On Fast Computation of a Circulant Matrix-Vector Product

TL;DR

This paper presents a novel method for multiplying circulant matrices and polynomials faster than FFT-based methods, achieving significant speedups without using FFT, with applications to large integer multiplication.

Contribution

It introduces an O(n log n) algorithm for circulant matrix-vector multiplication that does not rely on FFT, providing a faster alternative for polynomial and integer multiplication.

Findings

Achieves about 2.25 times speedup over FFT-based polynomial multiplication.

Provides an O(n log n) algorithm for circulant matrix-vector multiplication without FFT.

Discusses applications to large integer multiplication.

Abstract

This paper deals with circulant matrices. It is shown that a circulant matrix can be multiplied by a vector in time O(n log(n)) in a ring with roots of unity without making use of an FFT algorithm. With our algorithm we achieve a speedup of a factor of about 2.25 for the multiplication of two polynomials with integer coefficients compared to multiplication by an FFT algorithm. Moreover this paper discusses multiplication of large integers as further application.