Prime Factor Algorithm (PFA)

By the prime factorization theorem, every integer $N$ can be uniquely factored into a product of prime numbers $p_i$ raised to an integer power $m_i\ge 1$:

$$N = \prod_{i=1}^{n_p} p_i^{m_i}$$

As discussed above, a mixed-radix Cooley Tukey FFT can be used to implement a length $N$ DFT using DFTs of length $p_i$. However, for factors of $N$ that are mutually prime (such as $p_i^{m_i}$ and $p_j^{m_j}$ for $i\ne j$), a more efficient prime factor algorithm (PFA), also called the Good-Thomas FFT algorithm, can be used [24,78,81].^A.5The Chinese Remainder Theorem is used to re-index either the input or output samples for the PFA.^A.6Since the PFA is only applicable to mutually prime factors of $N$, it is ideally combined with a mixed-radix Cooley-Tukey FFT, which works for any integer factors.

It is interesting to note that the PFA actually predates the Cooley-Tukey FFT paper of 1965 [14], with Good's 1958 work on the PFA being cited in that paper [81].