Factorization problem Say, $X= a\cdot b$, where $(a, b) \in Z_q^*$ and $q$ is a large prime. If $X$ is given, then what is the complexity (or hardness) of finding $a$ and $b$?

Note that, either $a$ or $b$ can be reused to compute another $X'$ which is also public. 

Edited: (more details)

Let's say Alice chooses two random numbers $a, b\in Z^∗_q$ and computes $X=a\cdot b$. Alice publishes $X$. What is the complexity of Bob to guess (or compute) $a$ and $b$ from the known $X$ and $Z^∗_q$?
Say, $X= a\cdot b$, where $(a, b) \in Z_q^*$ and $q$ is a large prime. If $X$ is given, then what is the complexity (or hardness) of finding $a$ and $b$?


If the multiplication is done within $Z_q^*$, then it's easy - pick an arbitrary nonzero $a$ and compute $b = a^{-1}X$; you're done.  You can compute $a^{-1}$ by either the Extended Euclidean method, or by using $a^{-1} = a^{p-2}$

This will find a $(a, b)$ pair that satisfies the equation.  If you're looking for the unique one that someone else had in mind, well, you're out of luck - there are $q-1$ pairs that satisfy the equation, and with no other information, there is no way to determine which is the correct one.