Abstract: The BGN (Boneh-Goh-Nissim) cryptosystem is a cryptosystem that permits arbitrary number of additions and one multiplication of ciphertext without growing the size of ciphertext. The scheme of BGV12 is a fully homomorphic encryption from (G)LWE which needs key switching, modulus switching and other technologies for the multiplicative homomorphism. This paper describes a BGN scheme based on BGV12. Although our constructed scheme only permits one multiplication, it does not need other technologies, so it is more efficient. Comparing with the scheme of GVH10, our scheme has better size of parameter. In addition, we extend our scheme to a threshold encryption scheme, which allows parties to cooperatively decrypt a ciphertext without learning anything but the plaintext, and can be protected from related-key attacks.