Berlekamp–Zassenhaus algorithm: Difference between revisions

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In [[mathematics]], in particular in [[computer algebra|computational algebra]], the '''Berlekamp–Zassenhaus algorithm''' is an [[algorithm]] for factoring [[polynomial]]s over the [[integer]]s, named after [[Elwyn Berlekamp]] and [[Hans Zassenhaus]]. As a consequence of [[Gauss's lemma (number theory)|Gauss's lemma]], this amounts to solving the problem also over the rationals.

The algorithm starts by finding factorizations over suitable [[finite field]]s using [[Hensel's lemma]] to lift the solution from modulo a prime ''p'' to a convenient power of ''p''. After this the right factors are found as a subset of these.

The worst case of this algorithm is exponential in the number of factors.

{{harvtxt|vanVan Hoeij|2002}} improved this algorithm by using the [[LLL algorithm]], substantially reducing the time needed to choose the right subsets of mod ''p'' factors.

==See also==

*[[Berlekamp's algorithm]]

==References==

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| pages = 1853–1859

| title = Factoring polynomials over finite fields

| url = http://www.alcatel.hu/bstj/vol46-1967/articles/bstj46-8-1853.pdf

| volume = 46

| year = 1967}}.

| issue = 8 | doi=10.1002/j.1538-7305.1967.tb03174.x}}.

*{{citation

| last = Berlekamp | first = E. R. | authorlink = Elwyn Berlekamp

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| title = Factoring polynomials over large finite fields

| volume = 24

| year = 1970}}.| issue = 111 | doi-access = free

}}.

*{{citation

| last1 = Cantor | first1 = David G.

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| title = A new algorithm for factoring polynomials over finite fields

| volume = 36

| year = 1981}}.| doi-access = free

}}.

*{{citation

| last1 = Geddes
| first1 = K. O.

| last2 = Czapor
| first2 = S. R.

| last3 = Labahn
| first3 = G.

| doi = 10.1007/b102438

| isbn = 0-7923-9259-0

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| publisher = Kluwer Academic Publishers

| title = Algorithms for computer algebra

| year = 1992}}.

| bibcode = 1992afca.book.....G

| url-access = registration

| url = https://archive.org/details/algorithmsforcom0000gedd

}}.

*{{citation

| last = vanVan Hoeij | first = Mark

| doi = 10.1016/S0022-314X(01)92763-5

| issue = 2

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| title = Factoring polynomials and the knapsack problem

| volume = 95

| year = 2002}}.| doi-access = free

}}.

*{{citation

| last = Zassenhaus | first = Hans | authorlink = Hans Zassenhaus

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| title = On Hensel factorization. I

| volume = 1

| year = 1969}}| issue = 3 | bibcode = 1969JNT.....1..291Z | doi-access =

}}.

==External links==

*{{mathworld|id=Berlekamp-ZassenhausAlgorithm|title=Berlekamp-Zassenhaus Algorithm|author=Domazet, Haris}}

==See also==

*[[Berlekamp's algorithm]]

[[Category:Computer algebra]]