Black-box cyclic group
A black-box cyclic group is a black-box group with the promise that the group is cyclic. In other words, it is a cyclic group with an encoding, where we are not told anything about the inner functionings of the algorithms for multiplication, inverse and the identity element. Typically, when talking of a black-box cyclic group, one additional piece of information is furnished: the order of the group.
There are a number of practical situations where we come up with cyclic groups with an encoding, and although these may not be black-box, any approach that works for black-box cyclic groups will work for these. Particular cases are the multiplicative group of a finite field and in particular the multiplicative group of a prime field.