Constructing high order elements through subspace polynomials

Authors:
Qi Cheng;Shuhong Gao;Daqing Wan
Affiliations:
University of Oklahoma, Norman, OK;Clemson University, Clemson, SC;University of California, Irvine, CA
Venue:
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Year:
2012

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Abstract

Every finite field has many multiplicative generators. However, finding one in polynomial time is an important open problem. In fact, even finding elements of high order has not been solved satisfactorily. In this paper, we present an algorithm that for any positive integer c and prime power q, finding an element of order exp(Ω(√qc)) in the finite field [EQUATION] in deterministic time (qc)O(1). We also show that there are exp(Ω(√qc)) many weak keys for the discrete logarithm problems in those fields with respect to certain bases.