Polynomials: a new tool for length reduction in binary discrete convolutions

TL;DR

This paper introduces a novel polynomial-based length reduction method that enables efficient algorithms for sparse Walsh transforms and improves deterministic sparse polynomial multiplication.

Contribution

The paper presents the first polynomial-based length reduction technique for sparse Walsh transforms and enhances deterministic algorithms for sparse polynomial multiplication.

Findings

Developed a new polynomial-based length reduction method.

First efficient algorithm for sparse Walsh transform.

Faster deterministic algorithms for sparse polynomial multiplication.

Abstract

Efficient handling of sparse data is a key challenge in Computer Science. Binary convolutions, such as polynomial multiplication or the Walsh Transform are a useful tool in many applications and are efficiently solved. In the last decade, several problems required efficient solution of sparse binary convolutions. both randomized and deterministic algorithms were developed for efficiently computing the sparse polynomial multiplication. The key operation in all these algorithms was length reduction. The sparse data is mapped into small vectors that preserve the convolution result. The reduction method used to-date was the modulo function since it preserves location (of the "1" bits) up to cyclic shift. To date there is no known efficient algorithm for computing the sparse Walsh transform. Since the modulo function does not preserve the Walsh transform a new method for length reduction is needed. In this paper we present such a new method - polynomials. This method enables the development of an efficient algorithm for computing the binary sparse Walsh transform. To our knowledge, this is the first such algorithm. We also show that this method allows a faster deterministic computation of sparse polynomial multiplication than currently known in the literature.