### Abstract

Let F be a field of q=p^{n} elements, where p is prime. We present two new probabilisticalgorithms for factoring polynomials in F[X] that make particularly efficient use of random bits. They are easy to implement, and require no randomness beyond an initial seed whose length is proportional to the input size. The first algorithm is based on a procedure of Berlekamp; on input f in F[X] of degree d, it uses d log_{2}p random bits and produces in polynomial time a complete factorisation of f with a failure probability of no more than 1/p^{(1−e)d/2}. (Here ε denotes a fixed parameter between 0 and 1 that can be chosen by the implementer.) The second algorithm is based on a method of Cantor and Zassenhaus; it uses d log_{2}q random bits and fails to find a complete factorisation with probability no more than 1/q^{(1−e)d/4}. For both of these algorithms, the failure probability is exponentially small in the number of random bits used.

Original language | English (US) |
---|---|

Pages (from-to) | 229-239 |

Number of pages | 11 |

Journal | Journal of Symbolic Computation |

Volume | 9 |

Issue number | 3 |

DOIs | |

State | Published - 1990 |

### Fingerprint

### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics

### Cite this

*Journal of Symbolic Computation*,

*9*(3), 229-239. https://doi.org/10.1016/S0747-7171(08)80011-9

**Factoring polynomials using fewer random bits.** / Bach, Eric; Shoup, Victor.

Research output: Contribution to journal › Article

*Journal of Symbolic Computation*, vol. 9, no. 3, pp. 229-239. https://doi.org/10.1016/S0747-7171(08)80011-9

}

TY - JOUR

T1 - Factoring polynomials using fewer random bits

AU - Bach, Eric

AU - Shoup, Victor

PY - 1990

Y1 - 1990

N2 - Let F be a field of q=pn elements, where p is prime. We present two new probabilisticalgorithms for factoring polynomials in F[X] that make particularly efficient use of random bits. They are easy to implement, and require no randomness beyond an initial seed whose length is proportional to the input size. The first algorithm is based on a procedure of Berlekamp; on input f in F[X] of degree d, it uses d log2p random bits and produces in polynomial time a complete factorisation of f with a failure probability of no more than 1/p(1−e)d/2. (Here ε denotes a fixed parameter between 0 and 1 that can be chosen by the implementer.) The second algorithm is based on a method of Cantor and Zassenhaus; it uses d log2q random bits and fails to find a complete factorisation with probability no more than 1/q(1−e)d/4. For both of these algorithms, the failure probability is exponentially small in the number of random bits used.

AB - Let F be a field of q=pn elements, where p is prime. We present two new probabilisticalgorithms for factoring polynomials in F[X] that make particularly efficient use of random bits. They are easy to implement, and require no randomness beyond an initial seed whose length is proportional to the input size. The first algorithm is based on a procedure of Berlekamp; on input f in F[X] of degree d, it uses d log2p random bits and produces in polynomial time a complete factorisation of f with a failure probability of no more than 1/p(1−e)d/2. (Here ε denotes a fixed parameter between 0 and 1 that can be chosen by the implementer.) The second algorithm is based on a method of Cantor and Zassenhaus; it uses d log2q random bits and fails to find a complete factorisation with probability no more than 1/q(1−e)d/4. For both of these algorithms, the failure probability is exponentially small in the number of random bits used.

UR - http://www.scopus.com/inward/record.url?scp=84968508531&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84968508531&partnerID=8YFLogxK

U2 - 10.1016/S0747-7171(08)80011-9

DO - 10.1016/S0747-7171(08)80011-9

M3 - Article

VL - 9

SP - 229

EP - 239

JO - Journal of Symbolic Computation

JF - Journal of Symbolic Computation

SN - 0747-7171

IS - 3

ER -