### Abstract

We present a new deterministic algorithm for factoring polynomials over Z_{p} of degree n. We show that the worst-case running time of our algorithm is O(p^{ 1 2}(log p)^{2}n^{2+∈}), which is faster than the running times of previous determi nistic algorithms with respect to both n and p. We also show that our algorithm runs in polynomial time for all but at most an exponentially small fraction of the polynomials of degree n over Z_{p}. Specifically, we prove that the fraction of polynomials of degree n over Z_{p} for which our algorithm fails to halt in time O((log p)^{2}n^{2+∈}) is ((n log p)^{2}/p). Consequently, the average-case running time of our algorithm is polynomial in n and log p.

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

Pages (from-to) | 261-267 |

Number of pages | 7 |

Journal | Information Processing Letters |

Volume | 33 |

Issue number | 5 |

DOIs | |

State | Published - Jan 10 1990 |

### Fingerprint

### Keywords

- Factorization
- finite fields
- irreducible polynomials

### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

**On the deterministic complexity of factoring polynomials over finite fields.** / Shoup, Victor.

Research output: Contribution to journal › Article

*Information Processing Letters*, vol. 33, no. 5, pp. 261-267. https://doi.org/10.1016/0020-0190(90)90195-4

}

TY - JOUR

T1 - On the deterministic complexity of factoring polynomials over finite fields

AU - Shoup, Victor

PY - 1990/1/10

Y1 - 1990/1/10

N2 - We present a new deterministic algorithm for factoring polynomials over Zp of degree n. We show that the worst-case running time of our algorithm is O(p 1 2(log p)2n2+∈), which is faster than the running times of previous determi nistic algorithms with respect to both n and p. We also show that our algorithm runs in polynomial time for all but at most an exponentially small fraction of the polynomials of degree n over Zp. Specifically, we prove that the fraction of polynomials of degree n over Zp for which our algorithm fails to halt in time O((log p)2n2+∈) is ((n log p)2/p). Consequently, the average-case running time of our algorithm is polynomial in n and log p.

AB - We present a new deterministic algorithm for factoring polynomials over Zp of degree n. We show that the worst-case running time of our algorithm is O(p 1 2(log p)2n2+∈), which is faster than the running times of previous determi nistic algorithms with respect to both n and p. We also show that our algorithm runs in polynomial time for all but at most an exponentially small fraction of the polynomials of degree n over Zp. Specifically, we prove that the fraction of polynomials of degree n over Zp for which our algorithm fails to halt in time O((log p)2n2+∈) is ((n log p)2/p). Consequently, the average-case running time of our algorithm is polynomial in n and log p.

KW - Factorization

KW - finite fields

KW - irreducible polynomials

UR - http://www.scopus.com/inward/record.url?scp=0025258153&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0025258153&partnerID=8YFLogxK

U2 - 10.1016/0020-0190(90)90195-4

DO - 10.1016/0020-0190(90)90195-4

M3 - Article

VL - 33

SP - 261

EP - 267

JO - Information Processing Letters

JF - Information Processing Letters

SN - 0020-0190

IS - 5

ER -