## Searching for primitive roots in finite fields.(English)Zbl 0747.11060

An earlier version of this paper appeared in the 22nd Annual ACM Symposium on Theory of Computing, 1990, pp. 546-554.
In a finite field $$GF(p^ n)$$ (with $$p$$ prime and $$n\geq 1$$) a nonzero element is called a primitive root if it generates the multiplicative group of units. It is well known that the density of primitive roots in $$GF(p^ n)$$ is great enough so that the simple method of choosig a small number of elements in $$GF(p^ n)$$ at random is a probabilistic polynomial-time search procedure for primitive roots. In the present paper the author considers the problem of how to deterministically generate in polynomial-time a subset of $$GF(p^ n)$$ that contains a primitive root. Three results are presented. First, the author solves this problem in the case that $$p=n^{O(1)}$$. Second, he shows under the assumption of the Extended Riemann Hypothesis (ERH) that there is a deterministic polynomial-time search procedure for primitive roots in $$GF(p^ 2)$$. Third, he gives a quantitative improvement of a theorem of Wang on the least primitive root for $$GF(p)$$, assuming ERH.
Reviewer: J.Hinz (Marburg)

### MSC:

 11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects) 11Y40 Algebraic number theory computations 11Y16 Number-theoretic algorithms; complexity 68Q25 Analysis of algorithms and problem complexity
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### References:

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