Multimedia Tools and Applications

, Volume 78, Issue 2, pp 1219–1234 | Cite as

Construction of highly nonlinear S-boxes for degree 8 primitive irreducible polynomials over ℤ2

  • Tariq ShahEmail author
  • Dawood Shah


Over binary filed F2 there are 16 primitive irreducible polynomials of degree 8, and hence one can construct 16 Galois field extensions of order 256. In this paper, we provide a novel technique to design 16 different robust 8 × 8 substitution boxes (S-boxes) over the elements these 16 Galois fields. For the purpose, on these Galois fields we define 16 linear fractional transformations as: z ⟼ (az + b)/(cz + d), where z is any arbitrary element in any of Galois fields and a, b, c, d are fixed elements from any Galois field GF(28). Accordingly for fixed parameters a, b, c, d, we obtained 16 distinct S-boxes. The algebraic strength of the proposed S-boxes is analyzed by Nonlinearity test, Strict Avalanche Criterion (SAC), Linear Approximation Probability (LP), Bit Independent Criterion (BIC), and Differential Approximation Probability (DP). As an application, by the majority logic criterion (MLC), entropy, correlation, contrast, energy and homogeneity of a plain image and its encrypted image through newly proposed S-box are assessed. Further, to fix the rank of proposed S-boxes, a comparison of these analyses is given with AES S-box, APA S-box, Residue Prime S-box, Gray S-box, Xyi S-box, Skipjack S-box and S8 AES S-box.


Primitive polynomial S-box Galois field LFT Nonlinearity 


  1. 1.
    Altaleb A, Saeed MS, Hussain I, Aslam M (2016) An algorithm for the construction of substitution box for block ciphers based on projective general linear group. AIP Adv 7:035116. CrossRefGoogle Scholar
  2. 2.
    Biham E, Shamir A (1991) Differential cryptanalysis of DES-like cryptosystems. J Cryptol 4(1):3–72MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cameron P (2000) Queen Mary and Westfield College London E1 4NS U.K Notes on Classical Groups School of Mathematical SciencesGoogle Scholar
  4. 4.
    Cui L, Cao Y (2007) A new S-box structure named affine-power-affine. Int J Innova Comput, Info Contrl 3(3):751–759Google Scholar
  5. 5.
    Daemen J., Rijmen V (2002) The design of rijndael: Aes. The Advanced Encryption StandardGoogle Scholar
  6. 6.
    Dawson MH, Tavares SE (1991) An expanded set of S-box design criteria based on information theory and its relation to differential-like attacks. In Advances in Cryptology—EUROCRYPT’91 (pp. 352–367). Springer Berlin HeidelbergGoogle Scholar
  7. 7.
    Detombe J, Tavares S (1992). On the design of S-boxes. Advances in cryptology: proceedings of CRYPTO_92. Lecture notes in computer scienceGoogle Scholar
  8. 8.
    Farwa S, Shah T, Idrees L (2016) A highly nonlinear S-box based on a fractional linear transformation. Springer Plus 5:1658. CrossRefGoogle Scholar
  9. 9.
    Feng D, Wu W (2000) Design and analysis of block ciphersGoogle Scholar
  10. 10.
    Hussain I, Shah T (2013) Literature survey on nonlinear components and chaotic nonlinear components of block ciphers. Nonlinear Dynam 74:869–904MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hussain I, Shah T, Mahmood H (2010) A new algorithm to construct secure keys for AES. Int J Contemp Math Sci 5(26):1263–1270MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hussain I, Shah T, Gondal MA, Khan M, Khan WA (2011) Construction of new S-box using a linear fractional transformation. World Appl Sci J 14(12):1779–1785Google Scholar
  13. 13.
    Hussain I, Shah T, Mahmood H, Gondal MA (2013) A projective general linear group based algorithm for the construction of substitution box for block ciphers. Neural Comput Appl 22(6):1085–1093CrossRefGoogle Scholar
  14. 14.
    Kim J, Phan RCW (2009) Advanced differential-style cryptanalysis of the NSA's skipjack block cipher. Cryptologia 33(3):246–270CrossRefGoogle Scholar
  15. 15.
    Matsui M (1993) Linear cryptanalysis method for DES cipher. In Advances in Cryptology—EUROCRYPT’93 (pp. 386–397). Springer Berlin HeidelbergGoogle Scholar
  16. 16.
    Niederreiter H, Winterhof A (2003) On the distribution of points in orbits of PGL(2, q) acting on GF(qn) Finite field and their application 9/ 458–471Google Scholar
  17. 17.
    Shah T, Hussain I, Gondal MA, Mahmood H (2011) Statistical analysis of S-box in image encryption applications based on majority logic criterion. Int J Phys Sci 6(16):4110–4127Google Scholar
  18. 18.
    Tran MT, Bui DK, Duong AD (2008) Gray S-box for advanced encryption standard. In computational intelligence and security, 2008. CIS'08 Int Conf IEEE 1:253–258Google Scholar
  19. 19.
    Webster AF, Tavares SE (1985) On the design of S-boxes. In Advances in Cryptology—CRYPTO’85 Proceedings (pp. 523–534). Springer Berlin HeidelbergGoogle Scholar
  20. 20.
    Yi X, Cheng SX, You XH, Lam KY (1997) A method for obtaining cryptographically strong 8× 8 S-boxes. Global Telecommun Conf, 1997 GLOBECOM'97, IEEE 2:689–693Google Scholar
  21. 21.
    Zimmermann R, Curiger A, Bonnenberg H, Kaeslin H, Felber N, Fichtner W (1994) A 177 Mb/s VLSI implementation of the international data encryption algorithm. Solid-State Circ, IEEE J 29(3):303–307CrossRefGoogle Scholar

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

Authors and Affiliations

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

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