Concrete Multiplicative Complexity of Symmetric Functions

  • Joan Boyar
  • René Peralta
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)


The multiplicative complexity of a Boolean function f is defined as the minimum number of binary conjunction (AND) gates required to construct a circuit representing f, when only exclusive-or, conjunction and negation gates may be used. This article explores in detail the multiplicative complexity of symmetric Boolean functions. New techniques that allow such exploration are introduced. They are powerful enough to give exact multiplicative complexities for several classes of symmetric functions. In particular, the multiplicative complexity of computing the Hamming weight of n bits is shown to be exactly n − H (n), where H (n) is the Hamming weight of the binary representation of n. We also show a close relationship between the complexity of symmetric functions and fractals derived from the parity of binomial coefficients.


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  1. 1.
    Aleksanyan, A.A.: On realization of quadratic Boolean functions by systems of linear equations. Cybernetics 25(1), 9–17 (1989)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness Theorems for Non-Cryptographic Fault-Tolerant Distributed Computation. In: Proceedings of the 20th ACM Symposium on the Theory of Computing, pp. 1–10 (1988)Google Scholar
  3. 3.
    Boyar, J., Damgård, I., Peralta, R.: Short non-interactive cryptographic proofs. Journal of Cryptology 13, 449–472 (2000)zbMATHCrossRefGoogle Scholar
  4. 4.
    Boyar, J., Peralta, R., Pochuev, D.: On the multiplicative complexity of Boolean functions over the basis ( ∧ , ⊕ , 1). Theoretical Computer Science 235, 43–57 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Grundlehren der mathematischen Wissenschaften, vol. 315. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  6. 6.
    Cramer, R., Damgård, I.B., Nielsen, J.B.: Multiparty computation from threshold homomorphic encryption. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 280–300. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Chaum, D., Crépeau, C., Damgård, I.: Multi-party unconditionally secure protocols. In: Proceedings of the 20th ACM Symposium on the Theory of Computing, pp. 11–19 (1988)Google Scholar
  8. 8.
    Kummer, E.E.: Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen. J. Reine Angew. Math. 44, 93–146 (1852)zbMATHCrossRefGoogle Scholar
  9. 9.
    Hirt, M., Nielsen, J.B.: Upper bounds on the communication complexity of optimally resilient cryptographic multiparty computation. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 79–99. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Mihaĭljuk, M.V.: On the complexity of calculating the elementary symmetric functions over finite fields. Sov. Math. Dokl. 20, 170–174 (1979)Google Scholar
  11. 11.
    Mirwald, R., Schnorr, C.: The multiplicative complexity of quadratic Boolean forms. Theoretical Computer Science 102(2), 307–328 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Paul, W.J.: A 2.5n lower bound on the combinational complexity of boolean functions. In: Proceedings of the 7th ACM Symposium on the Theory of Computing, pp. 27–36 (1975)Google Scholar
  13. 13.
    Rueppel, R., Massey, J.: The knapsack as a nonlinear function. In: Abstracts of papers, IEEE Int. Symp. on Information Theory, p. 46 (1985)Google Scholar
  14. 14.
    Schnorr, C.P.: The multiplicative complexity of Boolean functions. In: Mora, T. (ed.) AAECC 1988. LNCS, vol. 357, pp. 45–58. Springer, Heidelberg (1989)Google Scholar
  15. 15.
    Stockmeyer, L.: On the combinational complexity of certain symmetric Boolean functions. Mathematical Systems Theory 10, 323–336 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    van der Waerden, B.L.: Algebra. Frederick Ungar PublishingGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Joan Boyar
    • 1
  • René Peralta
    • 2
  1. 1.Dept. of Math. and Computer ScienceUniversity of Southern Denmark 
  2. 2.Security Division, Information Technology LaboratoryNIST 

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