Computing Elementary Symmetric Polynomials with a Sublinear Number of Multiplications

Elementary symmetric polynomials $S_n^k$ are used as a benchmark for the bounded-depth arithmetic circuit model of computation. In this work we prove that $S_n^k$ modulo composite numbers $m=p_1p_2$ can be computed with much fewer multiplications than over any field, if the coefficients of monomials $x_{i_1}x_{i_2}... x_{i_k}$ are allowed to be 1 either mod $p_1$ or mod $p_2$ but not necessarily both. More exactly, we prove that for any constant $k$ such a representation of $S_n^k$ can be computed modulo $p_1p_2$ using only $\exp(O(\sqrt{\log n}\log\log n))$ multiplications on the most restricted depth-3 arithmetic circuits, for $\min({p_1,p_2})>k!$. Moreover, the number of multiplications remain sublinear while $k=O(\log\log n).$ In contrast, the well-known Graham-Pollack bound yields an $n-1$ lower bound for the number of multiplications even for the exact computation (not the representation) of $S_n^2$. Our results generalize for other non-prime power composite moduli as well. The proof uses the famous BBR-polynomial of Barrington, Beigel and Rudich.