Constructing finite field extensions with large order elements

In this paper, we present an algorithm that given a fixed prime power q and a positive integer N, finds an integer n [N, 2qN] and an element Fqn of order greater than 5.8n/ logq n , in time polynomial in N. We present another algorithm that find an integer n [N, N + O(N0.77 )] and an element Fqn of order at least 5.8 n , in time polynomial in N. Our result is inspired by the recent AKS primality testing algorithm [1] and the subsequent improvements [4, 5, 3].

• Type: Conference
• Year: 2004
• Where: SODA
• Authors: Qi Cheng
• Keywords: Algorithms, Element Fqn, Fixed Prime Power, SODA 2004, Time Polynomial