Prime Numbers

502 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
There is one more important aspect of the DGT convolution: All mod
operations are with respect to Mersenne primes, and so an implementation
can enjoy the considerable speed advantage we have previously encountered
for such special cases of the modulus.
9.5.6 Schönhage method
The pioneering work in [Schönhage and Strassen 1971], [Schönhage 1982],
based on Strassen’s ideas for using FFTs in large-integer multiplication,
focuses on the fact that a certain number-theoretical transform is possible—
using exclusively integer arithmetic—in the ring Z2 m +1. Thisissometimes
called a Fermat number transform (FNT) (see Exercise 9.52) and can be used
within a certain negacyclic convolution approach as follows (we are grateful
to P. Zimmermann for providing a clear exposition of the method, from which
description we adapted our rendition here):
Algorithm 9.5.23. [Fast multiplication (mod 2 n +1) (Schönhage)] Given
two integers 0 ≤ x, y < 2 n +1, this algorithm returns the product xy mod (2 n +
1).
1. [Initialize]
Choose FFT size D =2 k dividing n;
Writing n = DM, set a recursion length n ′ ≥ 2M + k such that D divides
n ′ , i.e., n ′ = DM ′ ;
2. [Decomposition]
Split x and y into D parts of M bits each, and store these parts, considered
as residues modulo (2n′ +1), in two respective arrays A0,...,AD−1 and
B0,...,BD−1, taking care that an array element could in principle have
n ′ +1bits later on;
3. [Prepare DWT by weighting the A, B signals]
for(0 ≤ j