Multiple product modulo arbitrary numbers

  • Claudia Bertram-Kretzberg
  • Thomas Hofmeister
Contributed Papers Lower Bounds
Part of the Lecture Notes in Computer Science book series (LNCS, volume 969)


Let n binary numbers of length n be given. The Boolean function ”Multiple Product” MP n asks for (some binary representation of) the value of their product. It has been shown in [SR],[SBKH] that this function can be computed in polynomial-size threshold circuits of depth 4. For a lot of other arithmetic functions, circuits of depth 3 are known. They are mostly based on the fact that the value of the considered function modulo some prime numbers p can be computed easily in threshold circuits.

In this paper, we show that the difficulty in constructing smaller depth circuits for MP n stems from the fact that for all numbers m which are divisible by a prime larger than 3, computing MP n modulo m already cannot be computed in depth 2 and polynomial size. This result still holds if we allow m to grow exponentially in n (m < 2cn, for some constant c). This improves upon recent results in [K1].

We also investigate moduli of the form 2i3j. In particular, we show that there are depth-2 polynomial-size threshold circuits for computing MP n modulo m if m ε {2,4,8}, and that no such circuits exist if m is divisible by 9 or 16.


Boolean Function Multiple Product Binary Number Arithmetic Function Chinese Remainder Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [HMPST]
    A. Hajnal, W. Maass, P. Pudlák, M. Szegedy, G. Turán, Threshold circuits of bounded depth, Proceedings of 28th FOCS, 1987, 99–110.Google Scholar
  2. [GHR]
    M. Goldmann, J. Håstad, A. Razborov, Majority gates vs. general weighted threshold gates, Proc. 7th Structures (1992), pp. 2–13.Google Scholar
  3. [GK]
    M. Goldmann, M. Karpinski, Simulating threshold circuits by majority circuits, Proc. 25th STOC, 1993, p. 551–560.Google Scholar
  4. [H]
    T. Hofmeister, Depth-efficient threshold circuits for arithmetic functions, Chap. 2 in: Theoretical Advances in Neural Computation and Learning, V. Roychowdhury, K-Y. Siu, and A. Orlitsky (eds.), Kluwer Academic Publ.Google Scholar
  5. [K1]
    M. Krause, On realizing iterated multiplication by small depth threshold circuits, Proc. of 12th STACS (1995), 83–94.Google Scholar
  6. [K2]
    M. Krause, pers. comm.Google Scholar
  7. [kw]
    M. Krause, S. Waack, Variation ranks of communication matrices and lower bounds for depth two circuits having symmetric gates with unbounded fan-in, Proc. of 32nd FOCS, 1991, 777–782.Google Scholar
  8. [R]
    A. Razborov, On small depth threshold circuits, In Proc. 3rd Scandinavian Workshop on Algorithm Theory, LNCS 621, 42–52, 1992.Google Scholar
  9. [S]
    A. Scholz, B. Schoeneberg, Einführung in die Zahlentheorie, Sammlung Göschen, Band 5131, Walter de Gruyter, 1973Google Scholar
  10. [SBKH]
    K.-Y. Siu, J. Bruck, T. Kailath, T. Hofmeister, Depth efficient neural networks for division and related problems, IEEE Transactions on Information Theory, May 1993, p. 946–956.Google Scholar
  11. [SR]
    K.-Y. Siu, V. Roychowdhury, On optimal depth threshold circuits for multiplication and related problems, SIAM Journal on Discrete Mathematics 7 (1994), p. 285–292.Google Scholar
  12. [W]
    I. Wegener, Optimal lower bounds on the depth of polynomial-size threshold circuits for some arithmetic functions, Inf. Proc. Letters 46 (1993), p. 85–87.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Claudia Bertram-Kretzberg
    • 1
  • Thomas Hofmeister
    • 1
  1. 1.Lehrstuhl Informatik IIUniversität DortmundDortmundGermany

Personalised recommendations