Can the Smart-Vercauteren Homomorphic Encryption be implemented as a practical fully homomorphic encryption scheme?
The SV scheme made some significant simplifications that turned out to hurt security.  Specifically, it's known how to break the scheme in subexponential time (roughly $2^{n^{2/3}}$) in the dimension $n$ of the lattice defined by the public key.  Moreover, it's known that the scheme can be broken in polynomial time by a quantum computer.  In short, the reason is that the SV scheme has an ad-hoc method of key generation, which tends to produce "weak" ideals that can be attacked more efficiently than arbitrary ideals.  See this paper for further details.