In this paper, we propose two new chosen-ciphertext (CCA) secure schemes from the computational Diffie-Hellman (CDH) and bilinear computational Diffie-Hellman (BCDH) assumptions. Our first scheme from the CDH assumption is constructed by extending Cash-Kiltz-Shoup scheme. This scheme yields the same ciphertext as that of Hanaoka-Kurosawa scheme (and thus Cramer-Shoup scheme) with cheaper computational cost for encryption. However, key size is still the same as that of Hanaoka-Kurosawa scheme. Our second scheme from the BCDH assumption is constructed by extending Boyen-Mei-Waters scheme. Though this scheme requires a stronger underlying assumption than the CDH assumption, it yields significantly shorter key size for both public and secret keys. Furthermore, ciphertext length of our second scheme is the same as that of the original Boyen-Mei-Waters scheme.