Prime Numbers

500 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
for(1 ≤ r ≤ q) {
h = hr; p = pr; d = ar; // Preparing for DFTs.
X (r) = DFT(x); // Via relation (9.33).
Y (r) = DFT(y);
3. [Dyadic product]
Z (r) = X (r) ∗ Y (r) ;
4. [Inverse transforms]
z (r) = DWT −1 (Z (r) ); // Via relation (9.34).
}
5. [Reconstruct elements]
From the now known relations zj ≡ z (r)
j (mod pr) find each (unambiguous)
element zj in [0,NM2 ) via CRT reconstruction, using such as Algorithm
2.1.7 or 9.5.26;
return z;
What this algorithm does is allow us to invoke length-2 m FFTs for the DFT
and its inverse, except that only integer arithmetic is to be used in the usual
FFT butterflies (and of course the butterflies are continually reduced (mod pr)
during the FFT calculations). This scheme has been used to good effect in
[Montgomery 1992a] in various factorization implementations. Note that if
the forward DFT (9.33) is performed with a decimation-in-frequency (DIF)
algorithm, and the reverse DFT (9.34) with a DIT algorithm, there is no
need to invoke the scramble function of Algorithm 9.5.5 in either of the FFT
functions shown there.
A second example of useful number-theoretical transforms has been called
the discrete Galois transform (DGT) [Crandall 1996a], with relevant field F p 2
for p =2 q − 1 a Mersenne prime. The delightful fact about such fields is that
the multiplicative group order is
|F ∗ p 2| = p2 − 1=2 q+1 (2 q−1 − 1),
so that in practice, one can find primitive roots of unity of orders N =2 k as
long as k ≤ q + 1. We can thus define discrete transforms of such lengths, as
Xk =
N−1
j=0
xjh −jk mod p, (9.35)
where now all arithmetic is presumed, due to the known structure of F p 2 for
primes p ≡ 3 (mod 4), to involve complex (Gaussian) integers (mod p) with
N =2 k ,
xj =Re(xj)+i Im(xj),
h =Re(h)+i Im(h),
the latter being an element of multiplicative order N in F p 2, with the
transform element Xk itself being a Gaussian integer (mod p). Happily, there