Constructing high order elements through subspace polynomials

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 Fq(qc−1)/(q−1) 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.

• Type: Journal
• Year: 2012
• Where: SODA
• Authors: Qi Cheng, Shuhong Gao, Daqing Wan
• Keywords: Algorithms, Discrete Logarithm, Polynomial Time, SODA 2012, Time Qc