On Symmetry and Differential Properties of Generalized Boolean Functions

  • Thor Martinsen
  • Wilfried Meidl
  • Alexander Pott
  • Pantelimon StănicăEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11321)


In this paper we investigate various differential properties of generalized Boolean functions defined on \({\mathbb F}_2^n\) with values in \({\mathbb Z}_{2^k}\), \(k\ge 2\). We characterize linear structures for the generalized Boolean functions in terms of their binary expansion components, and find all symmetric generalized bent functions. Next, we show that there are no symmetric balanced functions defined on \({\mathbb F}_2^n\) with values in a group of order \(2^k, k\ge 2\), a contrast to the classical case for \(k=1\), commonly known as the bisection of binomial coefficients. Further, we characterize the avalanche features of a generalized Boolean function in terms of differentials. Lastly, we show that a partially gbent function is plateaued.


Generalized Boolean functions Linear structures Generalized bent Semibent Partially bent Avalanche features Plateaued 



This paper was started while the second and fourth named authors visited the third named author at the Institute of Algebra and Geometry, of Otto von Guericke University Magdeburg. They thank the host and the institute for hospitality and excellent working conditions.


  1. 1.
    Budaghyan, L.: Construction and Analysis of Cryptographic Functions. Springer, Cham (2014). 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)zbMATHGoogle Scholar
  3. 3.
    Carlet, C.: Vectorial Boolean functions for cryptography. In: Crama, Y., Hammer, P. (eds.) Boolean Methods and Models, pp. 398–472. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  4. 4.
    Carlet, C.: Partially bent functions. Des. Codes Cryptogr. 3, 135–145 (1993)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Çeşmelioğlu, A., Meidl, W., Topuzoğlu, A.: Partially bent functions and their properties. Applied Algebra and Number Theory, pp. 22–38. Cambridge University Press, Cambridge (2014)CrossRefGoogle Scholar
  6. 6.
    Cusick, T.W., Li, Y.: \(k\)-th order symmetric SAC Boolean functions and bisecting binomial coefficients. Discrete Appl. Math. 149, 73–86 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cusick, T.W., Stănică, P.: Cryptographic Boolean Functions and Applications, 2nd edn. Academic Press, San Diego (2017)zbMATHGoogle Scholar
  8. 8.
    Dubuc, S.: Characterization of linear structures. Des. Codes Cryptogr. 22, 33–45 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Evertse, J.-H.: Linear structures in blockciphers. In: Chaum, D., Price, W.L. (eds.) EUROCRYPT 1987. LNCS, vol. 304, pp. 249–266. Springer, Heidelberg (1988). Scholar
  10. 10.
    Freiman, G.A.: On solvability of a system of two Boolean linear equations. In: Chudnovsky, D.V., Chudnovsky, G.V., Nathanson, M.B. (eds.) Number Theory: New York Seminar 1991–1995, pp. 135–150. Springer, New York (1996)CrossRefGoogle Scholar
  11. 11.
    von zur Gathen, J., Roche, J.: Polynomials with two values. Combinatorica 17, 345–362 (1997)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gopalakrishnan, K., Hoffman, D.G., Stinson, D.R.: A note on a conjecture concerning symmetric resilient functions. Inf. Process. Lett. 47, 139–143 (1993)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gouget, A.: On the propagation criterion of Boolean functions. In: Feng, K., Niederreiter, H., Xing, C. (eds.) Coding, Cryptography and Combinatorics, vol. 23, pp. 153–168. Birkhäuser, Basel (2004). Scholar
  14. 14.
    Hodžić, S., Meidl, W., Pasalic, E.: Full characterization of generalized bent functions as (semi)-bent spaces, their dual and the Gray image. IEEE Trans. Inf. Theory 64(7), 5432–5440 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ionascu, E.J., Martinsen, T., Stănică, P.: Bisecting binomial coefficients. Discrete Appl. Math. 227, 70–83 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jefferies, N.: Sporadic partitions of binomial coefficients. Electron. Lett. 27(15), 134–136 (1991)CrossRefGoogle Scholar
  17. 17.
    Kumar, P.V., Scholtz, R.A., Welch, L.R.: Generalized bent functions and their properties. J. Comb. Theory Ser. A 40, 90–107 (1985)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lai, X.: Additive and linear structures of cryptographic functions. In: Preneel, B. (ed.) FSE 1994. LNCS, vol. 1008, pp. 75–85. Springer, Heidelberg (1995). Scholar
  19. 19.
    Lechner, R.L.: Harmonic analysis of switching functions. In: Mukhopadhyay, A. (ed.) Recent Developments in Switching Theory. Academic Press, New York (1971)Google Scholar
  20. 20.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error Correcting Codes. North-Holland, Amsterdam (1977)zbMATHGoogle Scholar
  21. 21.
    Martinsen, T.: Correlation immunity, avalanche features, and other cryptographic properties of generalized Boolean functions. Ph.D. dissertation, Naval Postgraduate School, Monterey, CA (2017)Google Scholar
  22. 22.
    Martinsen, T., Meidl, W., Mesnager, S., Stanica, P.: Decomposing generalized bent and hyperbent functions. IEEE Trans. Inf. Theory 63(12), 7804–7812 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Martinsen, T., Meidl, W., Stănică, P.: Generalized bent functions and their Gray images. In: Duquesne, S., Petkova-Nikova, S. (eds.) WAIFI 2016. LNCS, vol. 10064, pp. 160–173. Springer, Cham (2016). Scholar
  24. 24.
    Martinsen, T., Meidl, W., Stănică, P.: Partial spread and vectorial generalized bent functions. Des. Codes Cryptogr. 85(1), 1–13 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mesnager, S.: Bent Functions: Fundamentals and Results. Springer, Cham (2016). Scholar
  26. 26.
    Savicky, P.: On the bent Boolean functions that are symmetric. Eur. J. Comb. 15, 407–410 (1994)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Schmidt, K.U.: Quaternary constant-amplitude codes for multicode CDMA. IEEE Trans. Inf. Theory 55(4), 1824–1832 (2009)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Solé, P., Tokareva, N.: Connections between quaternary and binary bent functions. Prikl. Diskr. Mat. 1, 16–18 (2009).
  29. 29.
    Stănică, P., Martinsen, T., Gangopadhyay, S., Singh, B.K.: Bent and generalized bent Boolean functions. Des. Codes Cryptogr. 69, 77–94 (2013)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Tang, C., Xiang, C., Qi, Y., Feng, K.: Complete characterization of generalized bent and \(2^k\)-bent Boolean functions. IEEE Trans. Inf. Theory 63(7), 4668–4674 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Tokareva, N.: Bent Functions, Results and Applications to Cryptography. Academic Press, San Diego (2015)zbMATHGoogle Scholar
  32. 32.
    Zheng, Y.L., Zhang, X.M.: On plateaued functions. IEEE Trans. Inf. Theory 47(9), 1215–1223 (2001)MathSciNetCrossRefGoogle Scholar

Copyright information

© This is a U.S. government work and not under copyright protection in the United States; foreign copyright protection may apply 2018

Authors and Affiliations

  • Thor Martinsen
    • 1
  • Wilfried Meidl
    • 2
  • Alexander Pott
    • 3
  • Pantelimon Stănică
    • 1
    Email author
  1. 1.Department of Applied MathematicsNaval Postgraduate SchoolMontereyUSA
  2. 2.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria
  3. 3.Institute of Algebra and Geometry, Faculty of MathematicsOtto von Guericke University MagdeburgMagdeburgGermany

Personalised recommendations