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Chaotic Maps: Applications to Cryptography and Network Generation for the Graph Laplacian Quantum States

  • Anoopa JoshiEmail author
  • Atul Kumar
Conference paper
  • 54 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 307)

Abstract

In this article, we proposed a new chaotic map and is compared with existing chaotic maps such as Logistic map and Tent map. The value of maximal Lyapunov exponent of the proposed chaotic map goes beyond 1 and shows more chaotic behaviour than existing one-dimensional chaotic maps. This shows that proposed chaotic maps are more effective for cryptographic applications. Further, we are using one-dimensional chaotic maps to generate random time series data and define a method to create a network. Lyapunov exponent and entropy of the data are considered to measure the randomness or chaotic behaviour of the time series data. We study the relationship between concurrence (for the two-qubit quantum states) and Lyapunov exponent with respect to initial condition and parameter of the logistic map which is showing how chaos can lead to concurrence based on such Lyapunov exponents.

Keywords

Cryptography Logistic map Lyapunov exponent. 

Notes

Acknowledgements

The authors are grateful to Satish Sangwan for valuable comments and suggestions.

References

  1. 1.
    M. Berezowski, M. Lawnik, Identification of fast-changing signals by means of adaptive chaotic transformations (2016), arXiv:1603.06763
  2. 2.
    M. Lawnik, M. Berezowski, Identification of the oscillation period of chemical reactors by chaotic sampling of the conversion degree. Chemical and Process Engineering 35(3), 387–393 (2014)CrossRefGoogle Scholar
  3. 3.
    S.H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (CRC Press, 2018)Google Scholar
  4. 4.
    W.F.H. Al-Shameri, M.A. Mahiub, Some dynamical properties of the family of tent maps. Int. J. Math. Anal. 7(29), 1433–1449 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    L. Shan, H. Qiang, J. Li, Z. Wang, Chaotic optimization algorithm based on tent map. Control Decis. 20(2), 179–182 (2005)zbMATHGoogle Scholar
  6. 6.
    T. Yoshida, H. Mori, H. Shigematsu, Analytic study of chaos of the tent map: band structures, power spectra, and critical behaviors. J. Stat. Phys. 31(2), 279–308 (1983)MathSciNetCrossRefGoogle Scholar
  7. 7.
    L. Kocarev, G. Jakimoski, Logistic map as a block encryption algorithm. Phys. Lett. A 289(4–5), 199–206 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    S.C. Phatak, S. Suresh Rao, Logistic map: a possible random-number generator. Phys. Rev. E 51(4), 3670 (1995)CrossRefGoogle Scholar
  9. 9.
    M.S. Baptista, Cryptography with chaos. Phys. Lett. A 240(1–2), 50–54 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    N.K. Pareek, V. Patidar, K.K. Sud, Image encryption using chaotic logistic map. Image Vis. Comput. 24(9), 926–934 (2006)CrossRefGoogle Scholar
  11. 11.
    T.P. Spiller, Quantum information processing: cryptography, computation, and teleportation. Proc. IEEE 84(12), 1719–1746 (1996)CrossRefGoogle Scholar
  12. 12.
    C.H. Bennett, G. Brassard, N.D. Mermin, Quantum cryptography without bell’s theorem. Phys. Rev. Lett. 68(5), 557 (1992)MathSciNetCrossRefGoogle Scholar
  13. 13.
    P.W. Shor, J. Preskill, Simple proof of security of the BB84 quantum key distribution protocol. Phys. Rev. Lett. 85(2), 441 (2000)CrossRefGoogle Scholar
  14. 14.
    A.K. Ekert, Quantum cryptography based on bell’s theorem. Phys. Rev. Lett. 67(6), 661 (1991)MathSciNetCrossRefGoogle Scholar
  15. 15.
    P.C. Müller, Calculation of lyapunov exponents for dynamic systems with discontinuities. Chaos, Solitons, Fractals, 5(9), 1671–1681 (1995)MathSciNetCrossRefGoogle Scholar
  16. 16.
    A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining lyapunov exponents from a time series. Phys. D: Nonlinear Phenom. 16(3), 285–317 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    C. Jin, H. Liu, A color image encryption scheme based on arnold scrambling and quantum chaotic. IJ Netw. Secur. 19(3), 347–357 (2017)Google Scholar
  18. 18.
    A.G. Radwan, S.K. Abd-El-Hafiz, Image encryption using generalized tent map, in 2013 IEEE 20th International Conference on Electronics, Circuits, and Systems (ICECS) (IEEE, 2013), pp. 653–656Google Scholar
  19. 19.
    X. Zhang, Y. Cao, A novel chaotic map and an improved chaos-based image encryption scheme. Sci. World J. 2014, (2014)Google Scholar
  20. 20.
    E. Ceyhan, Edge density of new graph types based on a random digraph family. Stat. Methodol. 33, 31–54 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Pradumn Kumar Pandey and Bibhas Adhikari, Context dependent preferential attachment model for complex networks. Phys. A: Stat. Mech. Its Appl. 436, 499–508 (2015)CrossRefGoogle Scholar
  22. 22.
    R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement. Rev. Mod. Phys. 81(2), 865 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80(10), 2245 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Indian Institute of Technology JodhpurJodhpurIndia

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