Faster polynomial multiplication via discrete fourier transforms

Abstract

We study the complexity of polynomial multiplication over arbitrary fields. We present a unified approach that generalizes all known asymptotically fastest algorithms for this problem and obtain faster algorithms for polynomial multiplication over certain fields which do not support DFTs of large smooth orders. We prove that the famous Schönhage-Strassen's upper bound cannot be improved over the field of rational numbers if we consider only algorithms based on consecutive applications of DFT, as all known fastest algorithms are. This work is inspired by the recent improvement for the closely related problem of complexity of integer multiplication by Fürer and its consequent modular arithmetic treatment due to De, Kurur et al. We explore the barriers in transferring the techniques for solutions of one problem to a solution of the other. © 2011 Springer-Verlag Berlin Heidelberg.