Similarity calculation based on homomorphic encryption

In recent years, some homomorphic encryption algorithms have been proposed to provide additive homomorphic encryption and multiplicative homomorphic encryption. However, similarity measures are required for searches and queries under homomorphic encrypted ciphertexts. Therefore, this study considers cosine similarity, angular similarity, Tanimoto similarity, and soft cosine similarity and combines homomorphic encryption algorithms for similarity calculation to propose homomorphic encryption-based cosine similarity (HE-CS), homomorphic encryption-based angular similarity (HE-AS), homomorphic encryption-based Tanimoto similarity (HE-TS), and homomorphic encryption-based soft cosine similarity (HE-SCS). This study proposes mathematical models to prove the proposed homomorphic encryption-based similarity calculation methods and gives practical cases to explain the feasibility of the proposed HE-CS, HE-AS, HE-TS, and HE-SCS. Furthermore, this study proposes normalized entropy and normalized Gini impurity as evaluation factors to measure the randomness and confusion of ciphertext. In experiments, the values of normalized entropy and normalized Gini impurity are higher than 0.999, which indicates significant differences between plaintexts and ciphertexts. Moreover, the encryption time and decryption time of the proposed homomorphic encryption-based similarity calculation methods have been evaluated under different security strengths.