International Conference on Theory and Practice of Natural Computing

Theory and Practice of Natural Computing pp 71-82 | Cite as

Evolutionary Approach for Finding Correlation Immune Boolean Functions of Order t with Minimal Hamming Weight

  • Stjepan Picek
  • Sylvain Guilley
  • Claude Carlet
  • Domagoj Jakobovic
  • Julian F. Miller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9477)

Abstract

The role of Boolean functions is prominent in several areas like cryptography, sequences and coding theory. Therefore, various methods to construct Boolean functions with desired properties are of direct interest. When concentrating on Boolean functions and their role in cryptography, we observe that new motivations and hence new properties have emerged during the years. It is important to note that there are still many design criteria left unexplored and this is where Evolutionary Computation can play a distinct role. One combination of design criteria that has appeared recently is finding Boolean functions that have various orders of correlation immunity and minimal Hamming weight. Surprisingly, most of the more traditionally used methods for Boolean function generation are inadequate in this domain. In this paper, we concentrate on a detailed exploration of several evolutionary algorithms and their applicability for this problem. Our results show that such algorithms are a viable choice when evolving Boolean functions with minimal Hamming weight and certain order of correlation immunity. This approach is also successful in obtaining Boolean functions with several values that were known previously to be theoretically optimal, but no one succeeded in finding actual Boolean functions with such values.

Keywords

Boolean functions Cryptography Correlation immunity Hamming weight Evolutionary algorithms 

References

  1. 1.
    Bhasin, S., Carlet, C., Guilley, S.: Theory of masking with codewords in hardware: low-weight \(d\)th-order correlation-immune boolean functions. Cryptology ePrint Archive, Report 2013/303 (2013). http://eprint.iacr.org/
  2. 2.
    Burnett, L.D.: Heuristic optimization of boolean functions and substitution boxes for cryptography. Ph.D. thesis, Queensland University of Technology (2005)Google Scholar
  3. 3.
    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, 1st edn, pp. 257–397. Cambridge University Press, New York (2010)CrossRefGoogle Scholar
  4. 4.
    Carlet, C., Danger, J.-L., Guilley, S., Maghrebi, H.: Leakage squeezing of order two. In: Galbraith, S., Nandi, M. (eds.) INDOCRYPT 2012. LNCS, vol. 7668, pp. 120–139. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  5. 5.
    Carlet, C., Guilley, S.: Side-channel Indistinguishability. In: Proceedings of the 2nd International Workshop on Hardware and Architectural Support for Security and Privacy, HASP 2013, pp. 9:1–9:8. ACM, New York (2013)Google Scholar
  6. 6.
    Carlet, C., Guilley, S.: Correlation-immune boolean functions for easing counter measures to side-channel attacks (Chapter 3). In: Niederreiter, H., Ostafe, A., Panario, D., Winterhof, A. (eds.) Algebraic Curves and Finite Fields Cryptography and Other Applications. Radon Series on Computational and Applied Mathematics, vol. 16, pp. 41–70. De Gruyter, Berlin (2014)Google Scholar
  7. 7.
    Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)CrossRefGoogle Scholar
  8. 8.
    Eiben, A.E., Smith, J.E.: Introduction to Evolutionary Computing. Springer, Heidelberg (2003)CrossRefMATHGoogle Scholar
  9. 9.
    Gammel, B.M., Mangard, S.: On the duality of probing and fault attacks. J. Electron. Test. 26(4), 483–493 (2010). http://dx.doi.org/10.1007/s10836-010-5160-0
  10. 10.
    Guo-Zhen, X., Massey, J.: A spectral characterization of correlation-immune combining functions. IEEE Trans. Inf. Theor. 34(3), 569–571 (1988)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hedayat, A.S., Sloane, N.J.A., Stufken, J.: Orthogonal Arrays—Theory and Applications. Springer Series in Statistics. Springer, New York (1999)CrossRefMATHGoogle Scholar
  12. 12.
    Koza, J.R.: Genetic Programming: On the Programming of Computers by Means of Natural Selection. MIT Press, Cambridge (1992)MATHGoogle Scholar
  13. 13.
    Mangard, S., Oswald, E., Popp, T.: Power Analysis Attacks: Revealing the Secrets of Smart Cards (Advances in Information Security). Springer, Secaucus (2007)MATHGoogle Scholar
  14. 14.
    McLaughlin, J., Clark, J.A.: Evolving balanced Boolean functions with optimal resistance to algebraic and fast algebraic attacks, maximal algebraic degree, and very high nonlinearity. Cryptology ePrint Archive, Report 2013/011 (2013). http://eprint.iacr.org/
  15. 15.
    Millan, W.L., Clark, A.J., 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) CrossRefGoogle Scholar
  16. 16.
    Miller, J.F. (ed.): Cartesian Genetic Programming. Natural Computing Series. Springer, Heidelberg (2011) MATHGoogle Scholar
  17. 17.
    Picek, S., Carlet, C., Jakobovic, D., Miller, J.F., Batina, L.: Correlation immunity of boolean functions: an evolutionary algorithms perspective. In: Proceedings of the Genetic and Evolutionary Computation Conference, GECCO 2015, Madrid, Spain, pp. 1095–1102, July 11–15, 2015Google Scholar
  18. 18.
    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. ACM, New York (2013)Google Scholar
  19. 19.
    Picek, S., Jakobovic, D., Miller, J.F., Marchiori, E., Batina, L.: Evolutionary methods for the construction of cryptographic boolean functions. In: Proceedings of Genetic Programming - 18th European Conference, EuroGP 2015, Copenhagen, Denmark, April 8–10, 2015, pp. 192–204 (2015)Google Scholar
  20. 20.
    Picek, S., Marchiori, E., Batina, L., Jakobovic, D.: Combining evolutionary computation and algebraic constructions to find cryptography-relevant boolean functions. In: Bartz-Beielstein, T., Branke, J., Filipič, B., Smith, J. (eds.) PPSN 2014. LNCS, vol. 8672, pp. 822–831. Springer, Heidelberg (2014) Google Scholar
  21. 21.
    Siegenthaler, T.: Correlation-immunity of nonlinear combining functions for cryptographic applications (corresp.). IEEE Trans. Inf. Theor. 30(5), 776–780 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Stjepan Picek
    • 1
  • Sylvain Guilley
    • 2
    • 3
  • Claude Carlet
    • 4
    • 5
  • Domagoj Jakobovic
    • 6
  • Julian F. Miller
    • 7
  1. 1.ESAT/COSIC and IMindsKU LeuvenLeuven-heverleeBelgium
  2. 2.TELECOM-ParisTechParis Cedex 13France
  3. 3.Secure-IC S.A.S.Cesson-SévignéFrance
  4. 4.LAGA, UMR 7539, CNRS, Department of MathematicsUniversity of Paris 8Saint-Denis CedexFrance
  5. 5.LAGA, UMR 7539, CNRS, Department of MathematicsUniversity of Paris 13VilletaneuseFrance
  6. 6.Faculty of Electrical Engineering and ComputingUniversity of ZagrebZagrebCroatia
  7. 7.Department of ElectronicsUniversity of YorkYorkUK

Personalised recommendations