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The connection between quadratic bent–negabent functions and the Kerdock code

  • Pantelimon StănicăEmail author
  • Bimal Mandal
  • Subhamoy Maitra
Original Paper
  • 12 Downloads

Abstract

In this paper we prove that all bent functions in the Kerdock code, except for the coset of the symmetric quadratic bent function, are bent–negabent. In this direction, we characterize the set of quadratic bent–negabent functions and show some results connecting quadratic bent–negabent functions and the Kerdock code. Further, we note that there are bent–negabent preserving nonsingular transformations outside the well known class of orthogonal ones that might provide additional functions in the bent–negabent set. This is the first time we could identify non-orthogonal (nonsingular) linear transformations that preserve bent–negabent property for a special subset.

Keywords

Boolean function Bent function Negabent function Kerdock code 

Notes

Acknowledgements

The authors would like to thank the reviewers for extraordinarily useful criticisms and suggestions, and for providing us with a better code of Fig. 1. The paper was partly written while the first author visited the second and third authors at the Indian Statistical Institute, Kolkata. He would like to thank the hosts and the institute for hospitality and excellent working conditions.

References

  1. 1.
    Andrews, G.E.: The Theory of Partitions. Encyclopedia of Mathematics and its Applications, vol. 2. Cambridge University Press, Cambridge (1976)Google Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    Dillon, J.F.: A survey of bent functions. NSA Tech. J. (Special Issue) 191, 215 (1972)Google Scholar
  4. 4.
    Delsarte, P., Goethals, J.M.: Alternating bilinear forms over \(GF(q)\). J. Combin. Theory Ser. A 19, 26–50 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hu, H., Feng, D.: On quadratic bent functions in polynomial forms. IEEE Trans. Inf. Theory 53(7), 2610–2615 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kerdock, A.M.: A class of low-rate nonlinear binary codes. Inf. Control 20(2), 182–187 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    van Lint, J.H.: Kerdock codes and Preparata codes. Congressus Numerantium 39, 25–41 (1983)MathSciNetzbMATHGoogle Scholar
  8. 8.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)zbMATHGoogle Scholar
  9. 9.
    Mykkeltveit, J.: A note on Kerdock codes. JPL Technical Report 32-1526, pp. 82–83Google Scholar
  10. 10.
    Parker, M.G., Pott, A.: On Boolean functions which are bent and negabent. In: Golomb, S.W., Gong, G., Helleseth, T., Song, H.Y. (eds) Sequences, Subsequences, and Consequences, International Workshop, SSC 2007 LNCS, vol. 4893, pp. 9–23 (2007)Google Scholar
  11. 11.
    Pott, A., Schmidt, K.-U., Zhou, Y.: Pairs of quadratic forms over finite fields. Electron. J. Comb. 23(2), P2.8 (2016)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Riera, C., Parker, M.G.: Generalized bent criteria for Boolean functions. IEEE Trans. Inf. Theory 52(9), 4142–4159 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rothaus, O.S.: On bent functions. J. Combin. Theory Ser. A 20, 300–305 (1976)CrossRefzbMATHGoogle Scholar
  14. 14.
    Schmidt, K.-U., Parker, M.G., Pott, A.: Negabent functions in Maiorana–McFarland class. In: SETA 2008, LNCS, vol. 5203, pp. 390–402 (2008)Google Scholar
  15. 15.
    Stănică, P., Gangopadhyay, S., Chaturvedi, A., Gangopadhyay, A.K., Maitra, S.: Investigations on bent and negabent functions via the nega–Hadamard transform. IEEE Trans. Inf. Theory 58(6), 4064–4072 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Su, W., Pott, A., Tang, X.: Characterization of negabent functions and construction of bent-negabent functions with maximum algebraic degree. IEEE Trans. Inf. Theory 59(6), 3387–3395 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yu, N.Y., Gong, G.: Constructions of quadratic bent functions in polynomial forms. IEEE Trans. Inf. Theory 52(7), 3291–3299 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Zhang, F., Wei, Y., Pasalic, E.: Constructions of bent–negabent functions and their relation to the completed Maiorana–McFarland class. IEEE Trans. Inf. Theory 61(3), 1496–1506 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsNaval Postgraduate SchoolMontereyUSA
  2. 2.R. C. Bose Centre for Security and Cryptology, Indian Statistical InstituteKolkataIndia
  3. 3.Applied Statistics UnitIndian Statistical InstituteKolkataIndia

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