. Prentice-Hall, Englewood Cliffs (2011)Google Scholar
  2. 2.
    Daemen, J., Rijmen, V.: The Design of Rijndael: AES-The Advanced Encryption Standard. Springer, Berlin (2002)CrossRefMATHGoogle Scholar
  3. 3.
    Wang, X., Xu, D.: A novel image encryption scheme based on Brownian motion and PWLCM chaotic system. Nonlinear Dyn. 75, 345–353 (2014)CrossRefGoogle Scholar
  4. 4.
    Souyah, A., Feraoun, K.M.: An image encryption scheme combining chaos-memory cellular automata and weighted histogram. Nonlinear Dyn. 86(1), 639–653 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Zhang, S., Gao, T.: A coding and substitution frame based on hyper-chaotic systems for secure communication. Nonlinear Dyn. 84(2), 833–849 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Wong, L., Song, H., Liu, P.: A novel hybrid color image encryption algorithm using two complex chaotic systems. Opt. Lasers Eng. 77, 118–125 (2016)CrossRefGoogle Scholar
  7. 7.
    Li, X., Wang, L., Yan, Y., Liu, P.: An improvement color image encryption algorithm based on DNA operations and real and complex chaotic systems. Optik. 127(5), 2558–2565 (2016)CrossRefGoogle Scholar
  8. 8.
    Liu, P., Song, H., Li, X.: Observe-based projective synchronization of chaotic complex modified Van Der Pol-Duffing oscillator with application to secure communication. J. Comput. Nonlinear Dyn. 10, 051015-7 (2015)Google Scholar
  9. 9.
    Hamdi, M., Rhouma, R., Belghith, S.: A selective compression-encryption of images based on SPIHT coding and Chirikov Standard Map. Signal Process. 131, 514–526 (2017)CrossRefGoogle Scholar
  10. 10.
    Hamiche, H., Lahdir, M., Tahanout, M., Djennoune, S.: Masking digital image using a novel technique based on a transmission chaotic system and SPIHT coding algorithm. Int. J. Adv. Comput. Sci. Appl. 3(12), 228–234 (2012)Google Scholar
  11. 11.
    Pecora, L.M., Carrol, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Yau, H.T., Pu, Y.C., Li, S.C.: Application of a chaotic synchronization system to secure communication. Inf. Technol. Control 41, 274–282 (2012)Google Scholar
  13. 13.
    Hamiche, H., Guermah, S., Saddaoui, R., Hannoun, K., Laghrouche, M., Djennoune, S.: Analysis and implementation of a novel robust transmission scheme for private digital communications using Arduino Uno board. Nonlinear Dyn. 81(4), 1921–1932 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Morgül, Ö., Solak, E.: Observer based synchronization of chaotic systems. Phys. Rev. E 5, 4803–4811 (1996)CrossRefGoogle Scholar
  15. 15.
    Nijmeijer, H., Mareels, I.M.Y.: An observer looks at synchronization. IEEE Trans. Circuits Syst. I. Fundam. Theory Appl. 44, 882–890 (1997)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Petráš, I.: Fractional-Order Nonlinear Systems. Modeling, Analysis and Simulation. Higher Education Press, Springer, Berlin (2011)CrossRefMATHGoogle Scholar
  17. 17.
    Guermah, S., Bettayeb, M., Djennoune, S.: Controllability and the observability of linear discrete-time fractional-order systems. Int. J. Appl. Math. Comput. Sci. 18, 213–222 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Atici, F.M., Senguel, S.: Modeling with fractional difference equations. J. Math. Anal. Appl. 369, 1–9 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and application of fractional differential equations. In: van Mill, J. (ed.) North Holland Mathematics Studies. Elsevier, Amsterdam (2006)Google Scholar
  20. 20.
    Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-Order Systems and Control. Fundamentals and Applications. Springer, Berlin (2010)CrossRefMATHGoogle Scholar
  21. 21.
    Wu, G.C., Baleanu, D.: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 75, 283–287 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Wu, G.C., Baleanu, D.: Chaos synchronization of the discrete fractional logistic map. Signal Process. 102, 96–99 (2014)CrossRefGoogle Scholar
  23. 23.
    Liu, Y.: Discrete Chaos in Fractional Henon Maps. Int. J. Nonlinear Sci. 18, 170–175 (2014)MathSciNetGoogle Scholar
  24. 24.
    Hu, J.B., Zhao, L.D.: Finite-time synchronization of fractional-order Chaotic Volta Systems with nonidentical orders. Math. Probl. Eng. (2013). doi: 10.1155/2013/264136 Google Scholar
  25. 25.
    Wu, G.C., Baleanu, D., Zeng, S.D.: Discrete chaos in fractional sine and standard maps. Phys. Lett. A 378, 484–487 (2014)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5, 367–386 (2002)MathSciNetMATHGoogle Scholar
  27. 27.
    El Gammoudi, I., Feki, M.: Synchronization of integer-order and fractional-order Chua’s systems using Robust observer. Commun. Nonlinear Sci. Numer. Simul. 18, 625–638 (2013)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kiani-B, A., Fallahi, K., Pariz, N., Leung, H.: A chaotic Secure communication scheme using fractional chaotic based on an extended fractional Kalman filter. Commun. Nonlinear Sci. Numer. Simul. 14, 863–879 (2009)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Hegazi, A.S., Ahmed, E., Matouk, A.E.: On chaos control and synchronization of the commensurate fractional-order Liu system. Commun. Nonlinear Sci. Numer. Simul. 18, 1193–1202 (2013)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Martinez-Guerra, R., Prez-Pinacho, C.A., Gómez-Corts, G.C.: Synchronization of Integral and Fractional Order Chaotic Systems: Adifferential Algebraic and Differential Geometric Approach Withselected Application in Real-Time. Understanding Complex System, Springer, Berlin (2015)CrossRefGoogle Scholar
  31. 31.
    Hamiche, H., Kassim, S., Djennoune, S., Guermah, S., Lahdir, M., Bettayeb, M.: Secure data transmission scheme based on fractional-order discrete chaotic system. In: International Conference on Control, Engineering and Information Technology (CEIT’2015). Tlemcen, Algeria (2015)Google Scholar
  32. 32.
    Xu, Y., Wang, H., Li, Y., Pei, B.: Image encryption based on synchronisation of fractional chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 19, 3735–3744 (2014)Google Scholar
  33. 33.
    Zhao, J., Wang, S., Chang, Y., Li, X.: A novel image encryption scheme based on an improper fractional-order chaotic system. Nonlinear Dyn. 80, 1721–1729 (2015)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Wang, Y.Q., Zhou, S.B.: Image encryption algorithm based on fractional-order Chen chaotic system. J. Comput. Appl. 33(4), 1043–1046 (2013)Google Scholar
  35. 35.
    Kassim, S., Megherbi, O., Hamiche, H., Djennoune, S., Lahdir, M., Bettayeb, M.: Secure image transmission scheme using hybrid encryption method. In: International Conference on Automatic Control. Telecommunications and Signals (ICATS’2015). Annaba, Algeria (2015)Google Scholar
  36. 36.
    Dzielinski, A., Sierociuk, D.: Adaptive feedback control of fractional order discrete state-space systems. In: Proceedings of the 2005 International Conference on Computational Intelligence for Modelling, Control and Automation, and International Conference on Intelligent Agents, Web Technologies and Internet Commerce (CIMCA,IAWTIC,05). Vienna Austria, pp. 804–809 (2005)Google Scholar
  37. 37.
    Nijmeijer, H., van der Schaft, A.J.: Nonlinear Dynamical Control Systems. Springer, New York (1990)CrossRefMATHGoogle Scholar
  38. 38.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica 16D, 285–317 (1985)MathSciNetMATHGoogle Scholar
  39. 39.
    Perruquetti, W., Barbot, J.-P.: Chaos in Automatic Control. CRC Press, Boca Raton (2006)Google Scholar
  40. 40.
    Sira-Ramirez, H., Rouchon, P.: Exact delayed reconstruction in nonlinear discrete-time system. In: European Union Nonlinear Control Network Workshop. June 25–27th. Sheffield. England (2001)Google Scholar
  41. 41.
    Hamiche, H., Ghanes, M., Barbot, J.P., Kemih, K., Djennoune, S.: Hybrid dynamical systems for private digital communications. Int. J. Model. Identif. Control 20, 99–113 (2013)CrossRefGoogle Scholar
  42. 42.
    Hamiche, H., Ghanes, M., Barbot, J.P., Kemih, K., Djennoune, S.: Chaotic synchronisation and secure communication via sliding-mode and impulsive observers. Int. J. Model. Identif. Control 20(4), 305–318 (2013)CrossRefGoogle Scholar
  43. 43.
    Djemaï, M., Barbot, J.-P., Belmouhoub, I.: Discrete-time normal form for left invertibility problem. Eur. J. Control 15, 194–204 (2009)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Zhao, L., Adhikari, A., Xiao, D., Sakurai, K.: On the security analysis of an image scrambling encryption of pixel bit and its improved scheme based on self-correlation encryption. Commun. Nonlinear Sci. Numer. Simul. 17(8), 3303–3327 (2012)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Jolfaei, A., Mirghadri, A.: Image encryption using chaos and block cipher. Comput. Inf. Sci. 4(1), 172–185 (2011)Google Scholar
  46. 46.
    El Assad, S., Farajallah, M.: A new chaos-based image encryption scheme. Signal Process. Image Commun. 41, 144–157 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • Sarah Kassim
    • 1
  • Hamid Hamiche
    • 1
  • Saïd Djennoune
    • 1
  • Maâmar Bettayeb
    • 2
  1. 1.Laboratoire de Conception et Conduite des Systèemes de Production (L2CSP)UMMTOTizi-OuzouAlgeria
  2. 2.Department of Electrical/Electronics and Computer EngineeringUniversity of SharjahSharjahUAE

Personalised recommendations