International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 5455

The set of congruence classes of integers modulo an integer N [see e.g. Apostol (1976), Chapter 5] inherits from not only the additive structure used in deriving the Cooley–Tukey factorization, but also a multiplicative structure in which the product of two congruence classes mod N is uniquely defined as the class of the ordinary product (in ) of representatives of each class. The multiplication can be distributed over addition in the usual way, endowing with the structure of a commutative ring.
If N is composite, the ring has zero divisors. For example, let , let mod N, and let mod N: then mod N. In the general case, a product of nonzero elements will be zero whenever these elements collect together all the factors of N. These circumstances give rise to a fundamental theorem in the theory of commutative rings, the Chinese Remainder Theorem (CRT), which will now be stated and proved [see Apostol (1976), Chapter 5; Schroeder (1986), Chapter 16].
Let be factored into a product of pairwise coprime integers, so that g.c.d. for . Then the system of congruence equationshas a unique solution mod N. In other words, each is associated in a onetoone fashion to the dtuple of its residue classes in .
The proof of the CRT goes as follows. LetSince g.c.d. there exist integers and such thatthen the integeris the solution. Indeed,because all terms with contain as a factor; andby the defining relation for .
It may be noted thatso that the are mutually orthogonal idempotents in the ring , with properties formally similar to those of mutually orthogonal projectors onto subspaces in linear algebra. The analogy is exact, since by virtue of the CRT the ring may be considered as the direct productvia the two mutually inverse mappings:
The mapping defined by (ii) is sometimes called the `CRT reconstruction' of from the .
These two mappings have the property of sending sums to sums and products to products, i.e:(the last proof requires using the properties of the idempotents ). This may be described formally by stating that the CRT establishes a ring isomorphism:
The CRT will now be used to factor the Npoint DFT into a tensor product of d transforms, the jth of length .
Let the indices k and be subjected to the following mappings:
ThenCross terms with vanish since they contain all the factors of N, henceDividing by N, which may be written as for each j, yieldsand henceTherefore, by the multiplicative property of ,
Let be described by a onedimensional array indexed by k. The index mapping (i) turns X into an element of described by a ddimensional array ; the latter may be transformed by into a new array . Finally, the onedimensional array of results will be obtained by reconstructing according to (ii).
The prime factor algorithm, like the Cooley–Tukey algorithm, reindexes a 1D transform to turn it into d separate transforms, but the use of coprime factors and CRT index mapping leads to the further gain that no twiddle factors need to be applied between the successive transforms (see Good, 1971). This makes up for the cost of the added complexity of the CRT index mapping.
The natural factorization of N for the prime factor algorithm is thus its factorization into prime powers: is then the tensor product of separate transforms (one for each prime power factor ) whose results can be reassembled without twiddle factors. The separate factors within each must then be dealt with by another algorithm (e.g. Cooley–Tukey, which does require twiddle factors). Thus, the DFT on a prime number of points remains undecomposable.
References
Apostol, T. M. (1976). Introduction to Analytic Number Theory. New York: SpringerVerlag.Good, I. J. (1971). The relationship between two fast Fourier transforms. IEEE Trans. C20, 310–317.
Schroeder, M. R. (1986). Number Theory in Science and Communication, 2nd ed. Berlin: SpringerVerlag.