Advertisement

New Bent Functions from Permutations and Linear Translators

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

Abstract

Starting from the secondary construction originally introduced by Carlet [“On Bent and Highly Nonlinear Balanced/Resilient Functions and Their Algebraic Immunities”, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 2006], that we shall call “Carlet‘s secondary construction”, Mesnager has showed how one can construct several new primary constructions of bent functions. In particular, she has showed that three tuples of permutations over the finite field \(\mathbb {F}_{2^m}\) such that the inverse of their sum equals the sum of their inverses give rise to a construction of a bent function given with its dual. It is not quite easy to find permutations satisfying such a strong condition \((\mathcal {A}_m)\). Nevertheless, Mesnager has derived several candidates of such permutations in 2015, and showed in 2016 that in the case of involutions, the problem of construction of bent functions amounts to solve arithmetical and algebraic problems over finite fields.

This paper is in the line of those previous works. We present new families of permutations satisfying \((\mathcal {A}_m)\) as well as new infinite families of permutations constructed from permutations in both lower and higher dimensions. Our results involve linear translators and give rise to new primary constructions of bent functions given with their dual. And also, we show that our new families are not in the class of Maiorana-McFarland in general.

Keywords

Boolean functions Bent functions Linear translators Permutations 

References

  1. 1.
    Canteaut, A., Naya-Plasencia, M.: Structural weakness of mappings with a low differential uniformity. In: Conference on Finite Fields and Applications, Dublin, 13–17 July 2009Google Scholar
  2. 2.
    Carlet, C.: On bent and highly nonlinear balanced/resilient functions and their algebraic immunities. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds.) AAECC 2006. LNCS, vol. 3857, pp. 1–28. Springer, Heidelberg (2006). doi: 10.1007/11617983_1 CrossRefGoogle Scholar
  3. 3.
    Carlet, C., Mesnager, S.: Four decades of research on bent functions. J. Des. Codes Crypt. 78(1), 5–50 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dillon, J.: Elementary hadamard difference sets. PhD Thesis, University of Maryland (1974)Google Scholar
  5. 5.
    Koçak, N., Mesnager, S., Özbudak, F.: Bent and semi-bent functions via linear translators. In: Groth, J. (ed.) IMACC 2015. LNCS, vol. 9496, pp. 205–224. Springer, Cham (2015). doi: 10.1007/978-3-319-27239-9_13 CrossRefGoogle Scholar
  6. 6.
    Kyureghyan, G.M.: Constructing permutations of finite fields via linear translators. J. Comb. Theory Ser. A 118, 1052–1061 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Mesnager, S.: Several new infinite families of bent functions and their duals. IEEE Trans. Inf. Theory 60(7), 4397–4407 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Mesnager, S.: Further constructions of infinite families of bent functions from new permutations and their duals. J. Crypt. Commun. (CCDS) 8(2), 1–18 (2016). SpringerMathSciNetzbMATHGoogle Scholar
  9. 9.
    Mesnager, S.: A Note on Constructions of Bent Functions from Involutions. IACR, Cryptology ePrint Archive, 982 (extended version of ISIT 2016) (2015)Google Scholar
  10. 10.
    Mesnager, S.: On constructions of bent functions from involutions. In Proceedings of 2015 IEEE International Symposium on Information Theory, ISIT (2016)Google Scholar
  11. 11.
    Mesnager, S.: Bent Functions: Fundamentals and Results. Springer, New York (2016)CrossRefzbMATHGoogle Scholar
  12. 12.
    Mesnager, S., Cohen, G., Madore, D.: On existence (based on an arithmetical problem) and constructions of bent functions. In: Groth, J. (ed.) IMACC 2015. LNCS, vol. 9496, pp. 3–19. Springer, Cham (2015). doi: 10.1007/978-3-319-27239-9_1 CrossRefGoogle Scholar
  13. 13.
    Rothaus, O.S.: On “bent” functions. J. Comb. Theory Ser. A 20, 300–305 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wang, Q., Johansson, T., Kan, H.: Some results on fast algebraic attacks and higher-order non-linearities. IET Inf. Secur. 6, 41–46 (2012)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Sihem Mesnager
    • 1
    • 2
    • 3
    Email author
  • Pınar Ongan
    • 4
  • Ferruh Özbudak
    • 4
    • 5
  1. 1.Department of MathematicsUniversity of Paris VIIISaint-DenisFrance
  2. 2.University of Paris XIII, LAGA, UMR 7539, CNRSVilletaneuseFrance
  3. 3.Telecom ParisTechParisFrance
  4. 4.Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  5. 5.Department of MathematicsMiddle East Technical UniversityAnkaraTurkey

Personalised recommendations