A Novel Technique to Improve Nonlinearity of Substitution Box Without Disturbing Its Mathematical Properties

  • Abdul RazaqEmail author
  • Atta Ullah
  • Adil Waheed


Substitution box is a basic nonlinear component in data encryption. It converts the plain text into an enciphered format. The formation of cryptographically secure S-boxes is the most important task for cryptographers to ensure the security of the system. In this paper, we have introduced a novel method to increase the working capability of an S-box without demolishing its basic mathematical structure, that is, Eigen values, characteristic polynomial and determinant of the S-box/\( 16 \times 16 \) matrix. We considered Skipjack, Xyi, Prime, Jakimoski, Tang and Iqtadar S-boxes and reshuffled their rows/columns in such a way that they become cryptographically stronger. Algebraically, this can be accomplished by using appropriate permutations of the symmetric group \( S_{16} \) over rows/columns of the S-boxes. The proposed technique is very constructive in enhancing the performance of S-boxes based on fundamental idea; bijection over the same set. The outcomes of different analyses demonstrated that the improved versions of S-boxes are more reliable than the original ones.


Symmetric key cryptography S-box Symmetric group Nonlinearity Bit independence criterion SAC 



  1. 1.
    Shannon, C. E. (1949). Communication theory of secrecy systems. Bell Systems Technical Journal,28, 656–715.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Memon, I., Arain, Q. A., Memon, M. H., Mangi, F. A., & Akhtar, R. (2017). Search me if you can: Multiple mix zones with location privacy protection for mapping services. International Journal of Communication Systems,30(16), e3312.CrossRefGoogle Scholar
  3. 3.
    Biham, E., & Shamir, A. (2012). Differential cryptanalysis of the data encryption standard. Berlin: Springer.zbMATHGoogle Scholar
  4. 4.
    Lindell, Y., & Katz, J. (2014). Introduction to modern cryptography. London: Chapman and Hall/CRC.zbMATHGoogle Scholar
  5. 5.
    Agrawal, V., Agrawal, S., & Deshmukh, R. (2014). Analysis and review of encryption and decryption for secure communication. International Journal of Scientific Engineering and Research,2(2), 2347–3878.Google Scholar
  6. 6.
    Pieprzyk, J., & Finkelstein, G. (1988). Towards effective nonlinear cryptosystem design. IEE Proceedings E-Computers and Digital Techniques,135(6), 325–335.CrossRefGoogle Scholar
  7. 7.
    Vergili, I., & Yücel, M. D. (2001). Avalanche and bit independence properties for the ensembles of randomly chosen n\times n S-boxes. Turkish Journal of Electrical Engineering & Computer Sciences,9(2), 137–146.Google Scholar
  8. 8.
    Webster, A., 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
  9. 9.
    Matsui, M. (1994). Linear cryptanalysis method of DES cipher. In Advances in cryptology, proceeding of the Eurocrypt’93. Lecture notes computer science (Vol. 765, pp. 386–397).Google Scholar
  10. 10.
    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,67a, 282–288.CrossRefGoogle Scholar
  11. 11.
    Anees, A., & Ahmed, Z. (2015). A technique for designing substitution box based on van der pol oscillator. Wireless Personal Communications,82(3), 1497–1503.CrossRefGoogle Scholar
  12. 12.
    Isa, H., Jamil, N., & Z’aba, M. R. (2016). Construction of cryptographically strong S-boxes inspired by bee waggle dance. New Generation Computing,34(3), 221–238.CrossRefGoogle Scholar
  13. 13.
    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(6–7), 827–833.zbMATHCrossRefGoogle Scholar
  14. 14.
    Farah, T., Rhouma, R., & Belghith, S. (2017). A novel method for designing S-box based on chaotic map and teaching–learning-based optimization. Nonlinear Dynamics,88(2), 1059–1074.CrossRefGoogle Scholar
  15. 15.
    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, 1–16.CrossRefGoogle Scholar
  16. 16.
    Zhang, T., Chen, C. P., Chen, L., Xu, X., & Hu, B. (2018). Design of highly nonlinear substitution boxes based on I-Ching operators. IEEE Transactions on Cybernetics,99, 1–10.Google Scholar
  17. 17.
    Lai, Q., Akgul, A., Li, C., Xu, G., & Çavuşoğlu, Ü. (2017). A new chaotic system with multiple attractors: Dynamic analysis, circuit realization and S-box design. Entropy,20(1), 12.CrossRefGoogle Scholar
  18. 18.
    Al Solami, E., Ahmad, M., Volos, C., Doja, M., & Beg, M. (2018). A new hyperchaotic system-based design for efficient bijective substitution-boxes. Entropy,20, 525.CrossRefGoogle Scholar
  19. 19.
    Ahmed, H. A., Zolkipli, M. F., & Ahmad, M. (2019). A novel efficient substitution-box design based on firefly algorithm and discrete chaotic map. Neural Computing and Applications, 31(11), 7201–7210.CrossRefGoogle Scholar
  20. 20.
    Firdousi, F., Batool, S. I., & Amin, M. (2019). A novel construction scheme for nonlinear component based on quantum map. International Journal of Theoretical Physics, 58(11), 3871–3898.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zahid, A. H., Arshad, M. J., & Ahmad, M. (2019). A novel construction of efficient substitution-boxes using cubic fractional transformation. Entropy,21(3), 245.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Shuai, L., Wang, L., Miao, L., & Zhou, X. (2019). S-boxes construction based on the Cayley graph of the symmetric group for UASNs. IEEE Access,7, 38826–38832.CrossRefGoogle Scholar
  23. 23.
    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, 78(22), 32467–32484.CrossRefGoogle Scholar
  24. 24.
    Lu, Q., Zhu, C., & Wang, G. (2019). A novel S-box design algorithm based on a new compound chaotic system. Entropy,21(10), 1004.CrossRefGoogle Scholar
  25. 25.
    Liu, L., & Lei, Z. (2019). An approach for constructing the S-box using the CML system. In Journal of physics: Conference series (Vol. 1303, No. 1, p. 012090). IOP Publishing.Google Scholar
  26. 26.
    Atta, U., Jamal, S. S., & Shah, T. (2018). A novel algebraic technique for the construction of strong substitution box. Wireless Personal Communications,99(1), 213–226.CrossRefGoogle Scholar
  27. 27.
    Akhtar, T., Din, N., & Uddin, J. (2019). Substitution box design based on chaotic maps and cuckoo search algorithm. In 2019 International conference on advanced communication technologies and networking (CommNet) (pp. 1–7). IEEE.Google Scholar
  28. 28.
    Hamermesh, M. (2012). Group theory and its application to physical problems. North Chelmsford: Courier Corporation.zbMATHGoogle Scholar
  29. 29.
    Skipjack and Kea: Algorithm specifications version 2 (pp. 1–23) (1998).
  30. 30.
    Shi, X. Y., Xiao, Hu, You, X. C., & Lam, K. Y. (2002). A method for obtaining cryptographically strong 8*8 S-boxes. International Conference on Advanced Information Networking and Applications,2(3), 14–20.Google Scholar
  31. 31.
    Guoping, T., Xiaofeng, L., & Yong, C. (2005). A novel method for designing S-boxes based on chaotic maps. Chaos, Solitons & Fractals,23, 413.zbMATHCrossRefGoogle Scholar
  32. 32.
    Jakimoski, G., & Kocarev, L. (2001). Chaos and cryptography: block encryption ciphers based on chaotic maps. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications,48(2), 163–169.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Abuelyman, E. S., & Alsehibani, A. A. S. (2008). An optimized implementation of the S-box using residue of prime numbers. International Journal of Computer Science and Network Security,8(4), 304–309.Google Scholar
  34. 34.
    Hussain, I., Shah, T., Gondal, M. A., Khan, W. A., & Mahmood, H. (2013). A projective general linear group based algorithm for the construction of substitution box for block ciphers. Neural Computing and Applications,22(6), 1085–1093.CrossRefGoogle Scholar
  35. 35.
    Meier, W., & Staffelbach, O. (1989). Nonlinearity criteria for cryptographic functions. In Workshop on the theory and application of cryptographic techniques (pp. 549–562). Berlin: Springer.Google Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics, Division of Science and TechnologyUniversity of Education LahoreLahorePakistan
  2. 2.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan
  3. 3.Department of Computer Science, Division of Science and TechnologyUniversity of Education LahoreLahorePakistan

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