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Multiple product modulo arbitrary numbers

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

Abstract

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.

Keywords

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.

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References

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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

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

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