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Searching for Primitive Roots in Finite Fields

Victor Shoup
Mathematics of Computation
Vol. 58, No. 197 (Jan., 1992), pp. 369-380
DOI: 10.2307/2153041
Stable URL: http://www.jstor.org/stable/2153041
Page Count: 12
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Searching for Primitive Roots in Finite Fields
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Abstract

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.

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