Subgroup Refinement Algorithms for Root Finding in GF(q)

This paper presents a generalization of Moenck’s root finding algorithm over $GF(q)$, for q a prime or prime power. The generalized algorithm, like its predecessor, is deterministic, given a primitive element $\omega $ for $GF(q)$. If $q - 1$ is b-smooth, where $b = (\log q)^{O(1)} $, then the algorithm runs in polynomial time. An analogue of this generalization which applies to extension fields $GF(q^m )$ is also considered. The analogue is a deterministic algorithm based on the recently introduced affine method for root finding in $GF(q^m )$, where $m > 1$; it is, however, less efficient that the affine method itself.