### Abstract

In this paper we develop a general technique to eliminate the assumption of the Generalized Riemann Hypothesis (GRH) from various deterministic polynomial factoring algorithms over finite fields. It is the first bona fide progress on that issue for more than 25 years of study of the problem. Our main results are basically of the following form: either we construct a nontrivial factor of a given polynomial or compute a nontrivial automorphism of the factor algebra of the given polynomial. Probably the most notable application of such automorphisms is efficiently finding zero divisors in noncommutative algebras. The proof methods used in this paper exploit virtual roots of unity and lead to efficient actual polynomial factoring algorithms in special cases.

Original language | English |
---|---|

Pages (from-to) | 493-531 |

Number of pages | 39 |

Journal | Mathematics of Computation |

Volume | 81 |

Issue number | 277 |

DOIs | |

Publication status | Published - 2011 |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics
- Computational Mathematics

### Cite this

*Mathematics of Computation*,

*81*(277), 493-531. https://doi.org/10.1090/S0025-5718-2011-02505-6

**Trading GRH for algebra : Algorithms for factoring polynomials and related structures.** / Ivanyos, Gábor; Karpinski, Marek; Rónyai, L.; Saxena, Nitin.

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 81, no. 277, pp. 493-531. https://doi.org/10.1090/S0025-5718-2011-02505-6

}

TY - JOUR

T1 - Trading GRH for algebra

T2 - Algorithms for factoring polynomials and related structures

AU - Ivanyos, Gábor

AU - Karpinski, Marek

AU - Rónyai, L.

AU - Saxena, Nitin

PY - 2011

Y1 - 2011

N2 - In this paper we develop a general technique to eliminate the assumption of the Generalized Riemann Hypothesis (GRH) from various deterministic polynomial factoring algorithms over finite fields. It is the first bona fide progress on that issue for more than 25 years of study of the problem. Our main results are basically of the following form: either we construct a nontrivial factor of a given polynomial or compute a nontrivial automorphism of the factor algebra of the given polynomial. Probably the most notable application of such automorphisms is efficiently finding zero divisors in noncommutative algebras. The proof methods used in this paper exploit virtual roots of unity and lead to efficient actual polynomial factoring algorithms in special cases.

AB - In this paper we develop a general technique to eliminate the assumption of the Generalized Riemann Hypothesis (GRH) from various deterministic polynomial factoring algorithms over finite fields. It is the first bona fide progress on that issue for more than 25 years of study of the problem. Our main results are basically of the following form: either we construct a nontrivial factor of a given polynomial or compute a nontrivial automorphism of the factor algebra of the given polynomial. Probably the most notable application of such automorphisms is efficiently finding zero divisors in noncommutative algebras. The proof methods used in this paper exploit virtual roots of unity and lead to efficient actual polynomial factoring algorithms in special cases.

UR - http://www.scopus.com/inward/record.url?scp=84856370192&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84856370192&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-2011-02505-6

DO - 10.1090/S0025-5718-2011-02505-6

M3 - Article

AN - SCOPUS:84856370192

VL - 81

SP - 493

EP - 531

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 277

ER -