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. 5960

Suppose that the decimation matrix N is diagonaland let each diagonal element be written in terms of its prime factors:where m is the total number of distinct prime factors present in the .
The CRT may be used to turn each 1D transform along dimension i into a multidimensional transform with a separate `pseudodimension' for each distinct prime factor of ; the number , of these pseudodimensions is equal to the cardinality of the set:The full ndimensional transform thus becomes μdimensional, with .
We may now permute the μ pseudodimensions so as to bring into contiguous position those corresponding to the same prime factor ; the m resulting groups of pseudodimensions are said to define `pprimary' blocks. The initial transform is now written as a tensor product of m pprimary transforms, where transform j is onpoints [by convention, dimension i is not transformed if ]. These pprimary transforms may be computed, for instance, by multidimensional Cooley–Tukey factorization (Section 1.3.3.3.1), which is faster than the straightforward row–column method. The final results may then be obtained by reversing all the permutations used.
The extra gain with respect to the multidimensional Cooley–Tukey method is that there are no twiddle factors between pprimary pieces corresponding to different primes p.
The case where N is not diagonal has been examined by Guessoum & Mersereau (1986).
References
Guessoum, A. & Mersereau, R. M. (1986). Fast algorithms for the multidimensional discrete Fourier transform. IEEE Trans. Acoust. Speech Signal Process. 34, 937–943.