On realizing iterated multiplication by small depth threshold circuits

  • Matthias Krause
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 900)


Using a lower bound argument based on probabilistic communication complexity it will be shown that iterated multiplication of n-bit numbers modulo polylog (n)-bit integers cannot be done efficiently by depth two threshold circuits. As a consequence we obtain that for iterated multiplication of n-bit numbers, in contrast to multiplication, powering, and division, decomposition via Chinese Remaindering does not yield efficient depth 3 threshold circuits.


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  1. [A89]
    Allender, E.: A note on the power of threshold circuits, Proceedings der 30. IEEE Symposium FOCS, 1989, 580–584.Google Scholar
  2. [AB91]
    Alon, N., J. Bruck: Explicit constructions of depth-2 majority circuits for comparison and addition, Technical Report RJ 8300 (75661) of the IBM Almaden Research Center, San Jose, 1991.Google Scholar
  3. [B90]
    Bruck, J. Harmonic analysis of polynomial threshold functions, SIAM Journal of Discrete Mathematics, 3, Nr. 22, 1990, 168–177.Google Scholar
  4. [BHK94]
    Bertram, C., Hofmeister, Th., Krause, M., Multiple product mod small numbers manuscript Dortmund 1994Google Scholar
  5. [BHKS92]
    Bruck, J., Th. Hofmeister, Th. Kailath, K.Y. Siu, Depth efficient networks for division and related problems. Technical Report 1992, to appear in IEEE Transactions on Information Theory.Google Scholar
  6. [GHR92]
    Goldmann, M., J. Håstad, A. A. Razborov: Majority Gates versus general weighted threshold gates, J. of Computational Complexity 2 (1992), 277–300.Google Scholar
  7. [GK93]
    Goldmann, M., M. Karpinski: Simulating Threshold Circuits by Majority Circuits. Proc. 25th ACM Conference STOC, 1993.Google Scholar
  8. [HMPST87]
    Hajnal, A., W. Maass, P. Pudlák, M. Szegedy, G. Turán: Threshold circuits of bounded depth, Proc. 28th IEEE Conf. FOCS, 1987, 99–110.Google Scholar
  9. [HR88]
    Halstenberg, B., R. Reischuk Relations between communication complexity classes Proc. of the 3. IEEE Structure in Complexity Theory Conference, 1988, 19–28.Google Scholar
  10. [H93]
    Hofmeister, Th. Depth-efficient threshold circuits for arithmetic functions in: Theoretical Advances in Neural Computation and Learning eds. Roychowdhury et. al, Kluwer Academic Publishers, ISBN 0-7923-9478-X.Google Scholar
  11. [HHK91]
    Hofmeister, Th., W. Hohberg, S. Köhling: Some notes on threshold circuits and multiplication in depth 4 IPL 39 (1991) 219–225.Google Scholar
  12. [K91]
    Krause, M. Geometric Arguments yield better bounds for threshold circuits and distributed computing Proc. of the 6. IEEE Structure in Complexity Theory Conference, 314–322.Google Scholar
  13. [KW91]
    Krause, M., S. Waack, Variation ranks of communication matrices and lower bounds for depth two circuits having symmetric gates with unbounded fanin, Proc. 32th IEEE Conference FOCS, 1991, 777–787.Google Scholar
  14. [RT92]
    Reif, J. H., S. R. Tate On threshold circuits and polynomial computation SIAM Journal of Computing, Vol. 21, Nr.5, pp. 896–908, 1992Google Scholar
  15. [Y90]
    Yao, A.C.: On ACC and Threshold Circuits, Proc. 31th IEEE Conference FOCS, 1990, 619–628.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Matthias Krause
    • 1
  1. 1.Lehrstuhl Informatik IIUniversität DortmundDeutschland

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