  At EUROCRYPT 2011, Gentry and Halevi implemented a variant of Gentry's fully
homomorphic encryption scheme. The core part in their key generation is to
generate an odd-determinant ideal lattice having a particular type of Hermite
Normal Form. However, they did not give a rigorous proof for the correctness.
We present a better key generation algorithm, improving their algorithm from
two aspects.
  -We show how to deterministically generate ideal lattices with odd
determinant, thus increasing the success probability close to 1.
  -We give a rigorous proof for the correctness. To be more specific, we
present a simpler condition for checking whether the ideal lattice has the
desired Hermite Normal Form. Furthermore, our condition can be checked more
efficiently.
  As a result, our key generation is about 1.5 times faster. We also give
experimental results supporting our claims. Our optimizations are based on the
properties of ideal lattices, which might be of independent interests.
