Multiple Product Modulo Arbitrary Numbers

Let n binary numbers of length n be given. The Boolean function ``Multiple Product'' MPn asks for (some binary representation of ) the value of their product. It has been shown (K.-Y. Siu and V. Roychowdhury, On optimal depth threshold circuits for multiplication and related problems, SIAM J. Discrete Math. 7, 285 292 (1994)) that this function can be computed in polynomial-size threshold circuits of depth 4. For many 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 of depth 2. In this paper, we investigate the complexity of computing MPn modulo m by depth-2 threshold circuits. It turns out that for all but a few integers m, exponential size is required. In particular, it is shown that for m # [2, 4, 8], polynomial-size circuits exist, for m # [3, 6, 12, 24], the question remains open and in all other cases, exponential-size circuits are required. The result still holds if we allow m to grow with n. ] 1996