Prime Numbers

9.5 Large-integer multiplication 489
and the acyclic convolution of x, y is a signal u = x ×A y having 2D elements
given by
un =
xiyj,
i+j=n
for n ∈{0,...,2D − 2}, together with the assignment u2D−1 = 0. Finally, the
half-cyclic convolution of x, y is the length-D signal x ×H y consisting of the
first D elements of the acyclic convolution u.
These fundamental convolutions are closely related, as is seen in the following
result. In such statements we interpret the sum of two signals c = a + b in
elementwise fashion; that is, cn = an + bn for relevant indices n. Likewise,
a scalar-signal product qa, withq a number and a a signal, is the signal
(qan). We shall require the notion of the splitting of signals (of even length)
into halves, so we denote by L(a),H(a), respectively, the lower-indexed and
higher-indexed halves of a. That is, from c = a ∪ b the natural, left-right,
concatenation of two signals of equal length, we shall have L(c) = a and
H(c) =b.
Theorem 9.5.10. Let signals x, y have the same length D. Then the various
convolutions are related as follows (it is assumed that in the relevant domain
to which signal elements belong, 2 −1 exists):
Furthermore,
x ×H y = 1
2 ((x × y)+(x ×− y)).
x ×A y =(x ×H y) ∪ 1
2 ((x × y) − (x ×− y)).
Finally, if the length D is even and xj,yj =0for j ≥ D/2, then
L(x) ×A L(y) =x × y = x ×− y.
These interrelations allow us to use certain algorithms more universally.
For example, a pair of algorithms for cyclic and negacyclic can be used to
extract both the half-cyclic or the acyclic, and so on. In the final statement
of the theorem, we have introduced the notion of “zero padding,” which in
practice amounts to appending D zeros to signals already of length D, sothat
the signals’ acyclic convolution is identical to the cyclic (or the negacyclic)
convolution of the two padded sequences.
The connection between convolution and the DFT of the previous section
is evident in the following celebrated theorem, wherein we refer to the dyadic
operator ∗, under which a signal z = x ∗ y has elements zn = xnyn:
Theorem 9.5.11 (Convolution theorem). Let signals x, y have the same
length D. Then the cyclic convolution of x, y satisfies
x × y = DFT −1 (DFT(x) ∗ DFT(y)),