# Multiple product modulo arbitrary numbers

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## 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* < 2^{cn}, for some constant *c*). This improves upon recent results in [K1].

We also investigate moduli of the form 2^{i}3^{j}. 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## Preview

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