In this paper we present a new lattice construction for a lattice based 
partial key exposure attack for the RSA cryptography.
We consider the situation that the RSA secret key $d$ is
small and a sufficient amount of the LSBs (least significant bits)
of $d$ are known by the attacker.
We show that our lattice construction is theoretically more efficient than known attacks
proposed in \cite{BD00,EJMW05}.
This is a full version of the paper that appears in the proceedings of PKC 2009.