Applications
An important practical application of smooth numbers is for fast Fourier transform (FFT) algorithms such as the Cooley–Tukey FFT algorithm that operate by recursively breaking down a problem of a given size n into problems the size of its factors. By using B-smooth numbers, one ensures that the base cases of this recursion are small primes, for which efficient algorithms exist. (Large prime sizes require less-efficient algorithms such as Bluestein's FFT algorithm.)
5-smooth or regular numbers play a special role in Babylonian mathematics. They are also important in music theory, (see Limit (music)) and the problem of generating these numbers efficiently has been used as a test problem for functional programming.
Smooth numbers have a number of applications to cryptography. Although most applications involve cryptanalysis (e.g. the fastest known integer factorization algorithms), the VSH hash function is one example of a constructive use of smoothness to obtain a provably secure design.
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