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.