The European Physical Journal Special Topics

, Volume 226, Issue 10, pp 2263–2280

Complexity and properties of a multidimensional Cat-Hadamard map for pseudo random number generation

Regular Article
Part of the following topical collections:
  1. Aspects of Statistical Mechanics and Dynamical Complexity


This paper presents a novel method to extend the Cat map from 2-dimension to higher dimension using the fast pseudo Hadamard Transform, and the resulted maps are called Cat-Hadamard maps. The complexity and properties of Cat-Hadamard maps are investigated under the point of view for cryptographic applications. In addition, we propose a method for constructing a pseudo random number generator using a novel design concept of the high dimensional Cat map. The simulation results show that the proposed generator fulfilled all the statistic tests of the NIST SP 800-90 A.


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  1. 1.
    V.I. Arnold, A. Avez, Ergodic Problems in Classical Mechanics (Benjamin, New York, 1968)Google Scholar
  2. 2.
    G. Peterson, Arnold’s cat map, Math45-Linear algebra (1997)Google Scholar
  3. 3.
    J. Fridrich, Symmetric ciphers based on two-dimensional chaotic maps, Int. J. Bifurc. Chaos. 8, 1259 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    S. Lian, J. Sun, Z. Wang, Security analysis of a chaos-based image encryption algorithm, Physica A 351, 645 (2005)ADSCrossRefGoogle Scholar
  5. 5.
    G. Ye, K.-W. Wong, An efficient chaotic image encryption algorithm based on a generalized Arnold map, Nonlinear Dyn. 69, 2079 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    M. Keyvanpour, F. Merrikh-Bayat, An effective chaos-based image watermarking scheme using fractal coding, Proc. Comput. Sci. 3, 89 (2011)CrossRefGoogle Scholar
  7. 7.
    W.K. Tang, Y. Liu, Formation of high-dimensional chaotic maps and their uses in cryptography, in Chaos-Based Cryptography (Springer, 2011), pp. 99–136Google Scholar
  8. 8.
    M. Falcioni, L. Palatella, S. Pigolotti, A. Vulpiani, Properties making a chaotic system a good pseudo random number generator, Phys. Rev. E 72, 016220 (2005)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    L. Kocarev, S. Lian, Chaos-based Cryptography (Springer, 2011)Google Scholar
  10. 10.
    N. Masuda, K. Aihara, Cryptosystems with discretized chaotic maps, Circuits and Systems I: Fundamental Theo. Appl. IEEE Trans. 49, 28 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    F. Chen, X. Liao, K.-w. Wong, Q. Han, Y. Li, Period distribution analysis of some linear maps, Commun. Nonlin. Sci. Numer. Simul. 17, 3848 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    F. Chen, K.-W. Wong, X. Liao, T. Xiang, Period distribution of the generalized discrete Arnold cat map for N = 2e, IEEE Trans. Inform. Theo. Acoustics Speech Signal Proc. 59, 3249 (2013)MathSciNetMATHGoogle Scholar
  13. 13.
    F.J. Dyson, H. Falk, Period of a discrete cat mapping, Amer. Math. Monthly 99, 603 (1992)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    F. Chen, K.-W. Wong, X. Liao, T. Xiang, Period distribution of generalized discrete Arnold cat map, Theor. Comp. Sci. 552, 13 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    J. Bao, Q. Yang, Period of the discrete Arnold cat map and general cat map, Nonlinear Dyn. 70, 1365 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    F. Svanström, Properties of a generalized Arnold’s discrete cat map,, 2014
  17. 17.
    S. Gao, A. Lauder, Hensel lifting and bivariate polynomial factorisation over finite fields, Math. Comput. 71, 1663 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    K.H. Sze, High-Dimensional Chaotic Map: Formulation, Nature and Applications, PhD dissertation, City University of Hong Kong, 2007Google Scholar
  19. 19.
    J. Houlrik, M. Jensen, Theory and Applications of Coupled Map Lattices (Wiley, 1993)Google Scholar
  20. 20.
    G. Grassi, S. Mascolo, A systematic procedure for synchronizing hyperchaos via observer design, J. Circ. Syst. Comput. 11, 1 (2002)CrossRefGoogle Scholar
  21. 21.
    J. Nance, Periods of the discretized Arnold Cat map and its extension to n dimensions, arXiv:1111.2984 (2011)
  22. 22.
    W. Just, Bifurcations in globally coupled map lattices, J. Stat. Phys. 79, 429 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    J. Kelsey, B. Schneier, D. Wagner, C. Hall, Cryptanalytic attacks on pseudorandom number generators, in International Workshop on Fast Software Encryption (Springer, 1998), pp. 168–188Google Scholar
  24. 24.
    T. St Denis, Fast pseudo-Hadamard transforms, Tech. Rep., Cryptology ePrint Archive, Report 2004-010, 2004Google Scholar
  25. 25.
    U. Schwengelbeck, F. Faisal, Definition of Lyapunov exponents and KS entropy in quantum dynamics, Phys. Lett. A 199, 281 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    H.V. Henderson, F. Pukelsheim, S.R. Searle, On the history of the Kronecker product, Linear and Multilinear Algebra 14, 113 (1983)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    A.N. Langville, W.J. Stewart, The Kronecker product and stochastic automata networks, J. Comput. Appl. Math. 167, 429 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    L. Kocarev, J. Szczepanski, J.M. Amigo, I. Tomovski, Discrete chaos-i: Theory, IEEE Trans. Circuits and Systems I: Regular Papers 53, 1300 (2006)MathSciNetCrossRefGoogle Scholar
  29. 29.
    P.D. Powell, Calculating determinants of block matrices, arXiv:1112.4379 (2011)
  30. 30.
    D.D. Wall, Fibonacci series modulo m, Amer. Mathematical Monthly 67, 525 (1960)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    D. Ruelle, Chaotic Evolution and Strange Attractors, Cambridge Books Online (Cambridge University Press, 1989)Google Scholar
  32. 32.
    D. Ruelle, Chaotic Evolution and Strange Attractors (Cambridge University Press, 1989)Google Scholar
  33. 33.
    P. L’Ecuyer, Uniform random number generation, Annals Operations Res. 53, 77 (1994)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    L.E. Bassham III, A.L. Rukhin, J. Soto, J.R. Nechvatal, M.E. Smid, E.B. Barker, S.D. Leigh, M. Levenson, M. Vangel, D.L. Banks, et al., Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications, National Institute of Standards & Technology, 2010Google Scholar

Copyright information

© EDP Sciences and Springer 2017

Authors and Affiliations

  1. 1.School of Electronics and Telecommunications, Hanoi University of Science and TechnologyHanoiVietnam

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