Hyper-bent Boolean Functions and Evolutionary Algorithms

  • Luca MariotEmail author
  • Domagoj Jakobovic
  • Alberto Leporati
  • Stjepan Picek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11451)


Bent Boolean functions play an important role in the design of secure symmetric ciphers, since they achieve the maximum distance from affine functions allowed by Parseval’s relation. Hyper-bent functions, in turn, are those bent functions which additionally reach maximum distance from all bijective monomial functions, and provide further security towards approximation attacks. Being characterized by a stricter definition, hyper-bent functions are rarer than bent functions, and much more difficult to construct. In this paper, we employ several evolutionary algorithms in order to evolve hyper-bent Boolean functions of various sizes. Our results show that hyper-bent functions are extremely difficult to evolve, since we manage to find such functions only for the smallest investigated size. Interestingly, we are able to identify this difficulty as not lying in the evolution of hyper-bent functions itself, but rather in evolving some of their components, i.e. bent functions. Finally, we present an additional parameter to evaluate the performance of evolutionary algorithms when evolving Boolean functions: the diversity of the obtained solutions.


Bent functions Hyper-bent functions Genetic programming Genetic algorithms Evolution strategies 



The authors wish to thank the anonymous referees for their useful comments on improving the presentation quality of the paper. This work has been supported in part by Croatian Science Foundation under the project IP-2014-09-4882. In addition, this work was supported in part by the Research Council KU Leuven (C16/15/058) and IOF project EDA-DSE (HB/13/020).


  1. 1.
    Hrbacek, R., Dvorak, V.: Bent function synthesis by means of cartesian genetic programming. In: Bartz-Beielstein, T., Branke, J., Filipič, B., Smith, J. (eds.) PPSN 2014. LNCS, vol. 8672, pp. 414–423. Springer, Cham (2014). Scholar
  2. 2.
    Picek, S., Jakobovic, D.: Evolving algebraic constructions for designing bent Boolean functions. In: Proceedings of the 2016 on Genetic and Evolutionary Computation Conference, Denver, CO, USA, 20–24 July 2016, pp. 781–788 (2016)Google Scholar
  3. 3.
    Picek, S., Sisejkovic, D., Jakobovic, D.: Immunological algorithms paradigm for construction of Boolean functions with good cryptographic properties. Eng. Appl. Artif. Intell. 62, 320–330 (2016)CrossRefGoogle Scholar
  4. 4.
    Youssef, A.M., Gong, G.: Hyper-bent functions. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 406–419. Springer, Heidelberg (2001). Scholar
  5. 5.
    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
  6. 6.
    Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
  7. 7.
    Carlet, C., Gaborit, P.: Hyper-bent functions and cyclic codes. J. Comb. Theory Ser. A 113(3), 466–482 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gong, G., Golomb, S.W.: Transform domain analysis of DES. IEEE Trans. Inf. Theory 45(6), 2065–2073 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Charpin, P., Gong, G.: Hyperbent functions, Kloosterman sums, and Dickson polynomials. IEEE Trans. Inf. Theory 54(9), 4230–4238 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Millan, W., Clark, A., Dawson, E.: An effective genetic algorithm for finding highly nonlinear Boolean functions. In: Proceedings of the First International Conference on Information and Communication Security, ICICS 1997, pp. 149–158 (1997)Google Scholar
  11. 11.
    Millan, W., Clark, A., Dawson, E.: Heuristic design of cryptographically strong balanced Boolean functions. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 489–499. Springer, Heidelberg (1998). Scholar
  12. 12.
    Millan, W., Fuller, J., Dawson, E.: New concepts in evolutionary search for Boolean functions in cryptology. Comput. Intell. 20(3), 463–474 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Picek, S., Jakobovic, D., Golub, M.: Evolving cryptographically sound Boolean functions. In: Proceedings of the 15th Annual Conference Companion on Genetic and Evolutionary Computation, GECCO 2013 Companion, pp. 191–192 (2013)Google Scholar
  14. 14.
    Mariot, L., Leporati, A.: Heuristic search by particle swarm optimization of Boolean functions for cryptographic applications. In: GECCO (Companion), pp. 1425–1426. ACM (2015)Google Scholar
  15. 15.
    Mariot, L., Leporati, A.: A genetic algorithm for evolving plateaued cryptographic Boolean functions. In: Dediu, A.-H., Magdalena, L., Martín-Vide, C. (eds.) TPNC 2015. LNCS, vol. 9477, pp. 33–45. Springer, Cham (2015). Scholar
  16. 16.
    Picek, S., Jakobovic, D., Miller, J.F., Batina, L., Cupic, M.: Cryptographic Boolean functions: one output, many design criteria. Appl. Soft Comput. 40, 635–653 (2016)CrossRefGoogle Scholar
  17. 17.
    Bäck, T., Fogel, D., Michalewicz, Z. (eds.): Evolutionary Computation 1: Basic Algorithms and Operators. Institute of Physics Publishing, Bristol (2000)zbMATHGoogle Scholar
  18. 18.
    Rozenberg, G., Bäck, T., Kok, J.N.: Handbook of Natural Computing. Springer, Heidelberg (2011). Scholar
  19. 19.
    Beyer, H.G., Schwefel, H.P.: Evolution strategies a comprehensive introduction. Nat. Comput. 1(1), 3–52 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Poli, R., Langdon, W.B., McPhee, N.F.: A field guide to genetic programming (2008).,
  21. 21.
    Dillon, J.F.: Elementary Hadamard difference sets. Ph.D. thesis, University of Maryland (1974)Google Scholar
  22. 22.
    Mariot, L., Picek, S., Jakobovic, D., Leporati, A.: Evolutionary search of binary orthogonal arrays. In: Auger, A., Fonseca, C.M., Lourenço, N., Machado, P., Paquete, L., Whitley, D. (eds.) PPSN 2018, Part I. LNCS, vol. 11101, pp. 121–133. Springer, Cham (2018). Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Luca Mariot
    • 1
    Email author
  • Domagoj Jakobovic
    • 2
  • Alberto Leporati
    • 1
  • Stjepan Picek
    • 3
  1. 1.DISCoUniversità degli Studi di Milano-BicoccaMilanItaly
  2. 2.Faculty of Electrical Engineering and ComputingUniversity of ZagrebZagrebCroatia
  3. 3.Cyber Security Research GroupDelft University of TechnologyDelftThe Netherlands

Personalised recommendations