From this discussion, I know that if I know $\varphi (N)$ and $N$ (where $N=pq$, $p$ and $q$ prime), then I can very easily get $p$ and $q$.
Suppose I have the encrypted message $c$. I want to get the exponent $d$ such that $c^d \pmod N$ is the original message $m,$ without knowing $e.$
I know that it is theoretically possible to go through all the exponents from $1$ to $\varphi (N)$, and then see if the resulting message makes sense. But since I know $p$ and $q$ and $N$, I think there should be a better way than brute force.
I don't know if trying to factor $\varphi (N)$ is that much easier than trying to factor $N$? That means that even though not all numbers between $1$ and $\varphi (N)$ will have inverses (since they will not all be relatively prime to $\varphi (N)$) I won't know based on the information I have.
Is there a better approach to breaking the encryption?