Designs, Codes and Cryptography

, Volume 82, Issue 1–2, pp 265–291 | Cite as

Idempotent and p-potent quadratic functions: distribution of nonlinearity and co-dimension

  • Nurdagül Anbar
  • Wilfried Meidl
  • Alev Topuzoğlu


The Walsh transform \(\widehat{Q}\) of a quadratic function \(Q:{\mathbb F}_{p^n}\rightarrow {\mathbb F}_p\) satisfies \(|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}\) for all \(b\in {\mathbb F}_{p^n}\), where \(0\le s\le n-1\) is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class \(\mathcal {C}_1\) is defined for arbitrary n as \(\mathcal {C}_1 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{\lfloor (n-1)/2\rfloor }a_ix^{2^i+1})\;:\; a_i \in {\mathbb F}_2\}\), and the larger class \(\mathcal {C}_2\) is defined for even n as \(\mathcal {C}_2 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{(n/2)-1}a_ix^{2^i+1}) + \mathrm{Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in {\mathbb F}_2\}\). For an odd prime p, the subclass \(\mathcal {D}\) of all p-ary quadratic functions is defined as \(\mathcal {D} = \{Q(x) = \mathrm{Tr_n}(\sum _{i=0}^{\lfloor n/2\rfloor }a_ix^{p^i+1})\;:\; a_i \in {\mathbb F}_p\}\). We determine the generating function for the distribution of the parameter s for \(\mathcal {C}_1, \mathcal {C}_2\) and \(\mathcal {D}\). As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case \(p > 2\), the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to \(\mathcal {C}_1\) and \(\mathcal {C}_2\) in terms of a generating function.


Quadratic functions Plateaued functions Bent functions Walsh transform Idempotent functions Rotation symmetric Reed-Muller code 

Mathematics Subject Classification

11T06 11T71 11Z05 



The first author gratefully acknowledges the support from The Danish Council for Independent Research (Grant No. DFF–4002-00367) and H.C. Ørsted COFUND Post-doc Fellowship from the project “Algebraic curves with many rational points”. The second author is supported by the Austrian Science Fund (FWF) Project No. M 1767-N26.


  1. 1.
    Berlekamp E.R.: The weight enumerators for certain subcodes of the second order binary Reed–Muller codes. Inf. Control 17, 485–500 (1970).Google Scholar
  2. 2.
    Berlekamp E.R., Sloane N.: The weight enumerator of second-order Reed–Muller codes. IEEE Trans. Inf. Theory 16, 745–751 (1970).Google Scholar
  3. 3.
    Carlet C., Gao G., Liu W.: A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions. J. Comb. Theory Ser. A 127, 161–175 (2014).Google Scholar
  4. 4.
    Çeşmelioğlu A., Meidl W.: Non-weakly regular bent polynomials from vectorial quadratic functions. In: Pott A. et al. (eds.) Proceedings of the 11th International Conference on Finite Fields and their Applications. Contemporary Mathematics, pp. 83–95 (2015).Google Scholar
  5. 5.
    Charpin P., Pasalic E., Tavernier C.: On bent and semi-bent quadratic Boolean functions. IEEE Trans. Inf. Theory 51, 4286–4298 (2005).Google Scholar
  6. 6.
    Fitzgerald R.W.: Trace forms over finite fields of characteristic 2 with prescribed invariants. Finite Fields Appl. 15, 69–81 (2009).Google Scholar
  7. 7.
    Fu F.W., Niederreiter H., Özbudak F.: Joint linear complexity of multisequences consisting of linear recurring sequences. Cryptogr. Commun. 1, 3–29 (2009).Google Scholar
  8. 8.
    Helleseth T., Kholosha A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 52, 2018–2032 (2006).Google Scholar
  9. 9.
    Hu H., Feng D.: On Quadratic bent functions in polynomial forms. IEEE Trans. Inf. Theory 53, 2610–2615 (2007).Google Scholar
  10. 10.
    Kasami T.: The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes. Inf. Control 18, 369–394 (1971).Google Scholar
  11. 11.
    Kaşıkcı C., Meidl W., Topuzoğlu A.: Spectra of quadratic functions: Average behaviour and counting functions. Cryptogr. Commun. 8, 191–214 (2016).Google Scholar
  12. 12.
    Khoo K., Gong G., Stinson D.: A new characterization of semi-bent and bent functions on finite fields. Des. Codes Cryptogr. 38, 279–295 (2006).Google Scholar
  13. 13.
    Kocak N., Kocak O., Özbudak F., Saygi Z.: Characterization and enumeration of a class of semi-bent Boolean functions. Int. J. Inf. Coding Theory 3, 39–57 (2015).Google Scholar
  14. 14.
    Li S., Hu L., Zeng X.: Constructions of \(p\)-ary quadratic bent functions. Acta Appl. Math. 100, 227–245 (2008).Google Scholar
  15. 15.
    Meidl W., Topuzoğlu A.: Quadratic functions with prescribed spectra. Des. Codes Cryptogr. 66, 257–273 (2013).Google Scholar
  16. 16.
    Meidl W., Roy S., Topuzoğlu A.: Enumeration of quadratic functions with prescribed Walsh spectrum. IEEE Trans. Inf. Theory 60, 6669–6680 (2014).Google Scholar
  17. 17.
    Yu N.Y., Gong G.: Constructions of quadratic bent functions in polynomial forms. IEEE Trans. Inf. Theory 52, 3291–3299 (2006).Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Nurdagül Anbar
    • 1
  • Wilfried Meidl
    • 2
  • Alev Topuzoğlu
    • 3
  1. 1.Technical University of DenmarkLyngbyDenmark
  2. 2.Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of SciencesLinzAustria
  3. 3.Sabancı University, MDBFIstanbulTurkey

Personalised recommendations