Computing Elementary Symmetric Polynomials with a Sublinear Number of Multiplications

TL;DR

This paper demonstrates that elementary symmetric polynomials can be computed with significantly fewer multiplications over composite moduli than over fields, especially for certain degrees, using novel circuit constructions.

Contribution

It introduces a sublinear multiplication method for computing symmetric polynomials modulo composite numbers, surpassing classical bounds for specific circuit depths and degrees.

Findings

Fewer multiplications needed for symmetric polynomials over composite moduli.

Sublinear complexity achieved for degrees up to O(log log n).

Generalizes to non-prime power composite moduli.

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 $x_{i_1}x_{i_2}... x_{i_k}$ 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.