We propose a lattice-based cryptosystem by modifying the Regev'05 cryptosystem (STOC 2005), 
and design a proof of secret-key knowledge. 
Lattice-based public-key identification schemes have already been proposed, however, 
it is unknown that their public keys can be used for the public keys of encryption schemes. 
Our modification admits the proof of knowledge on its secret key, however, we need a 
stronger assumption than that required by the original cryptosystem.