Berlekamp–Rabin algorithm: Difference between revisions

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{{short description|Method in number theory}}

In [[number theory]], '''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 |journal= Mathematics of Computation|type= |origyear= | agency = |edition= Mathematics of Computation |___location= |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= |last1= Berlekamp|first1= E. R.}}</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 |journal= Siam Journal on Computing|type= |origyear= | agency = |edition= SIAM Journal on Computing |___location= |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 | authorlink = Donald E Knuth | chapter = | chapter-url = | 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 |date = 1998 |publisher= | isbn = 978-0201896848| oclc = 900627019 }}</ref>

== History ==

The method was proposed by [[Elwyn Berlekamp]] in his 1970 work<ref name=":0" /> on polynomial factorization over finite fields. His original work lacked a formal [[Correctness (computer science)|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= https://doi.org/10.1090/s0025-5718-2011-02419-1 |title= On taking square roots without quadratic nonresidues over finite fields |journal= Mathematics of Computation|type= |origyear= | agency = |edition= Mathematics of Computation |___location= |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 =0812.2591 |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.) |journal= IEEE Transactions on Information Theory|type= |origyear= | agency = |edition= IEEE Transactions on Information Theory |___location= |date=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= https://doi.org/10.1016/s0893-9659(02)00031-9 |title= Taking cube roots in Zm |journal= Applied Mathematics Letters|type= |origyear= | agency = |edition= Applied Mathematics Letters |___location= |date=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 of problem==

Let <math>p</math> be an odd prime number. Consider the polynomial <math display="inline">f(x) = a_0 + a_1 x + \cdots + a_n x^n</math> over the field <math>\mathbb Z_p</math> of remainders modulo <math>p</math>. The algorithm should find all <math>\lambda</math> in <math>\mathbb Z_p</math> such that <math display="inline">f(\lambda)= 0</math> in <math>\mathbb Z_p</math>.<ref name=":1" /><ref name=":2">{{cite book| author = Alfred J. Menezes, Ian F. Blake, XuHong Gao, Ronald C. Mullin, Scott A. Vanstone | chapter = | chapter-url = | format = | url = https://www.springer.com/gp/book/9780792392828 | title = Applications of Finite Fields | orig-year = | agency = | edition = |___location= |date = 1993 |publisher= Springer US |at= |volume= |issue = | pages = | page = | series = The Springer International Series in Engineering and Computer Science | isbn = 9780792392828| ref = }}</ref>

== Algorithm ==

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== Correctness proof ==

The algorithm finds factorization of <math>f_z(x)</math> in all cases except for ones when all numbers <math>z+\lambda_1, z+\lambda_2, \ldots, z+\lambda_n</math> are quadratic residues or non-residues simultaneously. According to [[theory of cyclotomy]],<ref>{{cite book| author = Marshall Hall | chapter = | chapter-url = | format = | url = https://books.google.com/books?hl=en&lr=&id=__JCiiCfu2EC&oi=fnd&pg=PA1&dq=Combinatorial+Theory+hall&ots=WeNDZ7uCSM&sig=a6JwSPPen2C2EysEnkSTXpUNaxM&redir_esc=y#v=onepage&q=Combinatorial%20Theory%20hall&f=false | title = Combinatorial Theory | orig-year = | agency = | edition = |___location= |date = 1998 |publisher= John Wiley & Sons |at= |volume= |issue = | pages = | page = | series = | isbn = 9780471315186| ref = }}</ref> the probability of such an event for the case when <math>\lambda_1, \ldots, \lambda_n</math> are all residues or non-residues simultaneously (that is, when <math>z=0</math> would fail) may be estimated as <math>2^{-k}</math> where <math>k</math> is the number of distinct values in <math>\lambda_1, \ldots, \lambda_n</math>.<ref name=":0" /> In this way even for the worst case of <math>k=1</math> and <math>f(x)=(x-\lambda)^n</math>, the probability of error may be estimated as <math>1/2</math> and for modular square root case error probability is at most <math>1/4</math>.

== Complexity ==

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# Taking the <math>\gcd</math> of two polynomials via [[Euclidean algorithm]] works in <math>O(n^2)</math>.

Thus the whole procedure may be done in <math>O(n^2 \log p)</math>. Using the [[fast Fourier transform]] and Half-GCD algorithm,<ref>{{cite book | author = Aho, Alfred V. | chapter = | chapter-url = | format = | url = https://archive.org/details/designanalysisof00ahoarich | title = The design and analysis of computer algorithms | orig-year = | agency = | edition = | ___location = | date = 1974 | publisher = Addison-Wesley Pub. Co | at = | volume = | issue = | pages = | page = | series = | isbn = 0201000296 | ref = | url-access = registration }}</ref> the algorithm's complexity may be improved to <math>O(n \log n \log pn)</math>. For the modular square root case, the degree is <math>n = 2</math>, thus the whole complexity of algorithm in such case is bounded by <math>O(\log p)</math> per iteration.<ref name=":2" />

== References ==