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Characterizations of Partially Bent and Plateaued Functions over Finite Fields

  • Sihem Mesnager
  • Ferruh Özbudak
  • Ahmet SınakEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11321)

Abstract

Partially bent and plateaued functions over finite fields have significant applications in cryptography, sequence theory, coding theory, design theory and combinatorics. They have been extensively studied due to their various desirable cryptographic properties. In this paper, we study on characterizations of partially bent and plateaued functions over finite fields, with the aim of clarifying their structure. We first redefine the notion of partially bent functions over any finite field \({\mathbb {F}}_q\), with q a prime power, and then provide a few characterizations of these functions in terms of their derivatives, Walsh power moments and autocorrelation functions. We next characterize partially bent (vectorial) functions over \({\mathbb {F}}_p\), with p a prime, by means of their derivatives and Walsh power moments. We finally characterize plateaued functions over \({\mathbb {F}}_p\) in terms of their Walsh power moments, derivatives and autocorrelation functions.

Keywords

p-ary functions q-ary functions Partially bent Plateaued Additive character 

Notes

Acknowledgment

The authors would like to thank the anonymous reviewers of WAIFI-2018 for their valuable comments and suggestions. The third author is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK), program no: BİDEB 2214/A.

References

  1. 1.
    Ambrosimov, A.: Properties of bent functions of q-valued logic over finite fields (1994)Google Scholar
  2. 2.
    Anbar, N., Meidl, W.: Quadratic functions and maximal Artin-Schreier curves. Finite Fields Appl. 30, 49–71 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Canteaut, A., Carlet, C., Charpin, P., Fontaine, C.: On cryptographic properties of the cosets of r (1, m). IEEE Trans. Inf. Theory 47(4), 1494–1513 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Carlet, C.: Partially-bent functions. Des. Codes Crypt. 3(2), 135–145 (1993)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Carlet, C.: Boolean functions for cryptography and error correcting codes. Boolean Models Methods Math. Comput. Sci. Eng. 2, 257–397 (2010)CrossRefGoogle Scholar
  6. 6.
    Carlet, C.: Boolean and vectorial plateaued functions and APN functions. IEEE Trans. Inf. Theory 61(11), 6272–6289 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Carlet, C., Dubuc, S.: On generalized bent and q-ary perfect nonlinear functions. In: Jungnickel, D., Niederreiter, H. (eds.) Finite Fields and Applications, pp. 81–94. Springer, Heidelberg (2001).  https://doi.org/10.1007/978-3-642-56755-1_8CrossRefzbMATHGoogle Scholar
  8. 8.
    Carlet, C., Mesnager, S.: Four decades of research on bent functions. Des. Codes Crypt. 78(1), 5–50 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Carlet, C., Mesnager, S., Özbudak, F., Sınak, A.: Explicit characterizations for plateaued-ness of p-ary (vectorial) functions. In: El Hajji, S., Nitaj, A., Souidi, E.M. (eds.) C2SI 2017. LNCS, vol. 10194, pp. 328–345. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-55589-8_22CrossRefGoogle Scholar
  10. 10.
    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).  https://doi.org/10.1007/978-3-540-39887-5_6CrossRefGoogle Scholar
  11. 11.
    Çeşmelioğlu, A., Meidl, W.: A construction of bent functions from plateaued functions. Des. Codes Crypt. 66, 231–242 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Çesmelioglu, A., Meidl, W., Topuzoglu, A.: Partially bent functions and their properties. In: Applied Algebra and Number Theory. Cambridge University Press, Cambridge (2014)Google Scholar
  13. 13.
    Coulter, R.S., Matthews, R.W.: Bent polynomials over finite fields. Bull. Aust. Math. Soc. 56(3), 429–437 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dillon, J.F.: Elementary hadamard difference sets. Ph.D. thesis. University of Maryland (1974)Google Scholar
  15. 15.
    Hou, X.D.: q-ary bent functions constructed from chain rings. Finite Fields Appl. 4(1), 55–61 (1998)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hou, X.D.: p-ary and q-ary versions of certain results about bent functions and resilient functions. Finite Fields Appl. 10(4), 566–582 (2004)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hyun, J.Y., Lee, J., Lee, Y.: Explicit criteria for construction of plateaued functions. IEEE Trans. Inf. Theory 62(12), 7555–7565 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kumar, P.V., Scholtz, R.A., Welch, L.R.: Generalized bent functions and their properties. J. Comb. Theory Ser. A 40(1), 90–107 (1985)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Langevin, P.: On generalized bent functions. In: Camion, P., Charpin, P., Harari, S. (eds.) Eurocode 92. ICMS, vol. 339, pp. 147–152. Springer, Vienna (1993).  https://doi.org/10.1007/978-3-7091-2786-5_13CrossRefGoogle Scholar
  20. 20.
    Lidl, R., Niederreiter, H.: Finite Fields, vol. 20. Cambridge University Press, Cambridge (1997)Google Scholar
  21. 21.
    Logachev, O.A., Salnikov, A., Yashchenko, V.V.: Bent functions on a finite Abelian group. Discrete Math. Appl. 7(6), 547–564 (1997)MathSciNetCrossRefGoogle Scholar
  22. 22.
    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).  https://doi.org/10.1007/978-3-319-12325-7_6CrossRefGoogle Scholar
  23. 23.
    Mesnager, S.: Bent Functions: Fundamentals and Results. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-319-32595-8CrossRefzbMATHGoogle Scholar
  24. 24.
    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).  https://doi.org/10.1007/978-3-319-29172-7_2CrossRefzbMATHGoogle Scholar
  25. 25.
    Mesnager, S., Özbudak, F., Sınak, A.: On the p-ary (cubic) bent and plateaued (vectorial) functions. Des. Codes Crypt. 86, 1865–1892 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mesnager, S., Özbudak, F., Sınak, A., Cohen, G.: On q-ary plateaued functions over \(\mathbb{F}_q\) and their explicit characterizations. Eur. J. Comb. (2018). ElsevierGoogle Scholar
  27. 27.
    Nyberg, K.: Constructions of bent functions and difference sets. In: Damgård, I.B. (ed.) EUROCRYPT 1990. LNCS, vol. 473, pp. 151–160. Springer, Heidelberg (1991).  https://doi.org/10.1007/3-540-46877-3_13CrossRefGoogle Scholar
  28. 28.
    Rothaus, O.S.: On “bent” functions. J. Comb. Theory Ser. A 20(3), 300–305 (1976)CrossRefGoogle Scholar
  29. 29.
    Rudin, W.: Principles of Mathematical Analysis, vol. 3. McGraw-Hill, New York (1964)zbMATHGoogle Scholar
  30. 30.
    Wang, X., Zhou, J.: Generalized partially bent functions. In: Future Generation Communication and Networking (FGCN 2007), vol. 1, pp. 16–21. IEEE (2007)Google Scholar
  31. 31.
    Zheng, Y., Zhang, X.-M.: Plateaued functions. In: Varadharajan, V., Mu, Y. (eds.) ICICS 1999. LNCS, vol. 1726, pp. 284–300. Springer, Heidelberg (1999).  https://doi.org/10.1007/978-3-540-47942-0_24CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Sihem Mesnager
    • 1
    • 2
    • 3
    • 4
  • Ferruh Özbudak
    • 5
    • 6
  • Ahmet Sınak
    • 2
    • 3
    • 6
    • 7
    Email author
  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

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