The replacement of irreducible polynomial and affine mapping for the construction of a strong S-box

Original Paper

Abstract

Substitution box (S-box) is a critical part of the data encryption and decryption procedures. The primary function of the S-box in advanced encryption standard algorithm is to randomize the 8-bit input into 8-bit output. This paper presents a novel approach to S-box construction based on the replacement of irreducible polynomial and affine mapping. The strength of the created S-box is assessed by applying several standard tests, e.g., balance, bijective, nonlinearity, strict avalanche criterion, and bit independence criterion-nonlinearity. The strength of the S-box outperforms those of available S-boxes.

Keywords

AES S-box Irreducible polynomial Affine mapping Affine matrix 

Notes

Acknowledgements

We are grateful to the anonymous reviewers for helpful comments leading to the improvement of the exposition. Special thanks are also given to Overseas Seminar Assistance Program, Directorate General of Research and Development Strengthening, Ministry of Research, Technology, and Higher Education, Indonesia. We would also like to show our gratitude to the Directorate of Research and Community Service (Grants No 084/SP2H/LT/DRPM/IV/2017 and No. 075/SP2H/LT/DRPM/I/2018), Directorate General of Research and Development, Ministry of Research, Technology and Higher Education, Indonesia.

References

  1. 1.
    FIPS PUB 46-3, Data encryption standard (DES) (1999)Google Scholar
  2. 2.
    Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Paar, C., Pelzl, J.: Understanding Cryptography, vol. 1, 1st edn. Springer, Berlin, Heidelberg (2010)CrossRefMATHGoogle Scholar
  4. 4.
    Daemen, J., Rijmen, V.: The Design of Rijndael. Springer, Berlin, Heidelberg, New York (2002)CrossRefMATHGoogle Scholar
  5. 5.
    Daemen, J., Rijmen, V.: AES Proposal: Rijndael. [Online]. http://csrc.nist.gov/archive/aes/rijndael/Rijndael-ammended.pdf. Accessed 26 Jan 2018
  6. 6.
    Wu, C.K., Feng, D.: Boolean Functions and Their Applications in Cryptography. Springer, Berlin, Heidelberg (2016)CrossRefMATHGoogle Scholar
  7. 7.
    Wang, Q., Jin, C.: Upper bound of the length of truncated impossible differentials for AES. Des. Codes Cryptogr. 86, 1541–1542 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ahmad, M., Haleem, H.: A new chaotic substitution box design for block ciphers. In: International Conference on Signal Processing and Integrated Networks (SPIN), vol. 1, pp. 255–258 (2014)Google Scholar
  9. 9.
    Lambic, D.: A novel method of S-box design based on discrete chaotic map. Nonlinear Dyn. 87, 2407–2413 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Özkaynak, F., Yavuz, S.: Designing chaotic S-boxes based on time-delay chaotic system. Nonlinear Dyn. 74(3), 551–557 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Khan, M., Shah, T., Mahmood, H., Asif, M., Iqtadar, G.: A novel technique for the construction of strong S-boxes based on chaotic Lorenz systems. Nonlinear Dyn. 70, 2303–2311 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Belazi, A., Khan, M., El-Latif, A.A.A., Belghith, S.: Efficient cryptosystem approaches: S-boxes and permutation–substitution-based encryption. Nonlinear Dyn. 87, 337–361 (2017)CrossRefGoogle Scholar
  13. 13.
    Özkaynak, F., Çelik, V., Özer, A.B.: A new S-box construction method based on the fractional-order chaotic Chen system. Signal Image Video Process. 11, 659–664 (2017)CrossRefGoogle Scholar
  14. 14.
    Khan, M., Shah, T.: An efficient construction of substitution box with fractional chaotic system. Signal Image Video Process. 9(6), 1335–1338 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Liu, G., Yang, W., Liu, W., Dai, Y.: Designing S-boxes based on 3-D four-wing autonomous chaotic system. Nonlinear Dyn. 82(4), 1867–1877 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Khan, M., Azam, N.A.: S-boxes based on affine mapping and orbit of power function. 3D Res. 6(2), 1–15 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Çavusoglu, Ü., Zengin, A., Pehlivan, I., Kaçar, S.: A novel approach for strong S-Box generation algorithm design based on chaotic scaled Zhongtang system. Nonlinear Dyn. 87, 1081–1094 (2017)CrossRefMATHGoogle Scholar
  18. 18.
    Ullah, A., Shaukat, S., Tariq, J.: A novel construction of substitution box using a combination of chaotic maps with improved chaotic range. Nonlinear Dyn. 88, 2757–2769 (2017)CrossRefGoogle Scholar
  19. 19.
    Isa, H., Jamil, N., Aba, M.R.Z.: Construction of cryptographically strong S-boxes inspired by bee waggle dance. New Gener. Comput. 7, 221–238 (2016)CrossRefGoogle Scholar
  20. 20.
    Sahoo, O.B., Kole, D.K., Rahaman, H.: An optimized S-box for advanced encryption standard (AES) design. In: Proceedings—2012 International Conference on Advances in Computing and Communications ICACC 2012, pp. 154–157 (2012)Google Scholar
  21. 21.
    Waqas, U., Afzal, S., Mir, M.A., Yousaf, M.: Generation of AES-like S-boxes by replacing affine matrix. In: Proceedings—12th International Conference on Frontiers of Information Technology FIT 2014, pp. 159–164 (2015)Google Scholar
  22. 22.
    Stallings, W.: Cryptography and Network Security: Principles and Practice, 6th edn. Pearson, London (2014)Google Scholar
  23. 23.
    Gangadaril, B.R., Ahamed, S.R.: Analysis and algebraic construction of S-box for AES algorithm using irreducible polynomials. In: Eighth International Conference on Contemporary Computing (IC3) (2015)Google Scholar
  24. 24.
    Wang, D., SUN, S.-L.: Replacement and structure of S-boxes in Rijndael. In: International Conference on Computer Science and Software Engineering, pp. 782–784 (2008)Google Scholar
  25. 25.
    Alamsyah, Bejo, A., Bharata Adji, T.: AES S-box construction using different irreducible polynomial and constant 8-bit vector. In: 2017 IEEE Conference on Dependable and Secure Computing, pp. 366–369 (2017 )Google Scholar
  26. 26.
    Lambić, D.: Security analysis and improvement of a block cipher with dynamic S-boxes based on tent map. Nonlinear Dyn. 79(4), 2531–2539 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Farah, T., Rhouma, R., Belghith, S.: A novel method for designing S-box based on chaotic map and Teaching–Learning-Based Optimization. Nonlinear Dyn. 88(2), 1059–1074 (2017)CrossRefGoogle Scholar
  28. 28.
    Hussain, I., Shah, T.: Literature survey on nonlinear components and chaotic nonlinear components of block ciphers. Nonlinear Dyn. 74(4), 869–904 (2013)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Liu, J., Mesnager, S., Chen, L.: On the nonlinearity of S-boxes and linear codes. Cryptogr. Commun. 9(3), 345–361 (2017)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Adams, C., Tavares, S.: Good S-boxes are easy to find. In: Advances in Cryptology—CRYPTO’ 89 Proceedings. CRYPTO 1989. Lecture Notes in Computer Science book series, vol. 435, pp. 612–615 (1990)Google Scholar
  31. 31.
    Webster, A.F., Tavares, S.E.: On the design of S-boxes. In: Williams H.C. (eds.) Advances in Cryptology—CRYPTO ’85 Proceedings. CRYPTO 1985. Lecture Notes in Computer Science, vol. 218, pp. 523–534 (1986)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Electrical Engineering and Information TechnologyUniversitas Gadjah MadaYogyakartaIndonesia
  2. 2.Department of Computer ScienceUniversitas Negeri SemarangKota SemarangIndonesia

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