### Abstract

Let p be a prime and F be a polynomial with integer coefficients. Suppose that the discriminant of F is not divisible by p, and denote by m the degree of the splitting field of F over Q and by L the maximal size of the coefficients of F. Then, assuming the generalized Riemann hypothesis (GRH), it is shown that the irreducible factors of F modulo p can be found in (deterministic) time polynomial in deg F, m, log p, and L. As an application, it is shown that it is possible under GRH to solve certain equations of the form nP = R, where R is a given and P is an unknown point of an elliptic curve defined over GF(p) in polynomial time (n is counted in unary). An elliptic analog of results obtained recently about factoring polynomials with the help of smooth multiplicative subgroups of finite fields is proved.

Original language | English |
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Title of host publication | Annual Symposium on Foundations of Computer Science (Proceedings) |

Publisher | Publ by IEEE |

Pages | 99-104 |

Number of pages | 6 |

ISBN (Print) | 0818619821 |

Publication status | Published - Nov 1989 |

Event | 30th Annual Symposium on Foundations of Computer Science - Research Triangle Park, NC, USA Duration: Oct 30 1989 → Nov 1 1989 |

### Other

Other | 30th Annual Symposium on Foundations of Computer Science |
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City | Research Triangle Park, NC, USA |

Period | 10/30/89 → 11/1/89 |

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### ASJC Scopus subject areas

- Hardware and Architecture

### Cite this

*Annual Symposium on Foundations of Computer Science (Proceedings)*(pp. 99-104). Publ by IEEE.

**Galois groups and factoring polynomials over finite fields.** / Rónyai, L.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Annual Symposium on Foundations of Computer Science (Proceedings).*Publ by IEEE, pp. 99-104, 30th Annual Symposium on Foundations of Computer Science, Research Triangle Park, NC, USA, 10/30/89.

}

TY - GEN

T1 - Galois groups and factoring polynomials over finite fields

AU - Rónyai, L.

PY - 1989/11

Y1 - 1989/11

N2 - Let p be a prime and F be a polynomial with integer coefficients. Suppose that the discriminant of F is not divisible by p, and denote by m the degree of the splitting field of F over Q and by L the maximal size of the coefficients of F. Then, assuming the generalized Riemann hypothesis (GRH), it is shown that the irreducible factors of F modulo p can be found in (deterministic) time polynomial in deg F, m, log p, and L. As an application, it is shown that it is possible under GRH to solve certain equations of the form nP = R, where R is a given and P is an unknown point of an elliptic curve defined over GF(p) in polynomial time (n is counted in unary). An elliptic analog of results obtained recently about factoring polynomials with the help of smooth multiplicative subgroups of finite fields is proved.

AB - Let p be a prime and F be a polynomial with integer coefficients. Suppose that the discriminant of F is not divisible by p, and denote by m the degree of the splitting field of F over Q and by L the maximal size of the coefficients of F. Then, assuming the generalized Riemann hypothesis (GRH), it is shown that the irreducible factors of F modulo p can be found in (deterministic) time polynomial in deg F, m, log p, and L. As an application, it is shown that it is possible under GRH to solve certain equations of the form nP = R, where R is a given and P is an unknown point of an elliptic curve defined over GF(p) in polynomial time (n is counted in unary). An elliptic analog of results obtained recently about factoring polynomials with the help of smooth multiplicative subgroups of finite fields is proved.

UR - http://www.scopus.com/inward/record.url?scp=0024771049&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0024771049&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0024771049

SN - 0818619821

SP - 99

EP - 104

BT - Annual Symposium on Foundations of Computer Science (Proceedings)

PB - Publ by IEEE

ER -