Designs, Codes and Cryptography

, Volume 78, Issue 2, pp 391–408 | Cite as

More constructions of differentially 4-uniform permutations on \({\mathbb {F}}_{2^{2k}}\)

  • Longjiang Qu
  • Yin TanEmail author
  • Chao Li
  • Guang Gong


Differentially \(4\)-uniform permutations on \({\mathbb {F}}_{2^{2k}}\) with high nonlinearity are chosen as Substitution boxes in many block ciphers and some stream ciphers. Recently, Qu et al. (IEEE Trans Inf Theory, 59(7), 4675–4686, 2013) introduced a class of functions, which are called preferred functions, to construct a lot of infinite families of such permutations. In this paper, we propose a particular type of Boolean functions to characterize the preferred functions. On the one hand, such Boolean functions can be determined by solving linear equations, and they give rise to a huge number of differentially \(4\)-uniform permutations over \({\mathbb {F}}_{2^{2k}}\). Hence they may provide more choices for the design of Substitution boxes. On the other hand, by investigating the number of these Boolean functions, we show that the number of CCZ-inequivalent differentially \(4\)-uniform permutations over \({\mathbb {F}}_{2^{2k}}\) grows exponentially when \(k\) increases, which gives a positive answer to an open problem proposed in Qu et al.(IEEE Trans Inf Theory, 59(7), 4675–4686, 2013).


Differentially \(4\)-uniform permutation Substitution box Preferred function Preferred Boolean function 

Mathematics Subject Classification

06E30 11T60 94A60 



We would like to thank the anonymous reviewer for the valuable comments which significantly improve the quality and presentation of this paper. Part of this work was done when the first author visited Hong Kong University of Science and Technology. He would like to thank Prof. Cunsheng Ding for the kind hospitality and discussions during this period. The research of Chao Li is supported by the National Basic Research Program of China 2013CB338002 and the open research fund of Science and Technology on Information Assurance Laboratory (Grant No. KJ-12-02). The research of Longjiang Qu is supported by the National Natural Science Foundation of China (No. 61272484), the Research Project of National University Defense Technology under Grant CJ 13-02-01 and the Program for New Century Excellent Talents in University (NCET).


  1. 1.
    Aoki K., Ichikawa T., Kanda M., Matsui M., Moriai S., Nakajima J., Tokita T.: Camellia: a 128-bit block cipher suitable for multiple platforms | design and analysis. In: Proceedings of SAC ’00. Lecture Notes in Computer Science 2012, pp. 39–56. Springer, Berlin (2001).Google Scholar
  2. 2.
    Biham E., Shamir A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991).Google Scholar
  3. 3.
    Bracken C., Leander G.: A highly nonlinear differentially 4 uniform power mapping that permutes fields of even degree. Finite Fields Appl. 16(4), 231–242 (2010).Google Scholar
  4. 4.
    Bracken C., Tan C.H., Tan Y.: Binomial differentially \(4\)-uniform permutations with high nonlinearity. Finite Fields Appl. 18(3), 537–546 (2012).Google Scholar
  5. 5.
    Braeken A., Lano J., Mentens N., Preneel B.: SFINKS: a synchronous stream cipher for restricted hardware environments. In: eSTREAM, ECRYPT Stream Cipher Project, Report 2005/026 (2005).
  6. 6.
    Carlet C.: Vectorial Boolean functions for cryptography, Boolean models and methods. In: Crama Y., Hamme P.L. (eds.) Mathematics, Computer Science, and Engineering. Cambridge University Press, Cambridge (2010).Google Scholar
  7. 7.
    Carlet C.: On known and new differentially uniform functions. In: Proceedings of the 16th Australasian Conference on Information Security and Privacy. Lecture Notes in Computer Science, vol. 6812, pp. 1–15 (2011).Google Scholar
  8. 8.
    Carlet C.: More constructions of APN and differentially \(4\)-uniform functions by concatenation. Sci. China Math. 56(7), 1373–1384 (2013).Google Scholar
  9. 9.
    Carlet C., Charpin P., Zinoviev V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15(2), 125–156 (1998).Google Scholar
  10. 10.
    Daemen J., Rijmen V.: The Design of Rijndael: AES—The Advanced Encryption Standard. Springer, Berlin (2002).Google Scholar
  11. 11.
    Dib S.: Asymptotic nonlinearity of vectorial Boolean functions. Cryptogr. Commun. 6(2), 103–115 (2014).Google Scholar
  12. 12.
    Dillon J.F.: Slides from Talk Given at Polynomials Over Finite Fields and Applications Held at Ban International Research Station, Banff (2006).Google Scholar
  13. 13.
    Edel Y., Pott A.: A new almost perfect nonlinear function which is not quadratic. Adv. Math. Commun. 3(1), 59–81 (2009).Google Scholar
  14. 14.
    Hirschfeld J.W.P.: Projective geometries over finite fields, 2nd edn. Mathematical Monographs. Clarendon Press, Oxford (1998).Google Scholar
  15. 15.
    Knudsen L.: Truncated and higher order differentials. In: Fast Software Encryption (1994). Lecture Notes in Computer Science, vol. 1008, pp. 196–211 (1995).Google Scholar
  16. 16.
    Lachaud G., Wolfmann J.: The weights of the orthogonals of the extended quadratic binary Goppa codes. IEEE Trans. Inf. Theory 36(3), 686–692 (1990).Google Scholar
  17. 17.
    Li Y., Wang M., Yu Y.: Constructing differentially 4-uniform permutations over \({\mathbb{F}}_{2^{2k}}\) from the inverse function revisited (2012).
  18. 18.
    Li Y., Wang M.: Constructing differentially 4-uniform permutations over \({\mathbb{F}}_{2^{2m}}\) from quadratic APN permutations over \({\mathbb{F}}_{2^{2m+1}}\). Des. Codes Cryptogr. 72(2), 249–264 (2014).Google Scholar
  19. 19.
    Matsui M.: Linear cryptanalysis method for DES cipher. In: Advances in Cryptology—EUROCRYPT‘93. Lecture Notes in Computer Science 1994, vol. 765, pp. 386–397. Springer, Berlin (2001).Google Scholar
  20. 20.
    Qu L., Tan Y., Tan C., Li C.: Constructing differentially \(4\)-uniform permutations over \({\mathbb{F}}_{2^{2k}}\) via the switching method. IEEE Trans. Inf. Theory 59(7), 4675–4686 (2013).Google Scholar
  21. 21.
    Qu L., Xiong H., Li C.: A negative answer to Bracken-Tan-Tan’s problem on differentially \(4\)-uniform permutations over \({\mathbb{F}}_{2^n}\). Finite Fields Appl. 24, 55–65 (2013).Google Scholar
  22. 22.
    Tan Y., Qu L., Tan C., Li C.: New families of differentially 4-uniform permutations over \({\mathbb{F}}_{2^{2k}}\). In: Proceedings of Sequences and Their Applications. Lecture Notes in Computer Science, vol. 7280, pp. 25–39. Springer, Berlin (2012).Google Scholar
  23. 23.
    Tang D., Carlet C., Tang X.: Differentially 4-uniform bijections by permuting the inverse function. Des. Codes Cryptogr. (2014). doi: 10.1007/s10623-014-9992-y.
  24. 24.
    Weng G., Tan Y., Gong G.: On Quadratic APN functions and their related algebraic objects. In: Proceedings of International Workshop on Coding and Cryptography, pp. 48–57 (2013).Google Scholar
  25. 25.
    Yu Y., Wang M., Li Y.: A matrix approach for constructing quadratic APN functions. Des. Codes Cryptogr. 73, 587–600 (2014).Google Scholar
  26. 26.
    Yu Y., Wang M., Li Y.: Constructing differentially 4 uniform permutations from known ones. Chin. J. Electron. 22(3), 495–499 (2013).Google Scholar
  27. 27.
    Zha Z., Hu L., Sun S.: Constructing new differentially 4-uniform permutations from the inverse function. Finite Fields Appl. 25, 64–78 (2014).Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.College of ScienceNational University of Defense TechnologyChangShaChina
  2. 2.Department of Electrical and Computer EngineeringUniversity of WaterlooWaterlooCanada
  3. 3.Science and Technology on Information Assurance LaboratoryBeijingChina

Personalised recommendations