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Revision #1 to TR02-052 | 11th November 2002 00:00

#### Computing Elementary Symmetric Polynomials with a Sub-Polynomial Number of Multiplications Revision of: TR02-052

Revision #1
Authors: Vince Grolmusz
Accepted on: 11th November 2002 00:00
Keywords:

Abstract:

### Paper:

TR02-052 | 3rd September 2002 00:00

#### Computing Elementary Symmetric Polynomials with a Sub-Polynomial Number of Multiplications

TR02-052
Authors: Vince Grolmusz
Publication: 16th September 2002 18:47
Keywords:

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.

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