种基于 LWE 问题的布尔电路同态加密方案

Traditional cryptography can only protect the security of data in storage and transmission, and the ciphertext holders cannot operate encrypted data directly. In 2009, the emergence of the first FHE scheme makes it possible for the ciphertext holders to perform the ciphertext operation directly. Based on the GSW FHE scheme, this paper redesigns the key generation, encryption, decryption, and homomorphic operation functions, proposes an improved Boolean circuit homomorphic encryption scheme. The homomorphic addition and homomorphic multiplication of the improved scheme are corresponding to the addition and multiplication of the matrix, which does not result in the expansion of ciphertext dimension. In this paper, we propose a modules dimension reduction method aiming at designing the scheme and its corresponding correctness analysis. The homomorphic encryption scheme is proved to be correct and secure. The analysis shows that the security of the improved scheme depends on the LWE problem, and can resists CPA attack. Compared with the GSW scheme, the improved scheme can be changed more naturally to a FHE scheme with fast bootstrapping method proposed by Peikert et al in 2014. In addition, this paper provides the selection rules of the parameters, implements the scheme, the AND, OR, XOR and other homomorphic computing gates of the scheme, gives the main parameters and calculation time, which makes essential foreshadowing for the forthcoming applications of the scheme.