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TR02-052 | 3rd September 2002 00:00
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#### Computing Elementary Symmetric Polynomials with a Sub-Polynomial Number of Multiplications

**Abstract:**
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 $x_{i_1}x_{i_2}\cdots x_{i_k}$ 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 second elementary symmetric polynomial $S_n^2$.

Our results generalize for other non-prime power composite moduli as

well. The proof uses perfect hashing functions and the famous BBR-polynomial of Barrington, Beigel and

Rudich.