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Dynamics analysis and cryptographic application of fractional logistic map

  • Liguo YuanEmail author
  • Song Zheng
  • Zeeshan Alam
Original Paper
  • 4 Downloads

Abstract

Based on Lyapunov exponent, Schwarzian derivative, Shannon entropy and Kolmogorov entropy, we will firstly study chaos and bifurcation of fractional (semi-) logistic map (FLM). It is derived from fractional integration (not fractional derivative) of the classical logistic map. Then, this paper put forward a new accumulated coupled fractional (semi-) logistic map lattice (ACFLML) whose lattice function is the FLM. Local stability, pattern information, high-dimensional chaos and bifurcation of the ACFLML are analyzed based on stability theory, pattern simulation, 0–1 test for chaos, Lyapunov exponent spectrum and Kolmogorov entropy. Finally, the chaotic ACFLML is successfully applied to encryption and decryption of digital image. To evaluate security, histogram analysis, correlation analysis, differential attack, key space, key sensitivity, encryption time, computational complexity and chosen/known-plaintext attacks, analysis is performed. Simulation analysis shows that this encryption scheme is effective and has good statistical effect.

Keywords

Fractional logistic map Chaos Coupled map lattice Chaotic dynamics Image encryption 

Notes

Acknowledgements

The research is supported by the Open Project of Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing (No. 2017CSOBDP0302), the Science and Technology Program of Guangzhou (No. 201707010031), the National Natural Science Foundation of China (Nos. 11671149, 51777180, 11402226), the Natural Science Foundation of Zhejiang Province (No. LY17A020007), First Class Discipline of Zhejiang−A (Zhejiang University of Finance and Economics-Statistics) and the Preeminent Youth Fund of Zhejiang University of Finance and Economics and Higher Education Commission of Pakistan under Start-up Research Grant Project (SRGP) (\(\#\)1419). L. G. Yuan’s research is partially supported by China Scholarship Council.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Debnath, L.: Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 2003(54), 3413–3442 (2003)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, New York (2010)zbMATHGoogle Scholar
  3. 3.
    Petravs, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer, New York (2011)Google Scholar
  4. 4.
    Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D., Feliu-Batlle, V.: Fractional-Order Systems and Controls: Fundamentals and Applications. Springer, New York (2010)zbMATHGoogle Scholar
  5. 5.
    Yuan, L., Yang, Q., Zeng, C.: Chaos detection and parameter identification in fractional-order chaotic systems with delay. Nonlinear Dyn. 73(1–2), 439–448 (2013)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Alam, Z., Yuan, L., Yang, Q.: Chaos and combination synchronization of a new fractional-order system with two stable node-foci. IEEE/CAA J. Autom. Sin. 3(2), 157–164 (2016)MathSciNetGoogle Scholar
  7. 7.
    Yuan, L.G., Yang, Q.G.: Parameter identification and synchronization of fractional-order chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 17(1), 305–316 (2012)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chen, L., He, Y., Chai, Y., Wu, R.: New results on stability and stabilization of a class of nonlinear fractional-order systems. Nonlinear Dyn. 75(4), 633–641 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Yang, Q., Zeng, C.: Chaos in fractional conjugate Lorenz system and its scaling attractors. Commun. Nonlinear Sci. Numer. Simul. 15(12), 4041–4051 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Sweilam, N.H., Khader, M.M., Mahdy, A.M.: Numerical studies for fractional-order logistic differential equation with two different delays. J. Appl. Math. 2012 (2012)Google Scholar
  11. 11.
    El-Sayed, A., El-Mesiry, A., El-Saka, H.: On the fractional-order logistic equation. Appl. Math. Lett. 20(7), 817–823 (2007)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Nyamoradi, N., Javidi, M.: Dynamic analysis of a fractional-order Rikitake system. Dyn. Contin. Discrete Impuls. Syst. Ser. B 20(2), 189–204 (2013)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Zhen, W., Xia, H., Ning, L., Xiao-Na, S.: Image encryption based on a delayed fractional-order chaotic logistic system. Chin. Phys. B 21(5), 050,506 (2012)Google Scholar
  14. 14.
    Wu, G.C., Baleanu, D., Lin, Z.X.: Image encryption technique based on fractional chaotic time series. J. Vib. Control 22(8), 2092–2099 (2016)MathSciNetGoogle Scholar
  15. 15.
    Zhao, J., Wang, S., Chang, Y., Li, X.: A novel image encryption scheme based on an improper fractional-order chaotic system. Nonlinear Dyn. 80(4), 1721–1729 (2015)MathSciNetGoogle Scholar
  16. 16.
    Munkhammar, J.: Chaos in a fractional order logistic map. Fract. Calc. Appl. Anal. 16(3), 511–519 (2013)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kaneko, K.: Theory and Applications of Coupled Map Lattices. Wiley, New York (1993)zbMATHGoogle Scholar
  18. 18.
    Li, P., Li, Z., Halang, W.A., Chen, G.: Li–Yorke chaos in a spatiotemporal chaotic system. Chaos Solitons Fractals 33(2), 335–341 (2007)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Yuan, L.G., Yang, Q.G.: A proof for the existence of chaos in diffusively coupled map lattices with open boundary conditions. Discrete Dyn. Nat. Soc. 2011 (2011)Google Scholar
  20. 20.
    Vasegh, N.: Spatiotemporal and synchronous chaos in accumulated coupled map lattice. Nonlinear Dyn. 89(2), 1089–1097 (2017)zbMATHGoogle Scholar
  21. 21.
    Xie, F., Hu, G.: Spatiotemporal periodic pattern and propagated spatiotemporal on-off intermittency in the one-way coupled map lattice system. Phys. Rev. E 53(5), 4439 (1996)Google Scholar
  22. 22.
    Zhang, Y.Q., Wang, X.Y.: Spatiotemporal chaos in mixed linear-nonlinear coupled logistic map lattice. Phys. A Stat. Mech. Appl. 402, 104–118 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Zhang, Y.Q., He, Y., Wang, X.Y.: Spatiotemporal chaos in mixed linear-nonlinear two-dimensional coupled logistic map lattice. Phys. A Stat. Mech. Appl. 490, 148–160 (2018)MathSciNetGoogle Scholar
  24. 24.
    Zhang, Y.Q., Wang, X.Y.: A new image encryption algorithm based on non-adjacent coupled map lattices. Appl. Soft Comput. 26, 10–20 (2015)Google Scholar
  25. 25.
    May, R.M.: Simple mathematical models with very complicated dynamics. Nature 261(5560), 459 (1976)zbMATHGoogle Scholar
  26. 26.
    Zhang, Y.Q., Wang, X.Y., Liu, L.Y., He, Y., Liu, J.: Spatiotemporal chaos of fractional order logistic equation in nonlinear coupled lattices. Commun. Nonlinear Sci. Numer. Simul. 52, 52–61 (2017)MathSciNetGoogle Scholar
  27. 27.
    Dos Santos, A.M., Viana, R.L., Lopes, S.R., Pinto, SdS, Batista, A.M.: Chaos synchronization in a lattice of piecewise linear maps with regular and random couplings. Phys. A Stat. Mech. Appl. 367, 145–157 (2006)Google Scholar
  28. 28.
    Song, C.Y., Qiao, Y.L., Zhang, X.Z.: An image encryption scheme based on new spatiotemporal chaos. Optik Int. J. Light Electron Opt. 124(18), 3329–3334 (2013)Google Scholar
  29. 29.
    Ye, R., Zhou, W.: An image encryption scheme based on 2D tent map and coupled map lattice. Int. J. Inf. Commun. Technol. Res. 1(8) (2011)Google Scholar
  30. 30.
    Zhang, Y.Q., Wang, X.Y.: A symmetric image encryption algorithm based on mixed linear-nonlinear coupled map lattice. Inf. Sci. 273, 329–351 (2014)Google Scholar
  31. 31.
    Liu, S., Sun, F.: Spatial chaos-based image encryption design. Sci. China Ser. G: Phys. Mech. Astron. 52(2), 177–183 (2009)Google Scholar
  32. 32.
    Wang, X.Y., Wang, T.: A novel algorithm for image encryption based on couple chaotic systems. Int. J. Mod. Phys. B 26(30), 1250,175 (2012)Google Scholar
  33. 33.
    Fu-Yan, S., Zong-Wang, L.: Digital image encryption with chaotic map lattices. Chin. Phys. B 20(4), 040,506 (2011)Google Scholar
  34. 34.
    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. 1–14 (2019)Google Scholar
  35. 35.
    Ma, J., Wu, X., Chu, R., Zhang, L.: Selection of multi-scroll attractors in Jerk circuits and their verification using Pspice. Nonlinear Dyn. 76(4), 1951–1962 (2014)Google Scholar
  36. 36.
    Matthews, R.: On the derivation of a chaotic encryption algorithm. Cryptologia 13(1), 29–42 (1989)MathSciNetGoogle Scholar
  37. 37.
    Li, C., Li, S., Asim, M., Nunez, J., Alvarez, G., Chen, G.: On the security defects of an image encryption scheme. Image Vis. Comput. 27(9), 1371–1381 (2009)Google Scholar
  38. 38.
    Zhu, H., Zhang, X., Yu, H., Zhao, C., Zhu, Z.: An image encryption algorithm based on compound homogeneous hyper-chaotic system. Nonlinear Dyn. 89(1), 61–79 (2017)Google Scholar
  39. 39.
    Hua, Z., Jin, F., Xu, B., Huang, H.: 2D Logistic-Sine-coupling map for image encryption. Signal Process. 149, 148–161 (2018)Google Scholar
  40. 40.
    Li, C., Xie, T., Liu, Q., Cheng, G.: Cryptanalyzing image encryption using chaotic logistic map. Nonlinear Dyn. 78(2), 1545–1551 (2014)Google Scholar
  41. 41.
    Xie, E.Y., Li, C., Yu, S., Lü, J.: On the cryptanalysis of Fridrich’s chaotic image encryption scheme. Signal Process. 132, 150–154 (2017)Google Scholar
  42. 42.
    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), 401–412 (2016)MathSciNetGoogle Scholar
  43. 43.
    Ozkaynak, F.: Brief review on application of nonlinear dynamics in image encryption. Nonlinear Dyn. 92(2), 305–313 (2018)Google Scholar
  44. 44.
    Shannon, C.E.: A mathematical theory of communication. ACM SIGMOBILE Mob. Comput. Commun. Rev. 5(1), 3–55 (2001)MathSciNetGoogle Scholar
  45. 45.
    Arnold, L., Wihstutz, V.: Lyapunov Exponents: A Survey. Springer, New York (1986)zbMATHGoogle Scholar
  46. 46.
    Devaney, R.: An Introduction to Chaotic Dynamical Systems. Westview Press, Boulder (2008)Google Scholar
  47. 47.
    Schuster, H.G., Just, W.: Deterministic Chaos: An Introduction. Wiley, New York (2006)zbMATHGoogle Scholar
  48. 48.
    Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  49. 49.
    Garbaczewski, P., Olkiewicz, R.: Dynamics of Dissipation, vol. 597. Springer, New York (2002)zbMATHGoogle Scholar
  50. 50.
    Dorfman, J.R.: An Introduction to Chaos in Nonequilibrium Statistical Mechanics, vol. 14. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  51. 51.
    Andrecut, M., Ali, M.: Robust chaos in smooth unimodal maps. Phys. Rev. E 64(2), 025,203 (2001)MathSciNetGoogle Scholar
  52. 52.
    Banerjee, S., Verghese, G.C.: Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations, Chaos, and Nonlinear Control. IEEE Press, New York (2001)Google Scholar
  53. 53.
    Wang, X.: Period-doublings to chaos in a simple neural network: an analytical proof. Complex Syst. 5(4), 425–44 (1991)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (2013)zbMATHGoogle Scholar
  55. 55.
    Nusse, H.E., Yorke, J.A.: Period halving for \(x_n+1={M F}(x_n)\) where F has negative Schwarzian derivative. Phys. Lett. A 127(6–7), 328–334 (1988)MathSciNetGoogle Scholar
  56. 56.
    Devaney, R.L., Siegel, P.B., Mallinckrodt, A.J., McKay, S.: A first course in chaotic dynamical systems: theory and experiment. Comput. Phys. 7(4), 416–417 (1993)Google Scholar
  57. 57.
    Khellat, F., Ghaderi, A., Vasegh, N.: Li–Yorke chaos and synchronous chaos in a globally nonlocal coupled map lattice. Chaos Solitons Fractals 44(11), 934–939 (2011)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Gottwald, G.A., Melbourne, I.: A new test for chaos in deterministic systems. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 460(2042), 603–611 (2004)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Gottwald, G.A., Melbourne, I.: Testing for chaos in deterministic systems with noise. Physica D 212(1–2), 100–110 (2005)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Gottwald, G.A., Melbourne, I.: On the implementation of the 0–1 test for chaos. SIAM J. Appl. Dyn. Syst. 8(1), 129–145 (2009)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Gottwald, G.A., Melbourne, I.: On the validity of the 0–1 test for chaos. Nonlinearity 22(6), 1367 (2009)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Guedes, A.V., Savi, M.A.: Spatiotemporal chaos in coupled logistic maps. Phys. Scr. 81(4), 045,007 (2010)zbMATHGoogle Scholar
  63. 63.
    Zhang, Y.: Chaotic Digital Image Cryptosystem. Tsinghua University Press, Beijing (2016). (in Chinese)Google Scholar
  64. 64.
    Li, C., Lin, D., Lü, J.: Cryptanalyzing an image-scrambling encryption algorithm of pixel bits. IEEE MultiMedia 24(3), 64–71 (2017)Google Scholar
  65. 65.
    Ye, G.: Image scrambling encryption algorithm of pixel bit based on chaos map. Pattern Recognit. Lett. 31(5), 347–354 (2010)Google Scholar
  66. 66.
    Maniyath, S.R., Supriya, M.: An uncompressed image encryption algorithm based on DNA sequences. Comput. Sci. Inf. Technol. 2, 258–270 (2011)Google Scholar
  67. 67.
    Chen, G., Mao, Y., Chui, C.K.: A symmetric image encryption scheme based on 3D chaotic cat maps. Chaos Solitons Fractals 21(3), 749–761 (2004)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Wu, Y., Noonan, J.P., Agaian, S.: NPCR and UACI randomness tests for image encryption. Cyber J. Multidiscip. J. Sci. Technol. 1(2), 31–38 (2011)Google Scholar
  69. 69.
    Zhu, H., Zhang, X., Yu, H., Zhao, C., Zhu, Z.: A novel image encryption scheme using the composite discrete chaotic system. Entropy 18(8), 276 (2016)Google Scholar
  70. 70.
    Zhu, Z., Zhang, W., Wong, Kw, Yu, H.: A chaos-based symmetric image encryption scheme using a bit-level permutation. Inf. Sci. 181(6), 1171–1186 (2011)Google Scholar
  71. 71.
    Zhou, Y., Bao, L., Chen, C.P.: Image encryption using a new parametric switching chaotic system. Signal Process. 93(11), 3039–3052 (2013)Google Scholar
  72. 72.
    Alvarez, G., Li, S.: Some basic cryptographic requirements for chaos-based cryptosystems. Int. J. Bifurc. Chaos 16(08), 2129–2151 (2006)MathSciNetzbMATHGoogle Scholar
  73. 73.
    Li, C., Lin, D., Lü, J., Hao, F.: Cryptanalyzing an image encryption algorithm based on autoblocking and electrocardiography. IEEE MultiMedia 25(04), 46–56 (2018)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsSouth China Agricultural UniversityGuangzhouChina
  2. 2.Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data ProcessingYulin Normal UniversityYulinChina
  3. 3.Institute of Applied MathematicsZhejiang University of Finance and EconomicsHangzhouChina
  4. 4.Department of Mathematics, Statistics and Computer ScienceThe University of AgriculturePeshawarPakistan

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