### Abstract

New algorithms are presented for factoring polynomials of degree n over the finite field of q elements, where q is a power of 2. When log q = n^{1+a}, where a>0 is constant, these algorithms are asymptotically faster than previous known algorithms, the fastest of which required time Ω(n(log q)^{2}), or Ω(n^{3+2a}) in this case, which corresponds to the cost of computing x^{q} modulo an n degree polynomial. The new algorithms factor an arbitrary polynomial in time O(n^{3+a+1+o(1)}+n^{2.69+1.69a}). All measures are in fixed precision operations, that is in bit complexity. Moreover, in the special case where all the irreducible factors have the same degree, the new algorithms run in time O(n^{2.69+1.69a}). In particular, one may test a polynomial for irreducibility in O(n^{2.69+1.69a}) bit operations. These results generalize to the case where q = p^{k}, where p is a small, fixed prime.

Original language | English (US) |
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Title of host publication | Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC |

Publisher | ACM |

Pages | 184-188 |

Number of pages | 5 |

State | Published - 1997 |

Event | Proceedings of the 1997 22nd International Symposium on Symbolic and Algebraic Computation, ISSAC - Maui, HI, USA Duration: Jul 21 1997 → Jul 23 1997 |

### Other

Other | Proceedings of the 1997 22nd International Symposium on Symbolic and Algebraic Computation, ISSAC |
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City | Maui, HI, USA |

Period | 7/21/97 → 7/23/97 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC*(pp. 184-188). ACM.

**Fast polynomial factorization over high algebraic extensions of finite fields.** / Kaltofen, Erich; Shoup, Victor.

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

*Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC.*ACM, pp. 184-188, Proceedings of the 1997 22nd International Symposium on Symbolic and Algebraic Computation, ISSAC, Maui, HI, USA, 7/21/97.

}

TY - GEN

T1 - Fast polynomial factorization over high algebraic extensions of finite fields

AU - Kaltofen, Erich

AU - Shoup, Victor

PY - 1997

Y1 - 1997

N2 - New algorithms are presented for factoring polynomials of degree n over the finite field of q elements, where q is a power of 2. When log q = n1+a, where a>0 is constant, these algorithms are asymptotically faster than previous known algorithms, the fastest of which required time Ω(n(log q)2), or Ω(n3+2a) in this case, which corresponds to the cost of computing xq modulo an n degree polynomial. The new algorithms factor an arbitrary polynomial in time O(n3+a+1+o(1)+n2.69+1.69a). All measures are in fixed precision operations, that is in bit complexity. Moreover, in the special case where all the irreducible factors have the same degree, the new algorithms run in time O(n2.69+1.69a). In particular, one may test a polynomial for irreducibility in O(n2.69+1.69a) bit operations. These results generalize to the case where q = pk, where p is a small, fixed prime.

AB - New algorithms are presented for factoring polynomials of degree n over the finite field of q elements, where q is a power of 2. When log q = n1+a, where a>0 is constant, these algorithms are asymptotically faster than previous known algorithms, the fastest of which required time Ω(n(log q)2), or Ω(n3+2a) in this case, which corresponds to the cost of computing xq modulo an n degree polynomial. The new algorithms factor an arbitrary polynomial in time O(n3+a+1+o(1)+n2.69+1.69a). All measures are in fixed precision operations, that is in bit complexity. Moreover, in the special case where all the irreducible factors have the same degree, the new algorithms run in time O(n2.69+1.69a). In particular, one may test a polynomial for irreducibility in O(n2.69+1.69a) bit operations. These results generalize to the case where q = pk, where p is a small, fixed prime.

UR - http://www.scopus.com/inward/record.url?scp=0030674617&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030674617&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0030674617

SP - 184

EP - 188

BT - Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

PB - ACM

ER -