A Method of Generating 8 × 8 Substitution Boxes Based on Elliptic Curves

Article

Abstract

Elliptic curve cryptography provides better security and is more efficient as compared to other public key cryptosystems with identical key size. In this article, we present a new method for the construction of substitution boxes(S-boxes) based on points on elliptic curve over prime field. The resistance of the newly generated S-box against common attacks such as linear, differential and algebraic attacks is analyzed by calculating their non-linearity, linear approximation, strict avalanche, bit independence, differential approximation and algebraic complexity. The experimental results are further compared with some of the prevailing S-boxes presented in Shi et al. (Int Conf Inf Netw Appl 2:689–693, 1997), Jakimoski and Kocarev (IEEE Trans Circuits Syst I 48:163–170, 2001), Guoping et al. (Chaos, Solitons Fractals 23:413–419, 2005), Guo (Chaos, Solitons Fractals 36:1028–1036, 2008), Kim and Phan (Cryptologia 33: 246–270, 2009), Neural et al. (2010 sixth international conference on natural computation (ICNC 2010), 2010), Hussain et al. (Neural Comput Appl.  https://doi.org/10.1007/s00521-012-0914-5, 2012). Comparison reveals that the proposed algorithm generates cryptographically strong S-boxes as compared to some of the other exiting techniques.

Keywords

Elliptic curve Substitution box Non-linearity Differential approximation probability Algebraic complexity 

Notes

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Shi, X. Y., Xiao, H., You, X. C., & Lam, K. Y. (1997). A method for obtaining cryptographically strong 8 × 8 S-boxes. International Conference on Information Network and Application, 2, 689–693.Google Scholar
  2. 2.
    Jakimoski, G., & Kocarev, L. (2001). Chaos and cryptography: block encryption ciphers. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 48, 163–170.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Guoping, T., Xiaofeng, L., & Yong, C. (2005). A novel method for designing S-boxes based on chaotic maps. Chaos, Solitons & Fractals, 23, 413–419.CrossRefMATHGoogle Scholar
  4. 4.
    Guo, C. (2008). A novel heuristic method for obtaining S-boxes. Chaos, Solitons & Fractals, 36, 1028–1036.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Kim, J., & Phan, R. C. W. (2009). Advanced differential-style cryptanalysis of the NSA’s skipjack block cipher. Cryptologia, 33, 246–270.CrossRefMATHGoogle Scholar
  6. 6.
    Neural, Y. W., Li, Y., Min, L., & Sihong, S. A method for designing S-box based on chaotic neural network. In 2010 Sixth international conference on natural computation (ICNC 2010).Google Scholar
  7. 7.
    Hussain, I., Shah, T., Gondal, M. A., Khan, W. A., & Mehmood, H. (2012). A group theoretic approach to construct cryptographically strong substitution boxes. Neural Computing and Applications.  https://doi.org/10.1007/s00521-012-0914-5.Google Scholar
  8. 8.
    Hussain, I., Azam, N. A., & Shah, T. (2014). Stego optical encryption based on chaotic S-box transformation. Optics & Laser Technology, 61, 50–56.CrossRefGoogle Scholar
  9. 9.
    Shannon, C. E. (1949). Communications theory of secrecy systems. Bell Labs Technical Journal, 20, 656–715.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Willi, M., & Othmar, S. (1990). Nonlinearity criteria for cryptographic functions. Advances in Cryptology–EUROCRYPT ’89 LNCS, 434, 549–562.MathSciNetMATHGoogle Scholar
  11. 11.
    Mitsuru, M. (1994). Linear cryptanalysis method for DES cipher. Advances in Cryptology–EUROCRYPT ‘93 LNCS, 765, 386–397.MATHGoogle Scholar
  12. 12.
    Eli, B., & Adi, S. (1991). Differential crypt analysis of DES-like cryptosystems. Advances in Cryptology - CRYPTO ‘90 LNCS, 537, 2–21.MATHGoogle Scholar
  13. 13.
    Thomas, J., & Knudsen, L, R. (1997). The interpolation attack on block ciphers. In International workshop on fast software encription (FSE), Fast Software Encription (pp. 28–40).Google Scholar
  14. 14.
    Nicolas, C., Alexander, K., Jacques, P., & Adi, S. (2000). Effcient algorithms for solving overdefined systems of multivariate polynomial equations. In International conference on the theory and application of cryptographic techniques EUROCRYPT 2000: advances in cryptology-EUROCRYPT (pp. 392–407).Google Scholar
  15. 15.
    Courtois, N. T., & Josef, P. (2002). Cryptanalysis of block ciphers with overdefined systems of equations. ASIACRYPT 2002 LNCS, 2501, 267–287.MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Daemen, J., & Rijmen, V. (1999). AES proposal: Rijndael (Version 2). NIST AES, csrc.nist.gov/encryption/aes.Google Scholar
  17. 17.
    Ferguson, N., Schroeppel, R., & Whiting, D. A. (2001). Simple algebraic representation of Rijndael. In Selected areas in cryptography SAC 01, LNCS 2259 (pp. 103–111).Google Scholar
  18. 18.
    Murphy, S., & Robshaw, M. J. (2002). Essential algebraic structure within the AES. In Proceedings of the 22th annual international cryptology (pp. 1–16). Berlin: Springer.Google Scholar
  19. 19.
    Rosenthal, J. (2003). A polynomial description of the Rijndael advanced encryption standard. Journal of Algebra and its Applications, 2, 223–236.MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Liu, J., Wai, B., Cheng, X., & Wang, X. (2005). An AES S-box to increase complexity and cryptographic analysis. In Proceedings of the 19th international conference on advanced information networking and applications, Taiwan (pp. 724–728).Google Scholar
  21. 21.
    Cui, L., & Cao, Y. (2007). A new S-box structure named affine power-affine. International Journal of Innovative Computing, Information and Control, 3, 751–759.Google Scholar
  22. 22.
    Tran, M. T., Bui, D. K., & Doung, A. D. (2008). Gray S-box for advanced encryption standard. International Conference on Computational Intelligence and Security, 1, 253–258.Google Scholar
  23. 23.
    Khan, M., & Azam, N. A. (2014). Right translated AES Gray S-box. Security and Network Communication.  https://doi.org/10.1002/sec.1110.Google Scholar
  24. 24.
    Khan, M., & Azam, N. A. (2015) S-boxes based on affine mapping and orbit of power function. 3D Research.  https://doi.org/10.1007/s13319-015-0043-x.
  25. 25.
    Hao, Y., Longyan, L., & Yong, W. (2010). An S-box construction algorithm based on spatiotemporal chaos. In International conference on communications and mobile computing.Google Scholar
  26. 26.
    Yong, W., Kwok, W., Changbing, L., & Yang, L. (2012). A novel method to design S-box based on chaotic map and genetic algorithm. Physics Letters A, 376, 827–833.CrossRefMATHGoogle Scholar
  27. 27.
    Wang, Y., Wong, K. W., Li, C., & Li, Y. (2012). A novel method to design S-box based on chaotic map and genetic algorithm. Physics Letters A, 376(376), 827–833.CrossRefMATHGoogle Scholar
  28. 28.
    Hussain, I., Azam, N. A., & Shah, T. (2014). Stego optical encryption based on chaotic S-box transformation. Optics and Laser Technology, 61, 50–56.CrossRefGoogle Scholar
  29. 29.
    Khan, M., Shah, T., & Syeda, I. B. (2016). Construction of S-box based on chaotic Boolean functions and its application in image encryption. Neural Computing and Applications, 27, 677–685.  https://doi.org/10.1007/s00521-015-1887-y.CrossRefGoogle Scholar
  30. 30.
    Vijayan, P., Paul, V., & Wahi, A. (2017). Dynamic colour table: A novel S-box for cryptographic applications. International Journal of Communication Systems.  https://doi.org/10.1002/dac.3318.
  31. 31.
    Özkaynak, F., Çelik, V., & Özer, A. B. (2017). A new S-box construction method based on the fractional-order chaotic Chen system. SIViP, 11, 659.  https://doi.org/10.1007/s11760-016-1007-1.CrossRefGoogle Scholar
  32. 32.
    Miller, V. (1986). Uses of elliptic curves in cryptography. Advances in Cryptology, 85, 417–426.MathSciNetGoogle Scholar
  33. 33.
    Neal, K. (1987). Elliptic curve cryptosystems. Mathematics of Computation, 48(177), 203–209.MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Jung, H. C., Seongtaek, C., & Choonsik, P. (1999). S-boxes with controllable nonlinearity, EUROCRYPT’99. LNCS, 1592, 286–294.MATHGoogle Scholar
  35. 35.
    Neal, K., Alfred, M., & Scott, V. (2000). The state of elliptic curve cryptography. Designs, Codes and Cryptography, 19, 173–193.MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Amara, M., & Siad, A.(2011). Elliptic curve cryptography and its applications. In 7th international workshop on systems, signal processing and their applications (pp. 247–250).Google Scholar
  37. 37.
    Vansfone, S. A. (1997). Elliptic curve cryptography. The answer to strong, fast public-key cryptography for securing constrained environments. Information Security Technical Report, 2(2), 78–87.CrossRefGoogle Scholar
  38. 38.
    Williams, S. (2000). Cryptography and network security (4th ed.). New York: Prentice Hall.Google Scholar
  39. 39.
    Gong, G., Berson, T. A., & Stinson, D. R. (2000). Elliptic curve pseudorandom sequence generators. In Selected areas in cryptography (Kingston, ON, 1999), (pp. 34–48). Berlin: Springer.Google Scholar
  40. 40.
    Caragiu, M., Johns, R. A., & Gieseler, J. (2006). Quasi-random structures from elliptic curves. Journal of Algebra, Number Theory and Applications, 6, 561–571.MathSciNetMATHGoogle Scholar
  41. 41.
    Farashahi, R. R., & Sidorenko, S. B. A. (2007). Efficient pseudorandom generators based on the DDH assumption. In Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS (Vol. 4450, pp. 426–441). Heidelberg: Springer.Google Scholar
  42. 42.
    Omar, R., & Zbigniew, K. (2015). On pseudo-random number generators using elliptic curves and chaotic systems. Applied Mathematics and Information Sciences, 9(1), 31–38.MathSciNetCrossRefGoogle Scholar
  43. 43.
    Brown, D. R. L. (2009). SEC 1: Elliptic curve cryptography. Mossossaiga: Certicom Corp.Google Scholar
  44. 44.
    Webster, A. F., & Tavares, S. E. (1986). On the design of S-boxes. Advances in Cryptology–CRYPT0 ‘85 LNCS, 218, 523–534.Google Scholar
  45. 45.
    Lidl, R., & Niederreiter, H. (1994). Introduction to finite fields and their applications (2nd ed.). Cambridge: Cambridge University Press.CrossRefMATHGoogle Scholar
  46. 46.
    Bustamante, M. D., & Hayat, U. (2013). Complete classification of discrete resonant Rossby/drift wave triads on periodic domains. Communications in Nonlinear Science and Numerical Simulation, 18, 2402–2419.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Umar Hayat
    • 1
  • Naveed Ahmed Azam
    • 2
    • 3
  • Muhammad Asif
    • 1
  1. 1.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan
  2. 2.Department of Applied Mathematics and Physics, Graduate School of InformaticsKyoto UniversityKyotoJapan
  3. 3.Faculty of Engineering SciencesGIK InstituteTopiPakistan

Personalised recommendations