Fast Integer Multiplication using Modular Arithmetic
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by
Anindya De, Piyush P Kurur, Chandan Saha, Ramprasad Saptharishi
2008
Abstract
We give an O(N· N· 2^O(^*N)) algorithm for multiplying
two N-bit integers that improves the O(N· N· N)
algorithm by Schönhage-Strassen. Both these algorithms use modular
arithmetic. Recently, Fürer gave an O(N· N· 2^O(^*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 Fürer's algorithm to achieve this
improvement in the modular setting. Our algorithm can also be viewed as a
p-adic version of Fürer's algorithm. Thus, we show that the two seemingly
different approaches to integer multiplication, modular and complex arithmetic,
are similar.
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