Enhancing paillier to fully homomorphic encryption with semi-honest TEE

Homomorphic encryption (HE) is an innovative privacy protection technique supporting homomorphic addition and multiplication. It has been widely applied in the applications of peer-to-peer networks, such as secure data sharing and privacy-preserving search. Existing HE schemes can be roughly categorized into partially HE and fully HE (FHE). The former is computationally efficient but only supports either additive or multiplicative homomorphic operations. The latter can simultaneously support both additive and multiplicative operations, but the corresponding computational costs are intensive. Recently, some works leverage trusted execution environment (TEE) to optimize the efficiency of FHE. However, they suffer from the limitations of ciphertext expansion and the strong trust assumption for TEE. To address these limitations, we present a new fully homomorphic encryption scheme named Paillier FHE (PFHE) by employing TEE to extend the additive Paillier HE to support multiplicative operations and further optimizing the computational efficiency, where TEE is assumed to be semi-honest to avoid the strong trust assumption. Specifically, we first design a Paillier multiplication protocol (PMUL) to achieve the ciphertext multiplication without bootstrapping. Based on the protocol, we utilize the packing technique to design a Paillier inner product protocol (PVMUL) and a Paillier matrix multiplication protocol (PMMUL) to support the inner product and matrix multiplication operations efficiently. Moreover, we provide the detailed security analysis for our protocols. We compare our PFHE with typical fully homomorphic encryption libraries by experiments, and at the same security level, our scheme demonstrates significant advantages.