Advertisement

Designs, Codes and Cryptography

, Volume 87, Issue 1, pp 163–171 | Cite as

On the q-bentness of Boolean functions

  • Zhixiong ChenEmail author
  • Ting Gu
  • Andrew Klapper
Article
  • 50 Downloads

Abstract

For each non-constant q in the set of n-variable Boolean functions, the q-transform of a Boolean function f is related to the Hamming distances from f to the functions obtainable from q by nonsingular linear change of basis. Klapper conjectured that no Boolean function exists with its q-transform coefficients equal to \(\pm \, 2^{n/2}\) (such function is called q-bent) when q is non-affine balanced. In our early work, we only gave partial results to confirm this conjecture for small n. Here we prove thoroughly that the conjecture is true for all n by investigating the nonexistence of the partial difference sets in abelian groups with special parameters. We also introduce a new family of functions called \((\delta ,q)\)-bent functions, which give a measurement of q-bentness.

Keywords

Boolean function Walsh–Hadamard transform Bent function q-transform q-bent function Partial difference set 

Mathematics Subject Classification

94A60 06E30 05B10 

Notes

Acknowledgements

The authors wish to thank Prof. Cunsheng Ding for some suggestions on the theory of difference sets. Thanks also go to the anonymous referees and the editor for their time and useful comments. The work was partially supported by the National Natural Science Foundation of China under grant No. 61772292 and by the Provincial Natural Science Foundation of Fujian under Grant No.  2018J01425. A. Klapper was partially supported by the National Science Foundation under Grant No. CNS-1420227. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

References

  1. 1.
    Carlet C.: Boolean functions for cryptography and error correcting codes. In: Crama Y., Hammer P. (eds.) Boolean Methods and Models, pp. 257–397. Cambridge University Press, Cambridge (2010). http://www.math.univ-paris13.fr/~carlet/pubs.html.
  2. 2.
    Cusick T., Stănică P.: Cryptographic Boolean Functions and Applications. Academic Press, Amsterdam (2009).zbMATHGoogle Scholar
  3. 3.
    Ding C.: Codes from Difference Sets. World Scientific, Singapore (2015).Google Scholar
  4. 4.
    Ding C., Pott A., Wang Q.: Constructions of almost difference sets from finite fields. Des. Codes Cryptogr. 72(3), 581–592 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gu T., Chen Z., Klapper A.: Correlation immune functions with respect to the \(q\)-transform. Cryptogr. Commun. (2017).  https://doi.org/10.1007/s12095-017-0267-0.zbMATHGoogle Scholar
  6. 6.
    Klapper A.: A new transform related to distance from a Boolean function. IEEE Trans. Inf. Theory 62(5), 2798–2812 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Klapper A., Chen Z.: On the nonexistence of \(q\)-bent Boolean functions. IEEE Trans. Inf. Theory 64(4), 2953–2961 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lidl R., Niederreiter H.: Finite Fields, 2nd edn. Cambridge University Press, Cambridge (1997).zbMATHGoogle Scholar
  9. 9.
    Ma S.L.: Partial difference sets. Discret. Math. 52, 75–89 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ma S.L.: A survey of partial difference sets. Des. Codes Cryptogr. 4(3), 221–261 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ma S.L.: Some necessary conditions on the parameters of partial difference sets. J. Stat. Plan. Inference 62, 47–56 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Rothaus O.S.: On “bent” functions. J. Comb. Theory A 20, 300–305 (1976).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Provincial Key Laboratory of Applied MathematicsPutian UniversityPutianPeople’s Republic of China
  2. 2.Elizabethtown CollegeElizabethtownUSA
  3. 3.Department of Computer ScienceUniversity of KentuckyLexingtonUSA

Personalised recommendations