{{short description|Method in number theory}} [[File:Elwyn_R_Berlekamp_2005.jpg|thumb|right|Elwyn R. Berlekamp at conference on Combinatorial Game Theory at Banff International Research Station]] In number theory, '''Berlekamp's root finding algorithm''', also called the '''Berlekamp–Rabin algorithm''', is the probabilistic method of finding roots of polynomials over the field <math>\mathbb F_p</math> with <math>p</math> elements. The method was discovered by Elwyn Berlekamp in 1970<ref name=":0">{{cite journal |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 |year= 1970 |volume= 24 |issue= 111 |pages = 713–735 |issn = 0025-5718 |doi = 10.1090/S0025-5718-1970-0276200-X |language= en |last1= Berlekamp|first1= E. R.|doi-access= free|url-access= subscription }}</ref> as an auxiliary to the algorithm for polynomial factorization over finite fields. The algorithm was later modified by Rabin for arbitrary finite fields in 1979.<ref name=":1">{{cite journal |author = M. Rabin |title= Probabilistic Algorithms in Finite Fields |journal= SIAM Journal on Computing |year= 1980 |volume= 9 |issue= 2 |pages = 273–280 |issn = 0097-5397 |doi = 10.1137/0209024 |citeseerx= 10.1.1.17.5653 }}</ref> The method was also independently discovered before Berlekamp by other researchers.<ref>{{cite book| author = Donald E Knuth | author-link = Donald E Knuth | title = The art of computer programming. Vol. 2 Vol. 2 |date = 1998 | publisher = Addison-Wesley | 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 proof<ref name=":1" /> and was later refined and modified for arbitrary finite fields by Michael Rabin.<ref name=":1" /> In 1986 René Peralta proposed a similar algorithm<ref>{{cite journal |author = Tsz-Wo Sze |title= On taking square roots without quadratic nonresidues over finite fields |journal= Mathematics of Computation|year= 2011 |volume= 80 |issue= 275 |pages = 1797–1811 |issn = 0025-5718 |doi = 10.1090/s0025-5718-2011-02419-1 |arxiv =0812.2591 |s2cid= 10249895 }}</ref> for finding square roots in <math>\mathbb F_p</math>.<ref>{{cite journal |author = R. Peralta |title= A simple and fast probabilistic algorithm for computing square roots modulo a prime number (Corresp.) |journal= IEEE Transactions on Information Theory |date=November 1986 |volume= 32 |issue= 6 |pages = 846–847 |issn = 0018-9448 |doi = 10.1109/TIT.1986.1057236 }}</ref> In 2000 Peralta's method was generalized for cubic equations.<ref>{{cite journal |author = C Padró, G Sáez |title= Taking cube roots in Zm |journal= Applied Mathematics Letters |date=August 2002 |volume= 15 |issue= 6 |pages = 703–708 |issn = 0893-9659 |doi = 10.1016/s0893-9659(02)00031-9 |doi-access= }}</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 F_p\simeq \mathbb Z/p\mathbb Z</math> of remainders modulo <math>p</math>. The algorithm should find all <math>\lambda</math> in <math>\mathbb F_p</math> such that <math display="inline">f(\lambda)= 0</math> in <math>\mathbb F_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 | url = https://www.springer.com/gp/book/9780792392828 | title = Applications of Finite Fields |date = 1993 |publisher= Springer US | series = The Springer International Series in Engineering and Computer Science | isbn = 9780792392828}}</ref>
== Algorithm ==
=== Randomization === Let <math display="inline">f(x) = (x-\lambda_1)(x-\lambda_2)\cdots(x-\lambda_n)</math>. Finding all roots of this polynomial is equivalent to finding its factorization into linear factors. To find such factorization it is sufficient to split the polynomial into any two non-trivial divisors and factorize them recursively. To do this, consider the polynomial <math display="inline">f_z(x)=f(x-z) = (x-\lambda_1 - z)(x-\lambda_2 - z) \cdots (x-\lambda_n-z)</math> where <math>z</math> is some element of <math>\mathbb F_p</math>. If one can represent this polynomial as the product <math>f_z(x)=p_0(x)p_1(x)</math> then in terms of the initial polynomial it means that <math>f(x) =p_0(x+z)p_1(x+z)</math>, which provides needed factorization of <math>f(x)</math>.<ref name=":0" /><ref name=":2" />
=== Classification of <math>\mathbb F_p</math> elements === Due to Euler's criterion, for every monomial <math>(x-\lambda)</math> exactly one of following properties holds:<ref name=":0" />
# The monomial is equal to <math>x</math> if <math>\lambda = 0</math>, # The monomial divides <math display="inline">g_0(x)=(x^{(p-1)/2}-1)</math> if <math>\lambda</math> is quadratic residue modulo <math>p</math>, # The monomial divides <math display="inline">g_1(x)=(x^{(p-1)/2}+1)</math> if <math>\lambda</math> is quadratic non-residual modulo <math>p</math>.
Thus if <math>f_z(x)</math> is not divisible by <math>x</math>, which may be checked separately, then <math>f_z(x)</math> is equal to the product of greatest common divisors <math>\gcd(f_z(x);g_0(x))</math> and <math>\gcd(f_z(x);g_1(x))</math>.<ref name=":2" />
=== Berlekamp's method === The property above leads to the following algorithm:<ref name=":0" />
# Explicitly calculate coefficients of <math>f_z(x) = f(x-z)</math>, # Calculate remainders of <math display="inline">x,x^2, x^{2^2},x^{2^3}, x^{2^4}, \ldots, x^{2^{\lfloor \log_2 p \rfloor}}</math> modulo <math>f_z(x)</math> by squaring the current polynomial and taking remainder modulo <math>f_z(x)</math>, # Using exponentiation by squaring and polynomials calculated on the previous steps calculate the remainder of <math display="inline">x^{(p-1)/2}</math> modulo <math display="inline">f_z(x)</math>, # If <math display="inline">x^{(p-1)/2} \not \equiv \pm 1 \pmod{f_z(x)}</math> then <math>\gcd</math> mentioned below provide a non-trivial factorization of <math>f_z(x)</math>, # Otherwise all roots of <math>f_z(x)</math> are either residues or non-residues simultaneously and one has to choose another <math>z</math>.
If <math>f(x)</math> is divisible by some non-linear primitive polynomial <math>g(x)</math> over <math>\mathbb F_p</math> then when calculating <math>\gcd</math> with <math>g_0(x)</math> and <math>g_1(x)</math> one will obtain a non-trivial factorization of <math>f_z(x)/g_z(x)</math>, thus algorithm allows to find all roots of arbitrary polynomials over <math>\mathbb F_p</math>.
=== Modular square root === Consider equation <math display="inline">x^2 \equiv a \pmod{p}</math> having elements <math>\beta</math> and <math>-\beta</math> as its roots. Solution of this equation is equivalent to factorization of polynomial <math display="inline">f(x) = x^2-a=(x-\beta)(x+\beta)</math> over <math>\mathbb F_p</math>. In this particular case problem it is sufficient to calculate only <math>\gcd(f_z(x); g_0(x))</math>. For this polynomial exactly one of the following properties will hold:
# GCD is equal to <math>1</math> which means that <math>z+\beta</math> and <math>z-\beta</math> are both quadratic non-residues, # GCD is equal to <math>f_z(x)</math>which means that both numbers are quadratic residues, # GCD is equal to <math>(x-t)</math>which means that exactly one of these numbers is quadratic residue.
In the third case GCD is equal to either <math>(x-z-\beta)</math> or <math>(x-z+\beta)</math>. It allows to write the solution as <math display="inline">\beta = (t - z) \pmod{p}</math>.<ref name=":0" />
=== Example === Assume we need to solve the equation <math display="inline">x^2 \equiv 5\pmod{11}</math>. For this we need to factorize <math>f(x)=x^2-5=(x-\beta)(x+\beta)</math>. Consider some possible values of <math>z</math>:
# Let <math>z=3</math>. Then <math>f_z(x) = (x-3)^2 - 5 = x^2 - 6x + 4</math>, thus <math>\gcd(x^2 - 6x + 4 ; x^5 - 1) = 1</math>. Both numbers <math>3 \pm \beta</math> are quadratic non-residues, so we need to take some other <math>z</math>.
# Let <math>z=2</math>. Then <math>f_z(x) = (x-2)^2 - 5 = x^2 - 4x - 1</math>, thus <math display="inline">\gcd( x^2 - 4x - 1 ; x^5 - 1)\equiv x - 9 \pmod{11}</math>. From this follows <math display="inline">x - 9 = x - 2 - \beta</math>, so <math>\beta \equiv 7 \pmod{11}</math> and <math display="inline">-\beta \equiv -7 \equiv 4 \pmod{11}</math>.
A manual check shows that, indeed, <math display="inline">7^2 \equiv 49 \equiv 5\pmod{11}</math> and <math display="inline">4^2\equiv 16 \equiv 5\pmod{11}</math>.
== 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 | url = https://books.google.com/books?id=__JCiiCfu2EC&q=Combinatorial+Theory+hall&pg=PA1 | title = Combinatorial Theory |date = 1998 |publisher= John Wiley & Sons | isbn = 9780471315186}}</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 == Let a polynomial have degree <math>n</math>. We derive the algorithm's complexity as follows:
# Due to the binomial theorem <math display="inline">(x-z)^k = \sum\limits_{i=0}^k \binom{k}{i} (-z)^{k-i}x^i</math>, we may transition from <math>f(x)</math> to <math>f(x-z)</math> in <math>O(n^2)</math> time. # Polynomial multiplication and taking remainder of one polynomial modulo another one may be done in <math display="inline">O(n^2)</math>, thus calculation of <math display="inline">x^{2^k} \bmod f_z(x)</math> is done in <math display="inline">O(n^2 \log p)</math>. # Binary exponentiation works in <math>O(n^2 \log p)</math>. # 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. | url = https://archive.org/details/designanalysisof00ahoarich | title = The design and analysis of computer algorithms | date = 1974 | publisher = Addison-Wesley Pub. Co | isbn = 0201000296 | 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 == {{reflist}}
{{Number-theoretic algorithms}}
Category:Algorithms Category:Algebra Category:Number theoretic algorithms Category:Polynomials