For the NTRU cryptosystem (as described here http://en.wikipedia.org/wiki/NTRUEncrypt), why is it really easy for Eve to decrypt if $p$ divides $q$.
My answer was that when Eve sees $e(x)= p*h(x)*r(x) + m \mod q$, Eve can multiple $e$ by $k$, where $kp=q$, to obtain $km$, but this doesn't work because then she will only have $km$ which she can't decrypt any further.
Maybe she can't recover the plaintext from there, but having knowledge of the quantity $km \mbox{ mod }q$ can lead to many security issues (mainly, because the private key is not involved anymore). See for example the One Time Pad scheme when it's used more than once, the plaintext it's not revealed, but there's a big leak of information, enough to completely break the scheme