Discrete logarithm problem
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 of the group such that is a power of , find any integer such that .
In the context of a black-box cyclic group, this problem is typically asked with chosen to be a generator of the group.
Note that is unique modulo the order of . Usually, the problem is discussed in a context where the order of is known, i.e., we already know the answer to the element order-finding problem for . Given this information, finding one value of is equivalent to finding all values of .
However, the problem makes sense even when the order of is not known. In this case, an algorithm that gives any (not necessarily the smallest positive value) suffices.