Prime Numbers

9.5 Large-integer multiplication 477
might be sequences of polynomial coefficients, or sequences in general, and
will be denoted by x =(xn), n ∈ [0,D− 1] for some “signal length” D.
The first essential notion is that multiplication is a kind of convolution.
We shall make that connection quite precise later, observing for the moment
that the DFT is a natural transform to employ in convolution problems. For
the DFT has the unique property of converting convolution to a less expensive
dyadic product. We start with a definition:
Definition 9.5.3 (The discrete Fourier transform (DFT)). Let x be a signal
of length D consisting of elements belonging to some algebraic domain
in which D −1 exists, and let g be a primitive D-th root of unity in that domain;
that is, g k = 1 if and only if k ≡ 0(modD). Then the discrete Fourier
transform of x is that signal X = DFT(x) whose elements are
Xk =
D−1
j=0
with the inverse DFT −1 (X) =x given by
xjg −jk , (9.20)
xj = 1
D−1
Xkg
D
jk . (9.21)
k=0
That the transform DFT −1 iswell-definedasthecorrectinverseisleftasan
exercise. There are several important manifestations of the DFT:
Complex-field DFT: x, X ∈ C D , g aprimitiveD-throotof1suchase 2πi/D ;
Finite-field DFT: x, X ∈ F D
p k, g aprimitiveD-throotof1inthesamefield;
Integer-ring DFT: x, X ∈ ZD N , g a primitive D-th root of 1 in the ring,
D−1 , g−1 exist.
It should be pointed out that the above are common examples, yet there are
many more possible scenarios. As just one extra example, one may define a
DFT over quadratic fields (see Exercise 9.50).
In the first instance of complex fields, the practical implementations
involve floating-point arithmetic to handle complex numbers (though when
the signal has only real elements, significant optimizations apply, as we shall
see). In the second, finite-field, cases one uses field arithmetic with all terms
reduced (mod p). The third instance, the ring-based DFT, is sometimes
applied simultaneously for N = 2 n − 1andN ′ = 2 n + 1, in which cases
the assignments g = 2 and D = n, D ′ =2n, respectively, can be made when
n is coprime to both N,N ′ .
It should be said that there exists a veritable menagerie of alternative
transforms, many of them depending on basis functions other than the
complex exponential basis functions of the traditional DFT; and often, such
alternatives admit of fast algorithms, or assume real signals, and so on.
Though such transforms lie beyond the scope of the present book, we observe