Fast Integer Multiplication using Modular Arithmetic

TL;DR

This paper presents a faster algorithm for multiplying large integers using modular arithmetic, improving previous methods and bridging the gap between modular and complex arithmetic approaches.

Contribution

It introduces an $O(N imes \log N imes 2^{O(\log^* N)})$ algorithm that enhances integer multiplication efficiency by combining multivariate polynomial multiplication with ideas from F"urer's complex arithmetic algorithm.

Findings

Achieves faster integer multiplication complexity

Bridges modular and complex arithmetic approaches

Demonstrates similarity between different algorithmic strategies

Abstract

We give an $O(N\cdot \log N\cdot 2^{O(\log^*N)})$ algorithm for multiplying two $N$-bit integers that improves the $O(N\cdot \log N\cdot \log\log N)$ algorithm by Sch\"{o}nhage-Strassen. Both these algorithms use modular arithmetic. Recently, F\"{u}rer gave an $O(N\cdot \log N\cdot 2^{O(\log^*N)})$ algorithm which however uses arithmetic over complex numbers as opposed to modular arithmetic. In this paper, we use multivariate polynomial multiplication along with ideas from F\"{u}rer's algorithm to achieve this improvement in the modular setting. Our algorithm can also be viewed as a $p$-adic version of F\"{u}rer's algorithm. Thus, we show that the two seemingly different approaches to integer multiplication, modular and complex arithmetic, are similar.