On o-Equivalence of Niho Bent Functions

  • Lilya Budaghyan
  • Claude Carlet
  • Tor Helleseth
  • Alexander Kholosha
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9061)

Abstract

As observed recently by the second author and S. Mesnager, the projective equivalence of o-polynomials defines, for Niho bent functions, an equivalence relation called o-equivalence. These authors also observe that, in general, the two o-equivalent Niho bent functions defined from an o-polynomial \(F\) and its inverse \(F^{-1}\) are EA-inequivalent. In this paper we continue the study of o-equivalence. We study a group of order 24 of transformations preserving o-polynomials which has been studied by Cherowitzo 25 years ago. We point out that three of the transformations he included in the group are not correct. We also deduce two more transformations preserving o-equivalence but providing potentially EA-inequivalent bent functions. We exhibit examples of infinite classes of o-polynomials for which at least three EA-inequivalent Niho bent functions can be derived.

Keywords

Bent function Boolean function Maximum nonlinearity Niho bent function o-polynomials Walsh transform 

References

  1. 1.
    Rothaus, O.S.: On “bent” functions. J. Combin. Theory Ser. A 20(3), 300–305 (1976)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Carlet, C.: Boolean functions for cryptography and error-correcting codes. In: Crama, Y., Hammer, P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering. Encyclopedia of Mathematics and its Applications, vol. 134, ch. 8, pp. 257–397. Cambridge University Press, Cambridge (2010)Google Scholar
  3. 3.
    Budaghyan, L., Carlet, C.: CCZ-equivalence of single and multi output Boolean functions. In: Post-proceedings of the Conference Fq9. AMS Contemporary Math., vol. 518, pp. 43–54 (2010)Google Scholar
  4. 4.
    Kholosha, A., Pott, A.: Bent and related functions. In: Mullen, G.L., Panario, D. (eds.) Handbook of Finite Fields. Discrete Mathematics and its Applications, ch. 9.3, pp. 255–265. CRC Press, London (2013)Google Scholar
  5. 5.
    McFarland, R.L.: A family of difference sets in non-cyclic groups. J. Combin. Theory Ser. A 15(1), 1–10 (1973)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Dillon, J. F.: Elementary Hadamard difference sets, Ph.D. dissertation, University of Maryland (1974)Google Scholar
  7. 7.
    Carlet, C., Mesnager, S.: On Dillon’s class \(H\) of bent functions, Niho bent functions and o-polynomials. J. Combin. Theory Ser. A 118(8), 2392–2410 (2011)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Dobbertin, H., Leander, G., Canteaut, A., Carlet, C., Felke, P., Gaborit, P.: Construction of bent functions via Niho power functions. J. Combin. Theory Ser. A 113(5), 779–798 (2006)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Leander, G., Kholosha, A.: Bent functions with \(2^r\) Niho exponents. IEEE Trans. Inf. Theory 52(12), 5529–5532 (2006)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Helleseth, T., Kholosha, A., Mesnager, S.: Niho bent functions and Subiaco hyperovals. In: Lavrauw, M., Mullen, G.L., Nikova, S., Panario, D., Storme, L. (eds.) Theory and Applications of Finite Fields. Contemporary Mathematics, vol. 579, pp. 91–101. American Mathematical Society, Providence (2012)CrossRefGoogle Scholar
  11. 11.
    Budaghyan, L., Carlet, C., Helleseth, T., Kholosha, A., Mesnager, S.: Further results on Niho bent functions. IEEE Trans. Inf. Theory 58(11), 6979–6985 (2012)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Budaghyan, L., Kholosha, A., Carlet, C., Helleseth, T.: Niho bent functions from quadratic o-monomials. In: Proceedings of the 2014 IEEE International Symposium on Information Theory (2014)Google Scholar
  13. 13.
    Li, N., Helleseth, T., Kholosha, A., Tang, X.: On the Walsh transform of a class of functions from Niho exponents. IEEE Trans. Inf. Theory 59(7), 4662–4667 (2013)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Carlet, C., Helleseth, T., Kholosha, A., Mesnager, S.: On the dual of bent functions with \(2^r\) Niho exponents. In: Proceedings of the 2011 IEEE International Symposium on Information Theory, pp. 657–661. IEEE, July/August 2011Google Scholar
  15. 15.
    Glynn, D.: Two new sequences of ovals in finite Desarguesian planes of even order. Combinatorial Mathematics, Lecture Notes in Mathematics, vol. 1036, pp. 217–229 (1983)Google Scholar
  16. 16.
    Cherowitzo, W.E., Storme, L.: \(\alpha \)-Flocks with oval herds and monomial hyperovals. Finite Fields Appl. 4(2), 185–199 (1998)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Vis, T.L.: Monomial hyperovals in Desarguesian planes, Ph.D. dissertation, University of Colorado Denver (2010)Google Scholar
  18. 18.
    Cherowitzo, W.: Hyperovals in Desarguesian planes of even order. Ann. Discrete Math. 37, 87–94 (1988)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Lilya Budaghyan
    • 1
  • Claude Carlet
    • 2
  • Tor Helleseth
    • 1
  • Alexander Kholosha
    • 1
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.LAGA, UMR 7539, CNRS, Department of MathematicsUniversity of Paris 8 and University of Paris 13Saint-Denis CedexFrance

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