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In mathematics, in particular in computational algebra, the Berlekamp–Zassenhaus algorithm is an algorithm for factoring polynomials over the integers. As a consequence of Gauss's lemma, this amounts to solving the problem also over the rationals. The algorithm starts by finding factorizations over suitable finite fields 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.

van Hoeij (2002) improved this algorithm by using the LLL algorithm, substantially reducing the time needed to choose the right subsets of mod p factors.
References

Berlekamp, E. R. "Factoring Polynomials over Finite Fields." Bell System Technical J. 46, 1853–1859, 1967.

Berlekamp, E. R. "Factoring Polynomials over Finite Fields." Math. Comput. 24, 713–735, 1970.

Cantor, D. G. and Zassenhaus, H. "A New Algorithm for Factoring Polynomials over Finite Fields." Math. Comput. 36, 587–592, 1981.

Geddes, K. O.; Czapor, S. R.; and Labahn, G. Algorithms for Computer Algebra. Amsterdam, Netherlands: Kluwer, 1992.

van Hoeij, M. "Factoring Polynomials and the Knapsack Problem." J. Number Th. 95, 167–189, 2002.

Zassenhaus, H. "On Hensel Factorization, I." J. Number Th. 1, 291–311, 1969.

External links

Berlekamp–Zassenhaus' algorithm at WolframMathWorld. [1]