Prime Numbers

494 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
Furthermore, the weighted cyclic convolution of two signals is the signal
z = x ×a y having
zn = 1
an
j+k≡n (mod D)
(a ∗ x)j(a ∗ y)k. (9.31)
It is clear that the DWT is simply the DFT of the dyadic product signal a ∗ x
consisting of elements ajxj. The considerable advantage of the DWT is that
particular weight signals give rise to useful alternative convolutions. In some
cases, the DWT eliminates the need for the zero padding of the standard FFT
multiplication Algorithm 9.5.12. We first state an important result:
Theorem 9.5.16 (Weighted convolution theorem). Let signals x, y and
weight signal a have the same length D. Then the weighted cyclic convolution
of x, y satisfies
that is to say,
x ×a y = DWT −1 (DWT(x, a) ∗ DWT(y, a),a),
(x ×a y)n = 1
D−1
(X ∗ Y )kg
Dan
kn .
Thus FFT algorithms may be applied now to weighted convolution. In
particular, one may compute not just the cyclic, but also the negacyclic,
convolution in this manner, because the specific choice of weight signal
k=0
a = A j , j ∈ [0,D− 1]
yields, when A is a primitive 2D-th root of unity in the field, the identity:
x ×− y = x ×a y, (9.32)
which means that the weighted cyclic in this case is the negacyclic. Note that
when the D-th root g has a square root in the field, as is the case with the
complex field arithmetic, we can simply assign A2 = g to effect the negacyclic.
Another interesting instance of generator A,namelywhenAis a primitive 4Dth
root of unity, gives the so-called right-angle convolution [Crandall 1996a].
These observations lead in turn to an important algorithm that has been
used to advantage in modern factorization studies. By using the DWT, the
method obviates zero padding entirely. Consider the problem of multiplication
of two numbers, modulo a Fermat number Fn =22n + 1. This operation can
happen, of course, a great number of times in attempts to factor an Fn. There
are at least three ways to attempt (xy) modFnvia convolution of length-D
signals where D and a power-of-two base B are chosen such that Fn = BD +1:
(1) Zero-pad each of x, y up to length 2D, perform cyclic convolution, do carry
adjust as necessary, take the result (mod Fn).