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Construction of Non-linear Component of Block Cipher by Means of Chaotic Dynamical System and Symmetric Group

  • Adnan JaveedEmail author
  • Tariq Shah
  • Atta Ullah
Article

Abstract

The interesting features of chaos theory are utilized now a day’s in information security. The simplest chaotic dynamical system is the double pendulum. Here in this article, two double pendulums are used to enhance the chaotic behavior of a dynamical system. This system is sensitive to initial conditions and bears complex and chaotic trajectory. Moreover, being multi dimensional system it endures grander solution space for the generation of large number of S-boxes. Furthermore, a permutation comprising on only two cycles of symmetric group of order 256 is applied to generate integer values for the construction of desired substitution box. The algebraic analysis of suggested S-box emphasis on its application, thereafter, an image is encrypted with the help of this S-box, whose statistical analysis validates its efficacy.

Keywords

Chaotic dynamical system Symmetric group Substitution box (S-box) Image encryption 

Notes

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Shannon, C. E. (1949). Communication theory of secrecy systems. Bell Systems Technical Journal,28, 656–715.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Kocarev, L. (2001). Chaos-based cryptography: A brief overview. IEEE Circuits and Systems Magazine,1, 6–21.CrossRefGoogle Scholar
  3. 3.
    Dachselt, F., & Schwarz, W. (2001). Chaos and cryptography. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications,48(12), 1498–1509.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Khan, M., Shah, T., & Batool, S. I. (2016). Construction of S-box based on chaotic Boolean functions and its application in image encryption. Neural Computing and Applications,27(3), 677–685.CrossRefGoogle Scholar
  5. 5.
    Zhou, Y., Bao, L., & Chen, C. L. P. (2014). A new 1D chaotic system for image encryption. Signal Processing,97, 172–182.CrossRefGoogle Scholar
  6. 6.
    Jakimoski, G., & Kocarev, L. (2001). Chaos and cryptography: Block encryption ciphers. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications,48(2), 163–169.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ullah, A., Jamal, S. S., & Shah, T. (2017). A novel construction of substitution box using a combination of chaotic maps with improved chaotic range. Nonlinear Dynamics.  https://doi.org/10.1007/s11071-017-3409-1.CrossRefGoogle Scholar
  8. 8.
    Li, X., Wang, L., Yan, Y., & Liu, P. (2016). An improvement color image encryption algorithm based on DNA operations and real and complex chaotic systems. Optik-International Journal for Light and Electron Optics,127(5), 2558–2565.CrossRefGoogle Scholar
  9. 9.
    Hussain, I., Shah, T., & Gondal, M. A. (2012). A novel approach for designing substitution-boxes based on nonlinear chaotic algorithm. Nonlinear Dynamics,70(3), 1791–1794.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Khan, M., & Shah, T. (2014). A novel image encryption technique based on Henon chaotic map and S8 symmetric group. Neural Computing and Applications,25(7), 1717–1722.CrossRefGoogle Scholar
  11. 11.
    Zhang, Y., & Xiao, D. (2014). Self-adaptive permutation and combined global diffusion or chaotic color image encryption. International Journal of Electronics and Communications,68(4), 361–368.CrossRefGoogle Scholar
  12. 12.
    Zhang, W., Yu, H., Zhao, Y., & Zhu, Z. (2016). Image encryption based on three-dimensional bit matrix permutation. Signal Processing,118, 36–50.CrossRefGoogle Scholar
  13. 13.
    Özkaynak, F., & Özer, A. B. (2010). A method for designing strong S-boxes based on chaotic Lorenz system. Physics Letters A,374(36), 3733–3738.zbMATHCrossRefGoogle Scholar
  14. 14.
    Brown, R., & Chua, L. O. (1996). Clarifying chaos: examples and counter examples. International Journal of Bifurcation and Chaos,6(2), 219–242.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Fridrich, J. (1998). Symmetric ciphers based on two-dimensional chaotic maps. International Journal of Bifurcation and Chaos,8(6), 1259–1284.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Tang, G., Liao, X., & Chen, Y. (2005). A novel method for designing S-boxes based on chaotic maps. Chaos, Solitons & Fractals,23(2), 413–419.zbMATHCrossRefGoogle Scholar
  17. 17.
    Chen, G., Chen, Y., & Liao, X. (2007). An extended method for obtaining S-boxes based on 3-dimensional chaotic baker maps. Chaos, Solitons & Fractals,31(3), 571–579.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Arroyo, D., Diaz, J., & Rodriguez, F. B. (2013). Cryptanalysis of a one round chaos-based substitution permutation network. Signal Processing,93(5), 1358–1364.CrossRefGoogle Scholar
  19. 19.
    Ullah, A., Javeed, A., & Shah, T. (2019). A scheme based on algebraic and chaotic structures for the construction of substitution box. Multimedia Tools and Applications.  https://doi.org/10.1007/s11042-019-07957-8.CrossRefGoogle Scholar
  20. 20.
    Khan, M., Shah, T., Mahmood, H., Gondal, M. A., & Hussain, I. (2012). A novel technique for the construction of strong S-boxes based on chaotic Lorenz systems. Nonlinear Dynamics,70(3), 2303–2311.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Javeed, A., Shah, T., & Ullah, A. Design of an S-box using Rabinovich–Fabrikant system of differential equations perceiving third order nonlinearity. Multimedia Tools and Applications,  https://doi.org/10.1007/s11042-019-08393-4.
  22. 22.
    Khan, M., Shah, T., Mahmood, H., & Gondal, M. A. (2013). An efficient method for the construction of block cipher with multi chaotic systems. Nonlinear Dynamics,71(3), 489–492.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ullah, A., Jamal, S. S., & Shah, T. (2018). A novel scheme for image encryption using substitution box and chaotic system. Nonlinear Dynamics,91(1), 359–370.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Khan, M., & Asghar, Z. (2018). A novel construction of substitution box for image encryption applications with Gingerbreadman chaotic map and S8 permutation. Neural Comput & Applications,29(4), 993–999.CrossRefGoogle Scholar
  25. 25.
    Ahmad, M., Doja, M. N., & Beg, M. M. S. (2018). ABC optimization based construction of strong substitution-boxes. Wireless Personal Communications,101(3), 1715–1729.CrossRefGoogle Scholar
  26. 26.
    Razaq, A., Yousaf, A., Shuaib, U., Siddiqui, N., Ullah, A., & Waheed, A. (2017). A novel construction of substitution box involving coset diagram and a bijective map. Security and Communication Networks,2017, 5101934.CrossRefGoogle Scholar
  27. 27.
    Shah, T., & Shah, D. (2019). Construction of highly nonlinear S-boxes for degree 8 primitive irreducible polynomials over Z 2. Multimdeia Tools and Applications,78(2), 1219–1234.CrossRefGoogle Scholar
  28. 28.
    Khan, M., & Munir, N. (2019). A novel Image encryption technique based on generalized advanced encryption standard based on field of any characteristic. Wireless and Personal Communication,109(2), 849–867.CrossRefGoogle Scholar
  29. 29.
    Wang, X. Y., Feng, L., & Zhao, H. (2019). Fast image encryption algorithm based on parallel computing system. Information Sciences,486, 340–358.CrossRefGoogle Scholar
  30. 30.
    Khan, M., Hussain, I., Jamal, S. S., & Amin, M. (2019). A privacy scheme for digital images based on quantum particles. Intrnational Journal of Theoretical Physics.  https://doi.org/10.1007/s10773-019-04301-6.zbMATHCrossRefGoogle Scholar
  31. 31.
    Wang, X. Y., & Gao, S. (2020). Image encryption algorithm for synchronously updating Boolean networks based on matrix semi-tensor product theory. Information Sciences,507, 16–36.MathSciNetCrossRefGoogle Scholar
  32. 32.
    Webster, A. F., & Tavares, S. (1986). On the design of S-boxes. In: Advances in cryptology: Proceedings of CRYPTO’85. Lecture Notes in Computer Science, pp. 523–534.Google Scholar
  33. 33.
    Hussain, I., Shah, T., Gondal, M. A., & Mahmood, H. (2012). Generalized majority logic criterion to analyze the statistical strength of S-boxes. Zeitschrift für Naturforschung A,67, 282–288.CrossRefGoogle Scholar
  34. 34.
    Belazi, A., Khan, M., El-Latif, A. A., & Belghith, S. (2016). Efficient cryptosystem approaches: S-boxes and permutation–substitution-based encryption. Nonlinear Dynamics,87, 337–361.CrossRefGoogle Scholar
  35. 35.
    Daemen, J., & Rijmen, V. (2002). The design of Rijndael-AES: The advanced encryption standard. Berlin: Springer.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan

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