Prime Numbers

482 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
}
}
Y = Y + J;
m = m + J;
}
J =2∗ J;
X = X − D/2;
Y = Y − D;
(X, Y )=(Y,X); // Swap pointers!
3. [Make ping-pong parity decision]
if(d even) return (complex data at X);
return (complex data at Y );
The useful loop aspect of this algorithm is the fact that the loop variable j
runs contiguously (from J down), and so on a vector machine one may process
chunks of data all at once, by picking up, then putting back, data as vectors.
Incidentally, to perform an inverse FFT is extremely simple, once the
forward FFT is implemented. One approach is simply to look at Definition
9.5.3 and observe that the root g can be replaced by g −1 ,withafinaloverall
normalization 1/D applied to an inverse FFT. But when complex fields are
used, so that g −1 = g ∗ , the procedure for FFT −1 can be, if one desires, just
a sequence:
x = x ∗ ; // Conjugate the signal.
X = FFT(x); // The usual FFT, with usual root g.
X = X ∗ /D; // Final conjugate and normalize.
Though these Cooley–Tukey and Gentleman–Sande FFTs are most often
invoked over the complex numbers, so that the root is g = e 2πi/D ,say,they
are useful also as number-theoretical transforms, with operations carried out
in a finite ring or field. In either the complex or finite-field cases, it is common
that a signal to be transformed has all real elements, in which case we call
the signal “pure-real.” This would be so for complex signal x ∈ C D but such
that xj = aj +0i for each j ∈ [0,D− 1]. It is important to observe that the
analogous signal class can occur in certain fields, for example F p 2 when p ≡ 3
(mod 4). For in such fields, every element can be represented as xj = aj + bji,
and we can say that a signal x ∈ F D p 2 is pure-real if and only if every bj is zero.
The point of the pure-real signal class designation is that in general, an FFT
for pure-real signals has about 1/2 the usual complexity. This makes sense
from an information-theoretical standpoint: Indeed, there is “half as much”
data in a pure-real signal. A basic way to cut down thus the FFT complexity
for pure-real signals is to embed half of a pure-real signal in the imaginary
parts of a signal of half the length, i.e., to form a complex signal
yj = xj + ix j+D/2
for j ∈ [0,D/2 − 1]. Note that signal y now has length D/2. One then
performs a full, half-length, complex FFT and then uses some reconstruction