Abstract
We give an O(N • log N • 2O(log*N)) algorithm for multiplying two N-bit integers that improves the O(N • log N • log log N) algorithm by Schonhage-Strassen. Both these algorithms use modular arithmetic. Recently, Furer gave an O(N • log N • 2O(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 Furer's algorithm to achieve this improvement in the modular setting. Our algorithm can also be viewed as a p-adic version of Furer's algorithm. Thus, we show that the two seemingly different approaches to integer multiplication, modular and complex arithmetic, are similar.
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