Discrete logarithm problem

Definition

This problem usually makes sense in the context of an encoding of a group or a multi-encoding of a group. The problem asks for the following: given the code-words for elements $g,h$ of the group such that $h$ is a power of $g$, find any integer $x$ such that $g^{x}=h$.

In the context of a black-box cyclic group, this problem is typically asked with $g$ chosen to be a generator of the group.

Caveat

Note that $x$ is unique modulo the order of $g$. Usually, the problem is discussed in a context where the order of $g$ is known, i.e., we already know the answer to the element order-finding problem for $g$. Given this information, finding one value of $x$ is equivalent to finding all values of $x$.

However, the problem makes sense even when the order of $g$ is not known. In this case, an algorithm that gives any $x$ (not necessarily the smallest positive value) suffices.