The number theoretic transform is based on generalizing the th primitive root of unity (see §3.12) to a ``quotient ring'' instead of the usual field of complex numbers. Let denote a primitive th root of unity. We have been using in the field of complex numbers, and it of course satisfies , making it a root of unity; it also has the property that visits all of the ``DFT frequency points'' on the unit circle in the plane, as goes from 0 to .
In a number theory transform, is an integer which satisfies
When the number of elements in the transform is composite, a ``fast number theoretic transform'' may be constructed in the same manner as a fast Fourier transform is constructed from the DFT, or as the prime factor algorithm (or Winograd transform) is constructed for products of small mutually prime factors .
Unlike the DFT, the number theoretic transform does not transform to a meaningful ``frequency domain''. However, it has analogous theorems, such as the convolution theorem, enabling it to be used for fast convolutions and correlations like the various FFT algorithms.
An interesting feature of the number theory transform is that all computations are exact (integer multiplication and addition modulo a prime integer). There is no round-off error. This feature has been used to do fast convolutions to multiply extremely large numbers, such as are needed when computing to millions of digits of precision.
Existence of the Fourier Transform
The Discrete Cosine Transform (DCT)