With reference to the Euler's totient function $\phi(\cdot)$, given any $n \in \mathbb{Z}^+$, it's quite straightforward to find $\phi(n)$.

In contrast, given $n \in \mathbb{Z}^+$, even though there are to find the $k \in \mathbb{Z}^+$ such that $\phi(k) = n$, I'm not aware of any method to determine the number of such $k$ values beforehand. What are the progress that has been made in this regard ?