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Abstract: In numerous applications involving finite fields, we often need high-order elements. Ideally we should be able to obtain a primitive element for any finite field in reasonable time. However, if the prime factorization of the group order is unknown, we do not know how to achieve the goal. We thus turn our attentions to a less ambitious problem: constructing an element of provably high order. In this paper, we survey various algorithms that find an element of high order for general or special finite fields. [Copyright &y& Elsevier]