Constructing Finite Field Extensions with Large Order Elements

Qi Cheng
2007 SIAM Journal on Discrete Mathematics  
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 α ∈ F q n of order greater than 5.8 n/ log q n , in time polynomial on N . We present another algorithm that find an integer n ∈ [N, N + O(N 0.77 )] and an element α ∈ F q n of order at least 5.8 √ n , in time polynomial on N . Our result is inspired by the recent AKS primality testing algorithm [1] and the subsequent improvements [4, 5, 3] .
doi:10.1137/s0895480104445514 fatcat:3olpn3z2uvgwnb5nrpli2natm4