Cryptography and Communications

, Volume 4, Issue 1, pp 65–77 | Cite as

A new construction of highly nonlinear S-boxes

  • Peter Beelen
  • Gregor Leander


In this paper we give a new construction of highly nonlinear vectorial Boolean functions. This construction is based on coding theory, more precisely we use concatenation to construct Boolean functions from codes over \(\mathbb{F}_q\) containing a first-order generalized Reed–Muller code. As it turns out this construction has a very compact description in terms of Boolean functions, which is of independent interest. The construction allows one to design functions with better nonlinearities than known before.


Boolean functions Linear codes Nonlinearity Reed–Muller codes Concatenation 

Mathematics Subject Classifications (2010)

06E30 94A60 14G50 65T50 



We would like to thank Anne Canteaut and Claude Carlet for suggesting to investigate more explicitly the functions obtained by using concatenation of codes.


  1. 1.
    Beelen, P., Leander, G.: Reconstruction of highly non linear Sboxes from linear codes. In: Preneel, B., Dodunekov, S., Rijmen, V., Nikova, S. (eds.) Enhancing Cryptographic Primitives with Techniques from Error Correcting Codes, vol. 23. NATO Science for Peace and Security Series - D: Information and Communication Security (2009). ISBN: 978-1-60750-002-5Google Scholar
  2. 2.
    Bosma, W., Cannon, J., Playoust, C.: The magma algebra system I: the user language. J. Symb. Comput. 24(3–4), 235–265 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bracken, C., Byrne, E., Markin, N., McGuire, G.: New families of quadratic almost perfect nonlinear trinomials and multinomials. Finite Fields Their Appl. 14(3), 703–714 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15(2), 125–156 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Carlet, C., Ding, C.: Nonlinearities of S-boxes. Finite Fields Their Appl. 13(1), 121–135 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Markus, G.: (2009)
  7. 7.
    Kasami, T.: Weight distributions of Bose-Chaudhuri-Hocquenghem codes. In: Combinatorial Mathematics and its Applications (Proc. Conf., Univ. North Carolina, Chapel Hill, N.C., 1967), pp. 335–357. Univ. North Carolina Press, Chapel Hill, N.C. (1969)Google Scholar
  8. 8.
    Nyberg, K.: Perfect nonlinear s-boxes. In: EUROCRYPT, pp. 378–386 (1991)Google Scholar
  9. 9.
    Nyberg, K.: On the construction of highly nonlinear permutations. In: EUROCRYPT, pp. 92–98 (1992)Google Scholar
  10. 10.
    Wadayama, T., Hada, T., Wakasugi, K., Kasahara, M.: Upper and lower bounds on maximum nonlinearity of n-input m-output boolean function. Des. Codes Cryptogr. 23(1), 23–34 (2001)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2011

Authors and Affiliations

  1. 1.DTU MathematicsTechnical University DenmarkLyngbyDenmark

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