Prime Numbers

9.5 Large-integer multiplication 501
is a way to find immediately an element of suitable order, thanks to the
following result of [Creutzburg and Tasche 1989]:
Theorem 9.5.21 (Creutzburg and Tasche). Let p =2 q − 1 be a Mersenne
prime with q odd. Then
is an element of order 2 q+1 in F ∗ p 2.
g =2 2q−2
+ i(−3) 2q−2
These observations lead to the following integer convolution algorithm, in
which we indicate the enhancements that can be invoked to reduce the
complex arithmetic. In particular, we exploit the fact that integer signals
are real, so the imaginary components of their elements vanish in the field,
and thus the transform lengths are halved:
Algorithm 9.5.22 (Convolution via DGT (Crandall)). Given two signals
x, y each of length N =2 k ≥ 2 and whose elements are integers in the interval
[0,M], this algorithm returns the integer convolution x × y. The method used is
convolution via “discrete Galois transform” (DGT).
1. [Initialize]
Choose a Mersenne prime p =2q − 1 such that p>NM2 and q>k;
Use Theorem 9.5.21 to find an element g of order 2q+1 ;
h = g2q+2−k; // h is now an element of order N/2.
2. [Fold signals to halve their lengths]
x =
x2j + ix2j+1 , j =0,...,N/2− 1;
y =
y2j + iy2j+1 , j =0,...,N/2− 1;
3. [Length-N/2 transforms]
X = DFT(x); // Via, say, split-radix FFT (mod p), rooth.
Y = DFT(y);
4. [Special dyadic product]
for(0 ≤ k