Prime Numbers

498 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
3. [Dyadic product]
Z = X ∗ Y ;
4. [Inverse transform]
z = DWT −1 (Z, a);
5. [Round digits]
z = round(z); // Elementwise rounding to nearest integer.
6. [Adjust carry in variable base]
carry =0;
for(0 ≤ n 0 as a high digit of z;
z = z mod p; // Via carry loop or special-form mod.
return z;
As this scheme is somewhat intricate, an example is appropriate. Consider
multiplication modulo the Mersenne number p =2 521 − 1. We take q = 521
and choose signal length D = 16. Then the signal d of Theorem 9.5.18 can be
seen to be
d = (33, 33, 32, 33, 32, 33, 32, 33, 33, 32, 33, 32, 33, 32, 33, 32),
and the weight signal will be
a = 1, 2 7/16 , 2 7/8 , 2 5/16 , 2 3/4 , 2 3/16 , 2 5/8 , 2 1/16 , 2 1/2 , 2 15/16 ,
2 3/8 , 2 13/16 , 2 1/4 , 2 11/16 , 2 1/8 , 2 9/16 .
In a typical floating-point FFT implementation, this a signal is, of course,
given inexact elements. But in Theorem 9.5.18 the weighted convolution (as
calculated approximately, just prior to the [Round digits] step of Algorithm
9.5.19) consists of exact integers. Thus, the game to be played is to choose
signal length D to be as small as possible (the smaller, the faster the FFTs
that do the DWT), while not allowing the rounding errors to give incorrect
elements of z. Rigorous theorems on rounding error are hard to come by,
although there are some observations—some rigorous and some not so—in
[Crandall and Fagin 1994] and references therein. More modern treatments
include the very useful book [Higham 1996] and the paper [Percival 2003] on
generalized IBDWT; see Exercise 9.48.
9.5.5 Number-theoretical transform methods
The DFT of Definition 9.5.3 can be defined over rings and fields other than the
traditional complex field. Here we give some examples of transforms over finite