### Abstract

Gauss periods yield (self-dual) normal bases in finite fields, and these normal bases can be used to implement arithmetic efficiently. It is shown that for a small prime power q and infinitely many integersn , multiplication in a normal basis of Fqn over Fq can be computed with O(n logn loglog n), division with O(n log^{2}n loglog n) operations in Fq, and exponentiation of an arbitrary element in Fqn withO (n^{2}loglog n) operations in Fq. We also prove that using a polynomial basis exponentiation in F 2 n can be done with the same number of operations in F 2 for all n. The previous best estimates were O(n^{2}) for multiplication in a normal basis, andO (n^{2}log n loglog n) for exponentiation in a polynomial basis.

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

Pages (from-to) | 879-889 |

Number of pages | 11 |

Journal | Journal of Symbolic Computation |

Volume | 29 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2000 |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics

### Cite this

*Journal of Symbolic Computation*,

*29*(6), 879-889. https://doi.org/10.1006/jsco.1999.0309

**Algorithms for Exponentiation in Finite Fields.** / Gao, Shuhong; Von Zur Gathen, Joachim; Panario, Daniel; Shoup, Victor.

Research output: Contribution to journal › Article

*Journal of Symbolic Computation*, vol. 29, no. 6, pp. 879-889. https://doi.org/10.1006/jsco.1999.0309

}

TY - JOUR

T1 - Algorithms for Exponentiation in Finite Fields

AU - Gao, Shuhong

AU - Von Zur Gathen, Joachim

AU - Panario, Daniel

AU - Shoup, Victor

PY - 2000/6

Y1 - 2000/6

N2 - Gauss periods yield (self-dual) normal bases in finite fields, and these normal bases can be used to implement arithmetic efficiently. It is shown that for a small prime power q and infinitely many integersn , multiplication in a normal basis of Fqn over Fq can be computed with O(n logn loglog n), division with O(n log2n loglog n) operations in Fq, and exponentiation of an arbitrary element in Fqn withO (n2loglog n) operations in Fq. We also prove that using a polynomial basis exponentiation in F 2 n can be done with the same number of operations in F 2 for all n. The previous best estimates were O(n2) for multiplication in a normal basis, andO (n2log n loglog n) for exponentiation in a polynomial basis.

AB - Gauss periods yield (self-dual) normal bases in finite fields, and these normal bases can be used to implement arithmetic efficiently. It is shown that for a small prime power q and infinitely many integersn , multiplication in a normal basis of Fqn over Fq can be computed with O(n logn loglog n), division with O(n log2n loglog n) operations in Fq, and exponentiation of an arbitrary element in Fqn withO (n2loglog n) operations in Fq. We also prove that using a polynomial basis exponentiation in F 2 n can be done with the same number of operations in F 2 for all n. The previous best estimates were O(n2) for multiplication in a normal basis, andO (n2log n loglog n) for exponentiation in a polynomial basis.

UR - http://www.scopus.com/inward/record.url?scp=0000185837&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000185837&partnerID=8YFLogxK

U2 - 10.1006/jsco.1999.0309

DO - 10.1006/jsco.1999.0309

M3 - Article

VL - 29

SP - 879

EP - 889

JO - Journal of Symbolic Computation

JF - Journal of Symbolic Computation

SN - 0747-7171

IS - 6

ER -