The large-order conjecture for Kummer field generators

Let $q$ be a prime power and let $\textbf{F}_{q^n}$ be a finite field. Let $g\in\textbf{F}_{q^n}$ satisfy the conditions in Theorem~, and suppose that $n\geq\log q$. Large-order conjecture. The order of $g$ is greater than

$$q^{n/c}$$

for an absolute constant $c$. Numerical evidence suggests that the order of $g$ is close to the group order $q^n-1$, and proving a large order would address a principal obstacle in improving the efficiency of AKS-style primality-testing algorithms. The conjecture concerns the generator produced in the paper's Kummer-field construction; its status is not resolved in the supplied text.