Advertisement

Semi-bent Functions with Multiple Trace Terms and Hyperelliptic Curves

  • Sihem Mesnager
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7533)

Abstract

Semi-bent functions with even number of variables are a class of important Boolean functions whose Hadamard transform takes three values. Semi-bent functions have been extensively studied due to their applications in cryptography and coding theory. In this paper we are interested in the property of semi-bentness of Boolean functions defined on the Galois field \({\mathbb F}_2^n\) (n even) with multiple trace terms obtained via Niho functions and two Dillon-like functions (the first one has been studied by the author and the second one has been studied very recently by Wang et al. using an approach introduced by the author). We subsequently give a connection between the property of semi-bentness and the number of rational points on some associated hyperelliptic curves. We use the hyperelliptic curve formalism to reduce the computational complexity in order to provide an efficient test of semi-bentness leading to substantial practical gain thanks to the current implementation of point counting over hyperelliptic curves.

Keywords

Boolean function Symmetric cryptography Walsh-Hadamard transformation Semi-bent functions Dickson polynomial Hyperelliptic curves 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Canteaut, A., Carlet, C., Charpin, P., Fontaine, C.: On cryptographic properties of the cosets of R (1,m). IEEE Transactions on Information Theory 47, 1494–1513 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Carlet, C., Mesnager, S.: On Semi-bent Boolean functions. IEEE Transactions on Information Theory 58(5), 3287–3292 (2012)Google Scholar
  3. 3.
    Charpin, P., Pasalic, E., Tavernier, C.: On bent and semi-bent quadratic Boolean functions. IEEE Transactions on Information Theory 51(12), 4286–4298 (2005)Google Scholar
  4. 4.
    Chee, S., Lee, S., Kim, K.: Semi-bent Functions. In: Safavi-Naini, R., Pieprzyk, J.P. (eds.) ASIACRYPT 1994. LNCS, vol. 917, pp. 107–118. Springer, Heidelberg (1995)Google Scholar
  5. 5.
    Cheon, J.H., Chee, S.: Elliptic Curves and Resilient Functions. In: Won, D. (ed.) ICISC 2000. LNCS, vol. 2015, pp. 64–397. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. 6.
    Dobbertin, H., Leander, G., Canteaut, A., Carlet, C., Felke, P., Gaborit, P.: Construction of bent functions via Niho Power Functions. Journal of Combinatorial therory, Serie A 113, 779–798 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Enge, A.: How to distinguish hyperelliptic curves in even characteristic. In: Public-Key Cryptography and Computational Number Theory. de Gruyter Proceedings in Mathematics. DE GRUYTER (2011)Google Scholar
  8. 8.
    Flori, J., Mesnager, S.: An efficient characterization of a family of hyperbent functions with multiple trace terms (in preprint)Google Scholar
  9. 9.
    Hubrechts, H.: Point counting in families of hyperelliptic curves in characteristic 2. LMS J. Comput. Math. 10, 207–234 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Lisoněk, P.: An efficient characterization of a family of hyperbent functions. IEEE Transactions on Information Theory 57, 6010–6014 (2011)CrossRefGoogle Scholar
  11. 11.
    Matsui, M.: Linear Cryptanalysis Method for DES Cipher. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 386–397. Springer, Heidelberg (1994)Google Scholar
  12. 12.
    Meier, W., Staffelbach, O.: Fast Correlation Attacks on Stream Ciphers. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 301–314. Springer, Heidelberg (1988)Google Scholar
  13. 13.
    Mesnager, S.: Semi-bent functions from Dillon and Niho exponents, Kloosterman sums and Dickson polynomials. IEEE Transactions on Information Theory 57(11), 7443–7458 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mesnager, S.: A new class of bent and hyper-bent Boolean functions in polynomial forms. Des. Codes Cryptography 59, 265–279 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Mesnager, S.: Hyper-bent Boolean Functions with Multiple Trace Terms. In: Hasan, M.A., Helleseth, T. (eds.) WAIFI 2010. LNCS, vol. 6087, pp. 97–113. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  16. 16.
    Rothaus, O.S.: On ”bent” functions. J. Combin.Theory Ser. A 20, 300–305 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Sun, G., Wu, C.: Construction of Semi-Bent Boolean Functions in Even Number of Variables. Chinese Journal of Electronics 18(2) (2009)Google Scholar
  18. 18.
    Vercauteren, F.: Computing zeta functions of curves over finite fields. PhD thesis, Katholieke Universiteit Leuven (2003)Google Scholar
  19. 19.
    Wang, B., Tang, C., Qi, Y., Yang, Y., Xu, M.: A New Class of Hyper-bent Boolean Functions with Multiple Trace Terms. In: Cryptology ePrint Archive, Report 2011/600 (2011), http://eprint.iacr.org/
  20. 20.
    Zheng, Y., Zhang, X.-M.: Plateaued Functions. In: Varadharajan, V., Mu, Y. (eds.) ICICS 1999. LNCS, vol. 1726, pp. 284–300. Springer, Heidelberg (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sihem Mesnager
    • 1
  1. 1.LAGA (Laboratoire Analyse, Géometrie et Applications), UMR 7539, CNRS, Department of MathematicsUniversity of Paris XIII and University of Paris VIIISaint-Denis CedexFrance

Personalised recommendations