Editing Smooth number (section)

== Applications ==
An important practical application of smooth numbers is the [[fast Fourier transform]] (FFT) algorithms (such as the [[Cooley–Tukey FFT algorithm]]), which operates 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 number]]s play a special role in [[Babylonian mathematics]].<ref>
{{citation
 | last = Aaboe | first = Asger | author-link = Aaboe
 | title = Some Seleucid mathematical tables (extended reciprocals and squares of regular numbers)
 | journal = Journal of Cuneiform Studies
 | volume = 19
 | issue = 3
 | pages = 79–86
 | year = 1965
 | mr=0191779
 | doi = 10.2307/1359089| jstor = 1359089 | s2cid = 164195082
}}.</ref> They are also important in [[music theory]] (see [[Limit (music)]]),<ref>
{{citation
 | last = Longuet-Higgins | first = H. C.
 | year = 1962
 | title = Letter to a musical friend
 | journal = Music Review
 | issue = August
 | pages = 244–248
}}.</ref> and the problem of generating these numbers efficiently has been used as a test problem for [[functional programming]].<ref>
{{citation
 | author-link = Edsger W. Dijkstra | last = Dijkstra | first = Edsger W.
 | title = Hamming's exercise in SASL
 | year = 1981
 | url = http://www.cs.utexas.edu/users/EWD/ewd07xx/EWD792.PDF
 | id = Report EWD792. Originally a privately circulated handwritten note
}}.</ref>

Smooth numbers have a number of applications to cryptography.<ref>{{cite journal|first1=David|last1=Naccache|first2=Igor |last2=Shparlinski|title=Divisibility, Smoothness and Cryptographic Applications|url=http://eprint.iacr.org/2008/437.pdf|date=17 October 2008|website=eprint.iacr.org|arxiv=0810.2067 |access-date=26 July 2017}}f</ref> While most applications center around [[cryptanalysis]] (e.g. the fastest known [[integer factorization]] algorithms, for example: the [[general number field sieve]]), the [[Very smooth hash|VSH]] hash function is another example of a constructive use of smoothness to obtain a [[Provably secure cryptographic hash function|provably secure design]].