Fast integer multiplication using modular arithmetic

Abstract

We give an O(N ·log N ·2 O(log * N ) ) algorithm for multiplying two N -bit integers that improves the O(N · log N · log log N ) algorithm by . Both these algorithms use modular arithmetic. Recently, Fürer [Für07] gave an O(N · log N · 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 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.

Fast fourier transforms over poor fields

“…A different approach of the same complexity for the multiplication of integers was proposed in [11]. The latter algorithm is also based on the DFT.…”

“…At this point no factorization-based FFT algorithm can be applied to compute 2 m DFTs of prime order p, and the naive quadratic algorithm (5) for the DFTs of order p together with (11) gives…”

“…Now we generalize the example of Section 2.1. Let m and p be as defined in (11). Let n1 := 2 m p, and p1 be a prime such that 2p1 ≤ n1 < 4p1 (by Chebyshev's theorem p1 always exists, see, e.g., [3,).…”

“…The most popular FFT is the Cooley-Tukey algorithm [10], whose idea traces back to Gauss. It is used as an important building block for the fast integer and polynomial multiplication algorithms [24,23,6,12,11]. However, there exist certain limitations which make the Cooley-Tukey FFT algorithm in its original form not suitable for the improvement of the currently known upper bounds for the mentioned problems [20].…”

Implementation of the DKSS Algorithm for Multiplication of Large Numbers

“…Fürer's algorithm inspired De, Kurur, Saha and Saptharishi to their multiplication method [6] (see [7] for an expanded text), here called DKSS multiplication (or DKSSA for short). Both Fürer's and DKSS' algorithms require…”