Efficient construction of a substitution box based on a Mordell elliptic curve over a finite field

  • Naveed Ahmed AzamEmail author
  • Umar Hayat
  • Ikram Ullah


Elliptic curve cryptography has been used in many security systems due to its small key size and high security compared with other cryptosystems. In many well-known security systems, a substitution box (S-box) is the only non-linear component. Recently, it has been shown that the security of a cryptosystem can be improved using dynamic S-boxes instead of a static S-box. This necessitates the construction of new secure S-boxes. We propose an efficient method to generate S-boxes that are based on a class of Mordell elliptic curves over prime fields and achieved by defining different total orders. The proposed scheme is developed in such a way that for each input it outputs an S-box in linear time and constant space. Due to this property, our method takes less time and space than the existing S-box construction methods over elliptic curves. Computational results show that the proposed method is capable of generating cryptographically strong S-boxes with security comparable to some of the existing S-boxes constructed via different mathematical structures.

Key words

Substitution box Finite field Mordell elliptic curve Total order Computational complexity 

CLC number



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


Compliance with ethics guidelines

Naveed Ahmed AZAM, Umar HAYAT, and Ikram ULLAH declare that they have no conflict of interest.


  1. Agarwal P, Singh A, Kilicman A, 2018. Development of keydependent dynamic S-boxes with dynamic irreducible polynomial and affine constant. Adv Mech Eng, 10(7): 1–18. CrossRefGoogle Scholar
  2. Azam NA, 2017. A novel fuzzy encryption technique based on multiple right translated AES gray S-boxes and phase embedding. Secur Commun Netw, 2017:1–9. CrossRefGoogle Scholar
  3. Chen G, 2008. A novel heuristic method for obtaining S-boxes. Chaos Sol Fract, 36(4):1028–1036. MathSciNetCrossRefGoogle Scholar
  4. Chen G, Chen Y, Liao XF, 2007. An extended method for obtaining S-boxes based on three-dimensional chaotic baker maps. Chaos Sol Fract, 31(3):571–579. MathSciNetCrossRefGoogle Scholar
  5. Cheon JH, Chee S, Park C, 1999. S-boxes with controllable nonlinearity. Proc 17th Int Conf on Theory and Application of Cryptographic Techniques, p.286–294. Google Scholar
  6. Courtois NT, Pieprzyk J, 2002. Cryptanalysis of block ciphers with overdefined systems of equations. Proc 8th Int Conf on Theory and Application of Cryptology and Information Security, p.267–287. Google Scholar
  7. Cui LG, Cao YD, 2007. A new S-box structure named affinepower-affine. Int J Innov Comput Inform Contr, 3(3): 751–759.Google Scholar
  8. Daemen J, Rijmen V, 2002. The Design of Rijndael-AES: the Advanced Encryption Standard. Springer, Berlin, Germany.CrossRefGoogle Scholar
  9. Devaraj P, Kavitha C, 2016. An image encryption scheme using dynamic S-boxes. Nonl Dynam, 86(2):927–940. MathSciNetCrossRefGoogle Scholar
  10. Gautam A, Gaba GS, Miglani R, et al., 2015. Application of chaotic functions for construction of strong substitution boxes. Ind J Sci Technol, 8(28):1–5. Google Scholar
  11. Hayat U, Azam NA, 2019. A novel image encryption scheme based on an elliptic curve. Signal Process, 155:391–402. CrossRefGoogle Scholar
  12. Hayat U, Azam NA, Asif M, 2018. A method of generating 8×8 substitution boxes based on elliptic curves. Wirel Pers Commun, 101(1):439–451. CrossRefGoogle Scholar
  13. Hussain I, Azam NA, Shah T, 2014. Stego optical encryption based on chaotic S-box transformation. Opt Laser Technol, 61:50–56. CrossRefGoogle Scholar
  14. Jakobsen T, Knudsen LR, 1997. The interpolation attack on block ciphers. Proc 4th Int Workshop on Fast Software Encryption, p.28–40. CrossRefGoogle Scholar
  15. Katiyar S, Jeyanthi N, 2016. Pure dynamic S-box construction. Int J Comput, 1:42–46.Google Scholar
  16. Kazlauskas K, Kazlauskas J, 2009. Key-dependent S-box generation in AES block cipher system. Informatica, 20(1):23–34.MathSciNetzbMATHGoogle Scholar
  17. Khan M, Azam NA, 2015a. Right translated AES gray S-boxes. Secur Commun Netw, 8:1627–1635. CrossRefGoogle Scholar
  18. Khan M, Azam NA, 2015b. S-boxes based on affine mapping and orbit of power function. 3D Res, 6(2), Article 43.
  19. Kim J, Phan RCW, 2009. Advanced differential-style cryptanalysis of the NSA’s skipjack block cipher. Cryptologia, 33(3):246–270. CrossRefGoogle Scholar
  20. Liu JM, Wai BD, Cheng XG, et al., 2005. An AES S-box to increase complexity and cryptographic analysis. Proc 19th Int Conf on Advanced Information Networking and Applications, p.724–728. Google Scholar
  21. Liu Y, Wang J, Fan JH, et al., 2016. Image encryption algorithm based on chaotic system and dynamic S-boxes composed of DNA sequences. Multim Tools Appl, 75(8):4363–4382. CrossRefGoogle Scholar
  22. Manjula G, Mohan HS, 2013. Constructing key dependent dynamic S-box for AES block cipher system. Proc 2nd Int Conf on Applied and Theoretical Computing and Communication Technology, p.613–617. Google Scholar
  23. Maram B, Gnanasekar JM, 2016. Evaluation of key dependent S-box based data security algorithm using Hamming distance and balanced output. TEM J, 5(1):67–75. Google Scholar
  24. Meier W, Staffelbach O, 1990. Nonlinearity criteria for cryptographic functions. Proc Advances in Cryptology— EUROCRYPT, p.549-562. Google Scholar
  25. Miller VS, 1986. Use of elliptic curves in cryptography. Proc Advances in Cryptology—CRYPTO, p.417–426. CrossRefGoogle Scholar
  26. Murphy S, Robshaw MJB, 2002. Essential algebraic structure within the AES. Proc 22nd Annual Int Cryptology Conf, p.1–16. Google Scholar
  27. Rahnama B, Kıran Y, Dara R, 2013. Countering AES static S-box attack. Proc 6th Int Conf on Security of Information and Networks, p.256–260. Google Scholar
  28. Rosenthal J, 2003. A polynomial description of the Rijndael advanced encryption standard. J Algebr Appl, 2(2):223–236. MathSciNetCrossRefGoogle Scholar
  29. Shannon CE, 1949. Communication theory of secrecy systems. Bell Syst Tech J, 28(4):656–715. MathSciNetCrossRefGoogle Scholar
  30. Tang GP, Liao XF, Chen Y, 2005. A novel method for designing S-boxes based on chaotic maps. Chaos Sol Fract, 23(2):413–419. CrossRefGoogle Scholar
  31. Tran MT, Bui DK, Duong AD, 2008. Gray S-box for advanced encryption standard. Proc Int Conf on Computational Intelligence and Security, p.253–258. Google Scholar
  32. Wang XY, Wang Q, 2014. A novel image encryption algorithm based on dynamic S-boxes constructed by chaos. Nonl Dynam, 75(3):567–576. CrossRefGoogle Scholar
  33. Wang Y, Yang L, Li M, et al., 2010. A method for designing S-box based on chaotic neural network. Proc 6th Int Conf on Natural Computation, p.1033–1037. Google Scholar
  34. Washington LC, 2008. Elliptic Curves: Number Theory and Cryptography (2nd Ed.). Chapman & Hall/CRC, London, UK.CrossRefGoogle Scholar
  35. Zaibi G, Kachouri A, Peyrard F, et al., 2009. On dynamic chaotic S-Box. Proc Global Information Infrastructure Symp, p.1–5. Google Scholar

Copyright information

© Zhejiang University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Physics, Graduate School of InformaticsKyoto UniversityKyotoJapan
  2. 2.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan

Personalised recommendations