On the CCA-1 security of somewhat homomorphic encryption over the integers

The notion of fully homomorphic encryption is very important since it enables many important applications, such as the cloud computing scenario. In EUROCRYPT 2010, van Dijk, Gentry, Halevi and Vaikuntanathan proposed an interesting fully homomorphic encryption scheme based on a somewhat homomorphic encryption scheme using integers. In this paper, we demonstrate a very practical CCA-1 attack against this somewhat homomorphic encryption scheme. Given a decryption oracle, we show that within O(λ2) queries, we can recover the secret key successfully, where λ is the security parameter for the system.