Discrete Logarithm in a specific group I am considering a setting similar to the one of the Paillier cryptosystem, where we sample two distinct odd primes $p,q$, we set $n=pq$, we generate $a \leftarrow \mathbb{Z}_{n^2}^*$ and finally we set $g = (n+1) a^n \mod{n^2}$.

My question now is if the discrete log with the parameters $(g, \mathbb{Z}_{n^2}^*)$ is known to be hard, i.e. if given $(g, g^x)$, one can simply recover $x$, where let's say $x \in \mathbb{Z}_n$. 

I know that we can apply Chinese Remainder Theorem in this case, but is this alone sufficient for breaking the Discrete Log?

Thank you very much!
Discrete logarithm can be hard in Paillier group. One case for example is when $p$ and $q$ are safe primes, i.e. $p=2p'+1$ and $q=2q'+1$ where $p',q'$ are themselves prime.

You can find more information about the critieria in this paper (and from its references): On the Portability of Generalized Schnorr Proofs, by Camenisch et al, Eurocrypt 2009