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March 2021 Integer multiplication in time $O(n\mathrm{log}\, n)$
David Harvey, Joris van der Hoeven
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Ann. of Math. (2) 193(2): 563-617 (March 2021). DOI: 10.4007/annals.2021.193.2.4

Abstract

We present an algorithm that computes the product of two $n$-bit integers in $O(n \mathrm{log}\, n)$ bit operations, thus confirming a conjecture of Schönhage 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.

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David Harvey. Joris van der Hoeven. "Integer multiplication in time $O(n\mathrm{log}\, n)$." Ann. of Math. (2) 193 (2) 563 - 617, March 2021. https://doi.org/10.4007/annals.2021.193.2.4

Information

Published: March 2021
First available in Project Euclid: 23 December 2021

Digital Object Identifier: 10.4007/annals.2021.193.2.4

Subjects:
Primary: 11Y16 , 68W30

Keywords: Complexity , FFT , integer multiplication

Rights: Copyright © 2021 David Harvey and Joris van der Hoeven

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Vol.193 • No. 2 • March 2021
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