Explicit Characterizations for Plateaued-ness of p-ary (Vectorial) Functions

  • Claude Carlet
  • Sihem MesnagerEmail author
  • Ferruh Özbudak
  • Ahmet Sınak
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10194)


Plateaued (vectorial) functions have an important role in the sequence and cryptography frameworks. Given their importance, they have not been studied in detail in general framework. Several researchers found recently results on their characterizations and introduced new tools to understand their structure and to design such functions. In this work, we mainly extend some of the observations made in characteristic 2 and given in (Carlet, IEEE Trans. Inf. Theor. 61(11), 6272–6289, 2015) to arbitrary characteristic. We first extend to arbitrary characteristic the characterizations of plateaued (vectorial) Boolean functions by the autocorrelation functions, next their characterizations in terms of the second-order derivatives, and finally their characterizations via the moments of the Walsh transform.


Vectorial functions p-ary functions Bent functions Plateaued functions 



The fourth author is supported by the Scientific and Technological Research Council of Turkey (TÜBITAK)-BIDEB 2214-A program.


  1. 1.
    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, pp. 257–397. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  2. 2.
    Carlet, C.: Boolean and vectorial plateaued functions, and APN functions. IEEE Trans. Inf. Theor. 61(11), 6272–6289 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Carlet, C.: On the properties of vectorial functions with plateaued components and their consequences on APN functions. In: El Hajji, S., Nitaj, A., Carlet, C., Souidi, E.M. (eds.) C2SI 2015. LNCS, vol. 9084, pp. 63–73. Springer, Cham (2015). doi: 10.1007/978-3-319-18681-8_5 Google Scholar
  4. 4.
    Carlet, C.: Partially-bent functions. Des. Code Crypt. 3(2), 135–145 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Carlet, C., Prouff, E.: On plateaued functions and their constructions. In: Johansson, T. (ed.) FSE 2003. LNCS, vol. 2887, pp. 54–73. Springer, Heidelberg (2003). doi: 10.1007/978-3-540-39887-5_6 CrossRefGoogle Scholar
  6. 6.
    Çesmelioglu, A., Meidl, W.: A construction of bent functions from plateaued functions. Des. Code Crypt. 66(1–3), 231–242 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Çesmelioglu, A., Meidl, W., Topuzoglu, A.: Partially bent functions and their properties. In: Larcher, G., Pillichshammer, F., Winterhof, A., Xing, C. (eds.) Applications of Algebra and Number Theory, pp. 22–40. Cambridge University Press, Cambridge (2014)CrossRefGoogle Scholar
  8. 8.
    Hyun, J.Y., Lee, J., Lee, Y.: Explicit criteria for construction of plateaued functions. IEEE Trans. Inf. Theor. 62(12), 7555–7565 (2016)CrossRefGoogle Scholar
  9. 9.
    Kumar, P.V., Scholtz, R.A., Welch, L.R.: Generalized bent functions and their properties. J. Comb. Theor. A–40, 90–107 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mesnager, S.: Characterizations of plateaued and bent functions in characteristic \(p\). In: Schmidt, K.-U., Winterhof, A. (eds.) SETA 2014. LNCS, vol. 8865, pp. 72–82. Springer, Cham (2014). doi: 10.1007/978-3-319-12325-7_6 Google Scholar
  11. 11.
    Mesnager, S.: Bent Functions: Fundamentals and Results. Springer, Cham (2016)CrossRefzbMATHGoogle Scholar
  12. 12.
    Mesnager, S., Özbudak, F., Sınak, A.: On the \(p\)-ary (cubic) bent and plateaued (vectorial) functions. Des. Code Crypt. (2017, submitted)Google Scholar
  13. 13.
    Mesnager, S., Özbudak, F., Sınak, A.: Results on characterizations of plateaued functions in arbitrary characteristic. In: Pasalic, E., Knudsen, L.R. (eds.) BalkanCryptSec 2015. LNCS, vol. 9540, pp. 17–30. Springer, Cham (2016). doi: 10.1007/978-3-319-29172-7_2 CrossRefGoogle Scholar
  14. 14.
    Mullen, G.L., Panario, D.: Handbook of finite fields. CRC Press, New York (2013)CrossRefzbMATHGoogle Scholar
  15. 15.
    Rudin, W.: Principles of Mathematical Analysis. McGraw-Hill, New York (1964)zbMATHGoogle Scholar
  16. 16.
    Rothaus, O.S.: On bent functions. J. Comb. Theor. A. 20, 300–305 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zheng, Y., Zhang, X.-M.: Plateaued functions. In: Varadharajan, V., Mu, Y. (eds.) ICICS 1999. LNCS, vol. 1726, pp. 284–300. Springer, Heidelberg (1999). doi: 10.1007/978-3-540-47942-0_24 CrossRefGoogle Scholar
  18. 18.
    Zheng, Y., Zhang, X.-M.: Relationships between bent functions and complementary plateaued functions. In: Song, J.S. (ed.) ICISC 1999. LNCS, vol. 1787, pp. 60–75. Springer, Heidelberg (2000). doi: 10.1007/10719994_6 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Claude Carlet
    • 1
    • 3
  • Sihem Mesnager
    • 1
    • 3
    • 4
    Email author
  • Ferruh Özbudak
    • 5
    • 6
  • Ahmet Sınak
    • 2
    • 6
    • 7
  1. 1.Department of MathematicsUniversity of Paris VIIISaint-DenisFrance
  2. 2.LAGA, UMR 7539, CNRS, University of Paris VIIISaint-DenisFrance
  3. 3.LAGA, UMR 7539, CNRS, University of Paris XIIIVilletaneuseFrance
  4. 4.Telecom ParisTechParisFrance
  5. 5.Department of MathematicsMiddle East Technical UniversityAnkaraTurkey
  6. 6.Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  7. 7.Department of Mathematics and Computer SciencesNecmettin Erbakan UniversityKonyaTurkey

Personalised recommendations