A Multi-Bit Fully Homomorphic Encryption with Shorter Public Key from LWE

The efficiency of fully homomorphic encryption is a big question at present. To improve efficiency of fully homomorphic encryption, we use the technique of packed ciphertext to construct a multi-bit fully homomorphic encryption based on Learning with Errors problem. Our scheme has a short public key. Since our fully homomorphic encryption scheme builds on the basic encryption scheme that choose Learning with Errors samples from Gaussian distribution and add Gaussian error to it, which result in that the number of Learning with Errors samples decrease from 2nlogq to n+1. We prove that our fully homomorphic encryption scheme is feasible and its security relies on the hardness of Learning with Errors problem. In addition we adapt the optimization for the process of key switching from GHS13 and formal this new process of key switching for multi-bit fully homomorphic encryption. At last, we analyze the concert parameters and compare these parameters between our scheme and GHS13 scheme. The data show that our scheme has public key smaller by a factor of about logq than it in GHS13 scheme.