Decode message $m = p * q$, where $p ^ 5 \bmod N$, $q ^ 5 \bmod N$ and $N$ are known I been going through some cryptography exercises and stumbled across this problem.


  Discover message $m$. You know that $m = p * q$. Also $p ^ 5 \bmod N$,
  $q ^ 5 \bmod N$ and $N$ are known numbers.


I am really lost and don't know where to start. Could anyone give some hints or point me to right direction where to read up on this problem?
I don't have a full answer yet but here is what I have so far.  Since you know what $p^5 \bmod N$ is and what $q^5 \bmod N$ is then you know what $m^5 \bmod N$ is. Since we also know what $N$ is then we should be able to find out how many times we need multiply $m^5 \bmod N$ by $m^5 \bmod N$ till we get a full cycle (you can calculate it through $\phi(N)$ or you can multiply by m till you get back to your original number).  From that cycle, go back 4 terms and that should be what $m \bmod N$ is.
Update: Finding $\phi(N)$ can be difficult for sufficiently large $N$ but multiplication with mods is a fairly easy cheap.  First make sure that N and 5 are relatively prime, and then (for practicality purposes) repeatedly iterate $F: x\mapsto x^5\bmod N$, starting with the known $x_0=m^5\bmod N$ until $x_{j+1}=x_0$, thus making $x_{j}$ a possible $m$