Primitive root-finding problem

Definition

Suppose $G$ is a finite group specified by means of a suitable group description rule (typically, an encoding), and we are given a promise that $G$ is a finite cyclic group. In other words, we can think of $G$ as a black-box cyclic group (though there may be more contextual structure known about $G$). The goal is to obtain an explicit description of a single element $g\in G$ such that $G=\langle g\rangle$. Such an element is termed a primitive root in some contexts.