Evolutionary Search of Binary Orthogonal Arrays

  • Luca MariotEmail author
  • Stjepan Picek
  • Domagoj Jakobovic
  • Alberto Leporati
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11101)


Orthogonal Arrays (OA) represent an interesting breed of combinatorial designs that finds applications in several domains such as statistics, coding theory, and cryptography. In this work, we address the problem of constructing binary OA through evolutionary algorithms, an approach which received little attention in the combinatorial designs literature. We focus on the representation of a feasible solution, which we encode as a set of Boolean functions whose truth tables are used as the columns of a binary matrix, and on the design of an appropriate fitness function and variation operators for this problem. We finally present experimental results obtained with genetic algorithms (GA) and genetic programming (GP) on optimizing such fitness function, and compare the performances of these two metaheuristics with respect to the size of the considered problem instances. The experimental results show that GP outperforms GA at handling this type of problem, as it converges to an optimal solution in all considered problem instances but one.


Orthogonal arrays Genetic algorithms Genetic programming Boolean functions 



This work has been supported in part by Croatian Science Foundation under the project IP-2014-09-4882.


  1. 1.
    Carlet, C., Guilley, S.: Correlation-immune boolean functions for easing counter measures to side-channel attacks. Algebraic Curves Finite Fields: Cryptograph. Other Appl. 16, 41–70 (2014)MathSciNetGoogle Scholar
  2. 2.
    Colbourn, C.J., Dinitz, J.H.: Handbook of Combinatorial Designs. CRC Press, Boca Raton (2006)CrossRefGoogle Scholar
  3. 3.
    Hedayat, A.S., Sloane, N.J.A., Stufken, J.: Orthogonal Arrays: Theory and Applications. Springer, Heidelberg (2012). Scholar
  4. 4.
    Mariot, L., Leporati, A.: Heuristic search by particle swarm optimization of boolean functions for cryptographic applications. In: Genetic and Evolutionary Computation Conference, Companion Material Proceedings , GECCO 2015, Madrid, Spain, 11–15 July 2015, pp. 1425–1426 (2015)Google Scholar
  5. 5.
    Mariot, L., Picek, S., Jakobovic, D., Leporati, A.: Evolutionary algorithms for the design of orthogonal latin squares based on cellular automata. In: Proceedings of the Genetic and Evolutionary Computation Conference, GECCO 2017, Berlin, Germany, 15–19 July 2017, pp. 306–313 (2017)Google Scholar
  6. 6.
    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
  7. 7.
    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
  8. 8.
    Poli, R., Langdon, W.B., McPhee, N.F.: A Field Guide to Genetic Programming (2008). and freely available at (With contributions by J.R. Koza)
  9. 9.
    Safadi, R., Wang, R.: The use of genetic algorithms in the construction of mixed multilevel orthogonal arrays. Technical report, Olin Corp Cheshire CT Olin Research Center (1992)Google Scholar
  10. 10.
    Sloane, N.J.: A library of orthogonal arrays. Fixed-level arrays with more than three levels: OA 16(4.2) (2007)Google Scholar
  11. 11.
    Stinson, D.R.: Combinatorial Designs: Constructions and Analysis. Springer, Heidelberg (2007). Scholar
  12. 12.
    Wang, R., Safadi, R.: Generating mixed multilevel orthogonal arrays by simulated annealing. In: Page, C., LePage, R. (eds.) Computing Science and Statistics, pp. 557–560. Springer, New York (1992). Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

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

Personalised recommendations