Integer multiplication in time O(n log n)

• 成果类型: Article
• 署名作者: Harvey, David; van der Hoeven, Joris
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2021.193.2.4
• 发表日期: 2021
• 页码: 563-617
• 关键词: discrete fourier-transforms; least prime; computation; algorithms

摘要

We present an algorithm that computes the product of two n-bit integers in O (n log n) bit operations, thus confirming a conjecture of Schonhage and Strassen from 1971. Our complexity analysis takes place in the multitape Turing machine model, with integers encoded in the usual binary representation. Central to the new algorithm is a novel Gaussian resampling technique that enables us to reduce the integer multiplication problem to a collection of multidimensional discrete Fourier transforms over the complex numbers, whose dimensions are all powers of two. These transforms may then be evaluated rapidly by means of Nussbaumer's fast polynomial transforms.