Prime Numbers

490 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
whichistosay
(x × y)n = 1
D−1
D
k=0
XkYk g kn .
Armed with this mighty theorem we can effect the cyclic convolution of
two signals with three transforms (one of them being the inverse transform),
or the cyclic convolution of a signal with itself with two transforms. As the
known complexity of the DFT is O(D ln D) operations in the field, the dyadic
product implicit in Theorem 9.5.11, being O(D), is asymptotically negligible.
A direct and elegant application of FFTs for large-integer arithmetic is
to perform the DFTs of Theorem 9.5.11 in order to effect multiplication via
acyclic convolution. This essential idea—pioneered by Strassen in the 1960s
and later optimized in [Schönhage and Strassen 1971] (see Section 9.5.6)—runs
asfollows.Ifanintegerx is represented as a length-D signal consisting of the
(base-B) digits, and the same for y, then the integer product xy is an acyclic
convolution of length 2D. Though Theorem 9.5.11 pertains to the cyclic, not
the acyclic, we nevertheless have Theorem 9.5.10, which allows us to use zeropadded
signals and then perform the cyclic. This idea leads, in the case of
complex field transforms, to the following scheme, which is normally applied
using floating-point arithmetic, with DFT’s done via fast Fourier transform
(FFT) techniques:
Algorithm 9.5.12 (Basic FFT multiplication). Given two nonnegative integers
x, y, each having at most D digits in some base B (Definition 9.1.1), this
algorithm returns the base-B representation of the product xy. (The FFTs normally
employed herein would be of the floating-point variety, so one must beware
of roundoff error.)
1. [Initialize]
Zero-pad both of x, y until each has length 2D, so that the cyclic
convolution of the padded sequences will contain the acyclic convolution
of the unpadded ones;
2. [Apply transforms]
X = DFT(x); // Perform transforms via efficient FFT algorithm.
Y = DFT(y);
3. [Dyadic product]
Z = X ∗ Y ;
4. [Inverse transform]
z = DFT −1 (Z);
5. [Round digits]
z = round(z); // Elementwise rounding to nearest integer.
6. [Adjust carry in base B]
carry =0;
for(0 ≤ n