Abstract: Homomorphic encryption scheme is a powerful cryptographic system which allows for a variety of applications. Fully homomorphic encryption(FHE) permits arbitrary computations on encrypted data. The recent breakthrough work in 2009 by Craig Gentry has shown the possibility of FHE schemes, and has provided the first construction. Consequently, during the past five years, numerous FHE involving novel mathematical techniques and a number of application schemes have appeared. Indeed, the construction and application of homomorphic encryption schemes have great theoretic and practical meaning. Homomorphic encryption has important applications in cloud computing. However, almost all of the homomorphic encryption schemes share two common flaws that the multiplication depth must be set in advance and they all use secret keys of large scales. We construct a circularly secure re-linearization process based on the “special b” variant of the learning with errors problem(bLWE). Then, we present an efficient homomorphic encryption scheme. Compared with Brakerski et al’s scheme, our scheme reduces the L+1 secret keys to one and doesn’t need to know the multiplication depth in advance. Finally, we prove the chosen-plaintext attack(CPA) security of the homomorphic scheme and the circular security of the re-linearization process in standard model by reducing them into learning with errors problem(LWE) assumption.

Key words: circular security, homomorphic encryption, learning with errors problem(LWE), lattice, standard model