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### Paper:

TR08-023 | 10th January 2008 00:00

#### Fast Integer Multiplication using Modular Arithmetic

TR08-023
Authors: Anindya De, Piyush Kurur, Chandan Saha, Ramprasad Saptharishi
Publication: 11th March 2008 07:16
Keywords:

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.

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