In the article (https://eprint.iacr.org/2009/571.pdf, pag 8) of Smart and Vercauteren, it is mentioned that the recovery of the private key is an instance of the small principal ideal problem. But I dont see how this is possible, since this implies that we know the ring in which we work, i.e. we know the polynomial $F$ chosen in the KeyGen() algorithm. But this is not public information. Am I getting this the wrong way ?