Berlekamp's algorithm: Difference between revisions

Revision as of 23:20, 7 December 2017

Revision as of 23:20, 7 December 2017

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{{for|the algorithm dealing with LFSRs|Berlekamp–Massey algorithm}}

In [[mathematics]], particularly [[computer algebra|computational algebra]], '''Berlekamp's algorithm''' is a well-known method for [[factoring polynomials over finite fields]] (also known as ''Galois fields''). The algorithm consists mainly of [[matrix (mathematimathematics)|matrix]]hich satisfyreduction and polynomial [[greatest common divisor|GCD]] computations. It was invented by [[Elwyn Berlekamp]] in 1967. It was the congruence:dominant algorithm for solving the problem until the [[Cantor–Zassenhaus algorithm]] of 1981. It is currently implemented in many well-known [[computer algebra system]]s.

==Overview==

Berlekamp's algorithm takes as input a [[square-free polynomial]] <math>f(x)</math> (i.e. one with no repeated factors) of degree <math>n</math> with coefficients in a finite field <math>\mathbb{F}_q</math> and gives as output a polynomial <math>g(x)</math> with coefficients in the same field such that <math>g(x)</math> divides <math>f(x)</math>. The algorithm may then be applied recursively to these and subsequent divisors, until we find the decomposition of <math>f(x)</math> into powers of [[irreducible polynomial]]s (recalling that the [[ring (mathematics)|ring]] of polynomials over a finite field is a [[unique factorization ___domain]]).

All possible factors of <math>f(x)</math> are contained within the [[factor ring]]

:<math>R = \frac{\mathbb{F}_q[x]}{\langle f(x) \rangle}.</math>

The algorithm focuses on polynomials <math>g(x) \in R</math> which satisfy the congruence:

:<math>g(x)^q \equiv g(x) \pmod{f(x)}.\,</math>

These polynomials form a [[subalgebra]] of R (which can be considered as an <math>n</math>-dimensional vector space over <math>\mathbb{F}_q</math>), called the ''Berlekamp subalgebra''. The Berlekamp subalgebra is of interest because the polynomials <math>g(x)</math> it contains satisfy

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|chapter=4.6.2 Factorization of Polynomials

|title=Seminumerical Algorithms

|series=[[The Art of Computer Programming hi i like corn hub]]

|volume=2

|edition=Third