The European Physical Journal Special Topics

, Volume 226, Issue 10, pp 2281–2297 | Cite as

Cryptanalysis of a family of 1D unimodal maps

  • Mohamad Rushdan Md Said
  • Aliyu Danladi Hina
  • Santo Banerjee
Regular Article
Part of the following topical collections:
  1. Aspects of Statistical Mechanics and Dynamical Complexity

Abstract

In this paper, we proposed a topologically conjugate map, equivalent to the well known logistic map. This constructed map is defined on the integer domain [0, 2n) with a view to be used as a random number generator (RNG) based on an integer domain as is the required in classical cryptography. The maps were found to have a one to one correspondence between points in their respective defining intervals defined on an n-bits precision. The dynamics of the proposed map similar with that of the logistic map, in terms of the Lyapunov exponents with the control parameter. This similarity between the curves indicates topological conjugacy between the maps. With a view to be applied in cryptography as a Pseudo-Random number generator (PRNG), the complexity of the constructed map as a source of randomness is determined using both the permutation entropy (PE) and the Lempel-Ziv (LZ-76) complexity measures, and the results are compared with numerical simulations.

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References

  1. 1.
    T. Stojanovski, L. Kocarev, IEEE Trans. Fundam. Theory Applications: Circuits Syst. I 48, 281 (2001)Google Scholar
  2. 2.
    Z. Hong, D. Ji-Xue, 2010 International Conference on Computer Application and System Modeling (ICCASM, New York, 2010)Google Scholar
  3. 3.
    G. Heidari-Bateni, D.C. McGillem, IEEE Trans. Commun. 42, 1524 (1994)CrossRefGoogle Scholar
  4. 4.
    C. Grebogi, E. Ott, Phys. Rev. Lett. 70, 3031 (1993)ADSCrossRefGoogle Scholar
  5. 5.
    L. Kocarev, G. Jakimoski, Z. Tasev, Chaos Cont. 2003, 247 (2003)Google Scholar
  6. 6.
    R. Matthews, Cryptologia 13, 29 (1989)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D.W. Mitchell, Cryptologia, 14, 350 (1990)Google Scholar
  8. 8.
    T. Habutsu, Y. Nishio, I. Sasase, IEICE TRANSACTIONS (1976-1990), Inst. Elect. Inf. Commun. Eng. 73, 1041 (1990)Google Scholar
  9. 9.
    S. Li, X. Mou, Y. Cai, Phys. Lett. A 290, 127 (2001)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    J.C. Lagarias, Cryptology Comput. number Theory 42, 115 (1990)CrossRefGoogle Scholar
  11. 11.
    H. Nejati, A. Beirami, H.W. Ali, Analog Integrated Circuits Signal Proc. 73, 363 (2012)CrossRefGoogle Scholar
  12. 12.
    I. Cicek, A.E. Pusane, G. Dundar, Integration VLSI J. 47, 38 (2014)CrossRefGoogle Scholar
  13. 13.
    K.S. Park, W.K. Miller, Commun. ACM 31, 1192 (1998)CrossRefGoogle Scholar
  14. 14.
    C.E. Shannon, Bell Syst. Tech. J. 27, 379 (1948)CrossRefGoogle Scholar
  15. 15.
    Y. Sinai, Scholarpedia 4, 2034 (2009)ADSCrossRefGoogle Scholar
  16. 16.
    P. Collet, J.P. Crutchfield, J.-P. Eckmann, Commun. Math. Phys. 88, 257 (1983)ADSCrossRefGoogle Scholar
  17. 17.
    J. Smítal, Trans. Amer. Math. Soc. 297, 269 (1986)MathSciNetCrossRefGoogle Scholar
  18. 18.
    J. Amigó, Permutation Complexity in Dynamical Systems: Ordinal Patterns, Permutation Entropy and all that (Springer Science & Business Media, New York, 2010)Google Scholar
  19. 19.
    C. Bandt, B. Pompe, Phys. Rev. Lett. 88, 174102 (2002)ADSCrossRefGoogle Scholar
  20. 20.
    J.S. Cánovas, J.M. Rodrıguez, Marın, M. Ruiz, Int. J. Pure Appl. Math. 82, 163 (2013)Google Scholar
  21. 21.
    L. Liu, S. Miao, M. Cheng, X. Gao, Entropy 77, 8207 (2015)ADSCrossRefGoogle Scholar
  22. 22.
    A. Lempel, J. Ziv, IEEE Trans. Inf. Theory 22, 75 (1976)CrossRefGoogle Scholar
  23. 23.
    A.N. Kolmogorov, N. v Zhizni, T. Nauke, Ser. Matem. Kibern 1, 24 (1965)Google Scholar
  24. 24.
    C. Bandt, G. Keller, B. Pompe, Nonlinearity 15, 1595 (2002)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    A. Lasota, C.M. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics (Springer Science & Business Media, New York, 2013)Google Scholar
  26. 26.
    A. Baranovsky, D. Daems, Int. J. Bifurc. Chaos 5, 1585 (1995)CrossRefGoogle Scholar
  27. 27.
    Q. Ding, J. Pang, J. Fang, X.U. Peng, Int. Jour. Innov. Comput. Inform Control 3, 449 (2007)Google Scholar
  28. 28.
    K.A. Jain, NASA STI/Recon Tech. Rep. A 76, 42860 (1976)ADSGoogle Scholar
  29. 29.
    M. Unser, Signal Proc. 7, 231 (1984)CrossRefGoogle Scholar
  30. 30.
    M. Uenohara, T. Kanade, IEEE Trans. Image Proc. 7, 116 (1998)ADSCrossRefGoogle Scholar
  31. 31.
    D. Yang, H. Ai, C. Kyriakakis, C.-C.J. Kuo, IEEE Trans. Speech and Audio Proc. 11, 365 (2003)CrossRefGoogle Scholar
  32. 32.
    S. Zhu, S.-K.A. Yeung, B. Zeng, IEEE Signal Proc. Lett. 17, 961 (2010)ADSCrossRefGoogle Scholar
  33. 33.
    J.A. Menezes, C.P.V. Oorschot, A.S. Vanstone, Cryptography: An Introduction (McGraw-Hill, New York, 2003)Google Scholar
  34. 34.
    J.A. Menezes, C.P.V. Oorschot, A.S. Vanstone, Handbook of Applied Cryptography (CRC press, 1996)Google Scholar
  35. 35.
    M. Martelli, Introduction to Discrete Dynamical Systems and Chaos (John Wiley & Sons, New York, 2011)Google Scholar
  36. 36.
    M. Riedl, A. Müller, N. Wessel, Eur. Phys. J. Special Topics 222, 249 (2013)ADSCrossRefGoogle Scholar
  37. 37.
    J. Ziv, A. Lempel, IEEE Trans. Inf. Theory 24, 530 (1978)CrossRefGoogle Scholar
  38. 38.
    M.J. Amigó, J. Szczepański, E. Wajnryb, V.M. Sanchez-Vives, Neural Comput. 16, 717 (2004)CrossRefGoogle Scholar
  39. 39.
    C.A.J. Nunes, E. Estevez-Rams, B.A. Fernández, R.L. Serrano, arXiv:1311.0822 (2013)
  40. 40.
    E. Estevez-Rams, R.L. Serrano, B.A. Fernández, I.B. Reyes, Chaos: An Interdiscip. J. Nonlinear Sci. 23, 023118 (2013)CrossRefGoogle Scholar
  41. 41.
    Q. Ding, Y. Zhu, F. Zhang, X. Peng, Int. J. Commun. Inf. Tech. 2005. ISCIT 2005. IEEE Int. Symp. 2, 1043 (2005)CrossRefGoogle Scholar
  42. 42.
    T. Kohda, A. Tsuneda, Int. J. Inf. Theory, IEEE Trans. 43, 104 (1997)Google Scholar
  43. 43.
    D.U. Baoxiang, C. Geng, P. Fangyue, D. Jing, C. Qun, Chin. J. Elect. 22, 131 (2013)Google Scholar

Copyright information

© EDP Sciences and Springer 2017

Authors and Affiliations

  1. 1.Institute for Mathematical Research (INSPEM),Universiti Putra Malaysia (UPM)Serdang, SelangorMalaysia
  2. 2.Malaysia-Italy Centre Of Excellence For Mathematical Sciences (MICEMS), INSPEM, UPMSerdang, SelangorMalaysia

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