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The bit-reversal permutation of $\{0,\dots ,2^m-1\}$ writes each index using exactly $m$ binary digits, reverses those digits, and interprets the result as another index. Leading zeros matter. Reversing twice restores the original index, so this permutation is an involution.

For $m=3$, the index order becomes $0,4,2,6,1,5,3,7$: for example, $3=011_2$ maps to $110_2=6$. It occurs naturally in radix-two Fast Fourier transform implementations. Repeatedly separating even from odd indices consumes the low-order bits first; traversing the resulting recursion tree reverses the order in which the index bits are read. Some implementations permute inputs, others permute outputs, and other layouts avoid an explicit reversal. The discrete Fourier transform itself is independent of this storage choice.^[ref]

C. Sidney Burrus authored notes on the Fast Fourier transform. The treatment includes the prime-factor algorithm, which exploits coprime transform lengths to reorganize a discrete Fourier transform. A suitable index map separates the calculation into smaller transforms without the interstage twiddle multiplications required by the usual Cooley-Tukey arrangement.^[ref]

For even length $N=2m$, split the input of a discrete Fourier transform into its even and odd samples, $x_{2r}$ and $x_{2r+1}$. Let $E_k,O_k$ be their length-$m$ discrete Fourier transforms, with negative-exponent convention. Splitting the original sum by the parity of its index gives, for $0\le k<m$,The sign change follows from ${\omega }_N^m=-1$. Thus the same two short transforms supply both halves of the result through an FFT butterfly. For $N=2^p$, recursive splitting reaches length one after $p$ levels. Each level needs $O(N)$ arithmetic operations, so the time complexity is $O(N\log N)$. For example, a length-four transform uses the two length-two transforms of $(x_0,x_2)$ and $(x_1,x_3)$, then twiddle factors $1$ and $-i$.^[ref]

The Good-Thomas algorithm computes a discrete Fourier transform of size $N=ab$ when $a,b$ are coprime integers. The Chinese remainder theorem reindexes samples and frequencies by pairs of residues. With compatible input and output permutations, the transform kernel factors into an $a$-point kernel and a $b$-point kernel, so the computation becomes independent transforms along two array directions.

For $N=15$, one uses lengths three and five. Unlike the usual Cooley-Tukey FFT algorithm factorization, the coprime reindexing eliminates the intervening twiddle-factor multiplications. The required permutations still have computational cost, and a factorization such as $16=4\cdot 4$ does not satisfy the coprimality hypothesis.^[ref]