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 $Z^∗_q$?