### Abstract

Let GF(p^{n}) be the finite field with p^{n} elements, where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(p^{n}) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(p^{n}). We present three results. First, we present a solution to this problem for the case where p is small, i.e., p = n^{O(1)}. Second, we present a solution to this problem under the assumption of the Extended Riemann Hypothesis (ERH) for the case where p is large and n = 2. Third, we give a quantitative improvement of a theorem of Wang on the least primitive root for GF(p), assuming the ERH.

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

Pages (from-to) | 369-380 |

Number of pages | 12 |

Journal | Mathematics of Computation |

Volume | 58 |

Issue number | 197 |

DOIs | |

State | Published - 1992 |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

### Cite this

**Searching for primitive roots in finite fields.** / Shoup, Victor.

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 58, no. 197, pp. 369-380. https://doi.org/10.1090/S0025-5718-1992-1106981-9

}

TY - JOUR

T1 - Searching for primitive roots in finite fields

AU - Shoup, Victor

PY - 1992

Y1 - 1992

N2 - Let GF(pn) be the finite field with pn elements, where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(pn) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(pn). We present three results. First, we present a solution to this problem for the case where p is small, i.e., p = nO(1). Second, we present a solution to this problem under the assumption of the Extended Riemann Hypothesis (ERH) for the case where p is large and n = 2. Third, we give a quantitative improvement of a theorem of Wang on the least primitive root for GF(p), assuming the ERH.

AB - Let GF(pn) be the finite field with pn elements, where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(pn) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(pn). We present three results. First, we present a solution to this problem for the case where p is small, i.e., p = nO(1). Second, we present a solution to this problem under the assumption of the Extended Riemann Hypothesis (ERH) for the case where p is large and n = 2. Third, we give a quantitative improvement of a theorem of Wang on the least primitive root for GF(p), assuming the ERH.

UR - http://www.scopus.com/inward/record.url?scp=84968515930&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84968515930&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-1992-1106981-9

DO - 10.1090/S0025-5718-1992-1106981-9

M3 - Article

VL - 58

SP - 369

EP - 380

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 197

ER -