Homomorphic encryption can support meaningful operations on encrypted data, and hence it enables users to outsource their data in encrypted format to cloud services. However, homomorphic encryption cannot operate on ciphertexts with different keys in general. To resolve the problem, re-encryption allows operations on such ciphertexts by unifying the different keys into a new one. In this paper, we focus on the scheme proposed by Brakerski and Vaikuntanathan (CRYPTO 2011), and give a practical re-encryption method. Our strategy for efficient re-encryption is to simply rewrite the decryption circuit and then to evaluate the circuit homomorphically with auxiliary information. In particular, our method requires only a few homomorphic operations for re-encryption, and it can be applied to various applications such as secure key exchange for collaboratively computing multiple users' data in the cloud.