Multi-hop Multi-key Homomorphic Encryption with Less Noise Under CRS Model

The application of information technologies such as big data, artificial intelligence, and cloud computing in the Internet environment makes protecting personal privacy and preventing data leakage an important issue. Homomorphic encryption (HE) supports the calculation and processing on ciphertexts and provides a technical guarantee for the security of personal information. Multi-key homomorphic encryption (MKHE) can perform cooperative calculation on ciphertexts under different keys, and is one of the solutions for secure computing in multi-user scenario. But the homomorphic multiplication after ciphertext extension makes the ciphertext dimension increase quadratically with respect to the number of users, so relinearization with noise is needed to reduce the dimension. In this paper, we propose an efficient multi-hop MKHE scheme with less relinearization noise under the common reference string (CRS) model, which supports real-time joining of new users. First, we propose an auxiliary coding scheme by introducing a random matrix instead of the encryption with noise. On this basis, we further propose how to construct the relinearization key, which can be pre-computed before the ciphertext input. In our MKHE scheme, the preprocessing of the ciphertext is cancelled to reduce the dimension of the ciphertext and the number of non-linear keys after homomorphic multiplication. In addition, we prove the semantic security of the scheme under the RLWE assumption, noise analysis and storage analysis also show that our scheme has smaller noise growth and storage space.