Lattice Attacks and Protection of Homomorphic Encryption Algorithm in Association Rule Mining Privacy Protection Schemes

Homomorphic Encryption (HE) is a kind of algorithm which provides data processing but not data access. Since it was proposed in 1978, as one of the important tools in cryptography, it is broadly used in many scenarios, like privacy protection, cloud computing, federated learning, and so on. Especially in the association rules mining privacy protection schemes, it often used as a key technology to ensure data security. Recently, Li et al. and Rajasekaran et al. introduced a kind of symmetric HE algorithm in their privacy protection scheme. However, in this paper, we find that this symmetric HE algorithm has the possibility to recover its secret key in practical applications. We propose two attacking algorithms based on lattice to recover its secret key SK=(sd,q). The core of our attack is to construct a lattice basis using the transformation relation between ciphertexts so that the short vector in the lattice contains the secret key SK. Then we can use the LLL algorithm to recover the secret key. We prove the feasibility of our attacking algorithms with experiments and the experimental results suggest that all of our algorithms can recover the key within 0.1s. Besides, we also give some improvements for this symmetric HE algorithm so that the new HE algorithm can resist our attacking algorithms.