Multimedia Tools and Applications

, Volume 77, Issue 16, pp 21445–21462 | Cite as

A new simple one-dimensional chaotic map and its application for image encryption

  • Lingfeng Liu
  • Suoxia Miao


In this paper, we propose a new simple one-dimensional chaotic map. The chaotic characteristics have been declared by using bifurcation analysis and Lyapunov exponent analysis. Furthermore, we propose a new image encryption algorithm based on this new chaotic map. Both shuffling algorithm and substitution algorithm are related to this map. Many statistical tests and security analysis indicate that this algorithm has an excellent security performance, and can be competitive with some other recently proposed image encryption algorithms.


Chaos One-dimensional chaotic map Bifurcation Lyapunov exponent Image encryption 



This work is supported by the National Natural Science Foundation of China (61601215).


  1. 1.
    Alpar O (2014) Analysis of a new simple one dimensional chaotic map. Nonlinear Dyn 78:771–778MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cang SJ, Wang ZH, Chen ZQ, Jia HY (2014) Analytical and numerical investigation of a new Lorenz-like chaotic attractor with compound structures. Nonlinear Dyn 75:745–760MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen G, Mao Y, Chui CK (2004) A symmetric image encryption scheme based on 3D chaotic cat maps. Chaos, Solitons Fractals 21:749–761MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen CS, Wang T, Kou YZ, Chen XC, Li X (2013) Improvement of trace-driven I-Cache timing attack on the RSA algorithm. J Syst Softw 86(1):100–107CrossRefGoogle Scholar
  5. 5.
    Coppersmith D (1994) The data encryption standard (DES) and its strength against attacks. IBM J Res Dev 38(3):243–250CrossRefzbMATHGoogle Scholar
  6. 6.
    Devaney R (1984) A piecewise linear model for zones of instability of an area-preserving map. Phys D 10(3):387–393MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Henon M (1976) A two-dimensional mapping with a strange attractor. Math Phys 50:69–77MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hua ZY, Zhou YC, Pun CM, Philip Chen CL (2015) 2D Sine Logistic modulation map for image encryption. Inf Sci 297:80–94CrossRefGoogle Scholar
  9. 9.
    Kaneko K (1993) Theory and application of coupled map lattices. John Wiley and Sons, Nwe YorkzbMATHGoogle Scholar
  10. 10.
    Li CQ, Xie T, Liu Q, Cheng G (2014) Cryptanalyzing image encryption using chaotic logistic map. Nonlinear Dyn 78:1545–1551CrossRefGoogle Scholar
  11. 11.
    Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20:130–141CrossRefzbMATHGoogle Scholar
  12. 12.
    Lu JH, Chen GR (2002) A new chaotic attractor coined. Int J Bifurcat Chaos 12:659–661MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261:459–465CrossRefzbMATHGoogle Scholar
  14. 14.
    Sprott J (2003) Chaos and time series analysis. Oxford University Press, OxfordzbMATHGoogle Scholar
  15. 15.
    Sun F, Lu Z, Liu S (2010) A new cryptosystem based on spatial chaotic system. Opt Commun 283:2066–2073CrossRefGoogle Scholar
  16. 16.
    Tong XJ, Wang Z, Zhang M, Liu Y, Xu H, Ma J (2015) An image encryption algorithm based on the perturbed high-dimensional chaotic map. Nonlinear Dyn 80:1493–1508MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wang XY, Guo K (2014) A new image alternate encryption algorithm based on chaotic map. Nonlinear Dyn 76:1943–1950CrossRefzbMATHGoogle Scholar
  18. 18.
    Ye G, Wong KW (2013) An image encryption scheme based on time-delay and hyperchaotic system. Nonlinear Dyn 71(1–2):259–267MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zhang YQ, Wang XY (2014) Spatiotemporal chaos in mixed linear-nonlinear coupled logisitic map lattice. Physica A 402:104–118MathSciNetCrossRefGoogle Scholar
  20. 20.
    Zhou Q, Liao X (2012) Collision-based flexible image encryption algorithm. J Syst Softw 85:400–407CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of SoftwareNanchang UniversityNanchangPeople’s Republic of China
  2. 2.Faculty of ScienceNanchang Institute of TechnologyNanchangPeople’s Republic of China

Personalised recommendations