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In [[number theory]], the '''Berlekamp's root finding algorithm''', also called the '''Berlekamp–Rabin algorithm''', is the [[Randomized algorithm|probabilistic]] method of [[Root-finding algorithm|finding roots]] of [[Polynomial|polynomials]] over a [[Finite field|field]] <math>\mathbb Z_p</math>. The method was discovered by [[Elwyn Berlekamp]] in 1970<ref name=":0">{{cite journal |author = |editor= |format= |url= https://www.ams.org/mcom/1970-24-111/S0025-5718-1970-0276200-X/ |title= Factoring polynomials over large finite fields |type= |origyear= | agency = |edition= Mathematics of Computation |___location= |date= 1970 |year= 1970 |publisher= |at= |volume= 24 |issue= 111 |number= |pages = 713–735 |page= |series= |isbn = |issn = 00255718 |doi = 10.1090/S0025-5718-1970-0276200-X |bibcode = |arxiv = |pmid = |ref= |archiveurl = |archivedate = |language= en |quote= }}</ref> as an auxiliary to the [[Berlekamp's algorithm|algorithm]] for polynomial factorization over finite fields. The algorithm was later modified by [[Michael O. Rabin|Rabin]] for arbitrary finite fields in 1979.<ref name=":1">{{cite journal |author = M. Rabin |editor= |format= |url= https://epubs.siam.org/doi/10.1137/0209024 |title= Probabilistic Algorithms in Finite Fields |type= |origyear= | agency = |edition= SIAM Journal on Computing |___location= |date= 1980 |year= 1980 |publisher= |at= |volume= 9 |issue= 2 |number= |pages = 273–280 |page= |series= |isbn = |issn = 00975397 |doi = 10.1137/0209024 |bibcode = |arxiv = |pmid = |ref= |archiveurl = |archivedate = |language= |quote= }}</ref> The method was also independently discovered before Berlekamp by other researchers.<ref>{{cite book| author = Donald E Knuth | chapter = | chapter-url = | format = | url = https://www.worldcat.org/title/art-of-computer-programming-vol-2/oclc/900627019&referer=brief_results | title = The art of computer programming. Vol. 2 Vol. 2 | orig-year = | agency = | edition = |___location= |date = 1998 |publisher= |at= |volume= |issue = | pages = | page = | series = | isbn = 9780201896848| ref = }}</ref>
== History ==
The method was proposed by [[Elwyn Berlekamp]] in his work<ref name=":0" /> on polynomial factorization over finite fields. His original work lacked a formal correctness proof<ref name=":1" /> and was later refined and modified for arbitrary finite fields by [[Michael O. Rabin|Michael Rabin]].<ref name=":1" /> In 1986 René Peralta proposed a similar algorithm<ref>{{cite journal |author = Tsz-Wo Sze |editor= |format= |url= http://dx.doi.org/10.1090/s0025-5718-2011-02419-1 |title= On taking square roots without quadratic nonresidues over finite fields |type= |origyear= | agency = |edition= Mathematics of Computation |___location= |date= 2011 |year= 2011 |publisher= |at= |volume= 80 |issue= 275 |number= |pages = 1797–1811 |page= |series= |isbn = |issn = 00255718 |doi = 10.1090/s0025-5718-2011-02419-1 |bibcode = |arxiv = |pmid = |ref= |archiveurl = |archivedate = |language= |quote= }}</ref> for finding square roots in <math>\mathbb Z_p</math><ref>{{cite journal |author = R. Peralta |editor= |format= |url= https://ieeexplore.ieee.org/document/1057236 |title= A simple and fast probabilistic algorithm for computing square roots modulo a prime number (Corresp.) |type= |origyear= | agency = |edition= IEEE Transactions on Information Theory |___location= |date= 1986 |year=November 1986 |publisher= |at= |volume= 32 |issue= 6 |number= |pages = 846–847 |page= |series= |isbn = |issn = 00189448 |doi = 10.1109/TIT.1986.1057236 |bibcode = |arxiv = |pmid = |ref= |archiveurl = |archivedate = |language= |quote= }}</ref>. In 2000 Peralta's method was generalized for cubic equations.<ref>{{cite journal |author = C Padró, G Sáez |editor= |format= |url= http://dx.doi.org/10.1016/s0893-9659(02)00031-9 |title= Taking cube roots in Zm |type= |origyear= | agency = |edition= Applied Mathematics Letters |___location= |date= 2002 |year=August 2002 |publisher= |at= |volume= 15 |issue= 6 |number= |pages = 703–708 |page= |series= |isbn = |issn = 08939659 |doi = 10.1016/s0893-9659(02)00031-9 |bibcode = |arxiv = |pmid = |ref= |archiveurl = |archivedate = |language= |quote= }}</ref>
== Statement ==
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