Editing Hamming weight (section)

== History and usage ==
The Hamming weight is named after the American mathematician [[Richard Hamming]], although he did not originate the notion.<ref name="Thompson_1983"/> The Hamming weight of binary numbers was already used in 1899 by [[James Whitbread Lee Glaisher|James W. L. Glaisher]] to give a formula for [[Gould's sequence|the number of odd binomial coefficients]] in a single row of [[Pascal's triangle]].<ref name="Glaisher_1899"/> [[Irving S. Reed]] introduced a concept, equivalent to Hamming weight in the binary case, in 1954.<ref name="Reed_1954"/>

Hamming weight is used in several disciplines including [[information theory]], [[coding theory]], and [[cryptography]]. Examples of applications of the Hamming weight include:
* In modular [[exponentiation by squaring]], the number of modular multiplications required for an exponent ''e'' is log<sub>2</sub> ''e'' + weight(''e''). This is the reason that the public key value ''e'' used in [[RSA (algorithm)|RSA]] is typically chosen to be a number of low Hamming weight.<ref name="CLNZ"/>
* The Hamming weight determines path lengths between nodes in [[Chord (distributed hash table)|Chord distributed hash tables]].<ref name="Stoica_2003"/>
* [[IrisCode]] lookups in biometric databases are typically implemented by calculating the [[Hamming distance]] to each stored record.<ref>{{cite journal
 | last1 = Kong | first1 = A.W.K.
 | last2 = Zhang | first2 = D.
 | last3 = Kamel | first3 = M.S.
 | date = February 2010
 | doi = 10.1109/tip.2009.2033427
 | issue = 2
 | journal = IEEE Transactions on Image Processing
 | pages = 522–532
 | title = An analysis of IrisCode
 | volume = 19}}</ref>
* In [[computer chess]] programs using a [[bitboard]] representation, the Hamming weight of a bitboard gives the number of pieces of a given type remaining in the game, or the number of squares of the board controlled by one player's pieces, and is therefore an important contributing term to the value of a position.<ref>{{cite journal
 | last = Heinz | first = E.A.
 | date = September 1997
 | doi = 10.3233/icg-1997-20304
 | issue = 3
 | journal = ICGA Journal
 | pages = 166–176
 | title = How Darkthought plays chess
 | volume = 20}} Updated and reprinted in ''Scalable Search in Computer Chess'' (Vieweg+Teubner Verlag, 2000), pp. 185–198, {{doi|10.1007/978-3-322-90178-1_13}}</ref>
* Hamming weight can be used to efficiently compute [[find first set]] using the identity ffs(x) = pop(x ^ (x - 1)). This is useful on platforms such as [[SPARC]] that have hardware Hamming weight instructions but no hardware find first set instruction.<ref name="SPARC_1992"/><ref name="Warren_2013"/>
* The Hamming weight operation can be interpreted as a conversion from the [[unary numeral system]] to [[binary number]]s.<ref name="Blaxell_1978"/>