How to Securely Collaborate on Data: Decentralized Threshold HE and Secure Key Update

Threshold homomorphic encryption (Threshold HE) schemes are modified homomorphic encryption schemes to be suitable for privacy-preserving data integration and analysis. In actual usage of it, one should take it care into consideration who manages secret keys. In Eurocrypt 2012, Asharov et al. proposed decentralized (n, n)-threshold HE schemes in bottom-up approach for which all n parties must allow by doing a partial decryption to decrypt successfully a ciphertext. To support more general threshold structure for HE, Boneh et al. presented (t, n)-threshold HE schemes using secret sharing schemes in top-down approach with a central key dealer. In this article, decentralized (t, n)-threshold HE schemes in bottom-up approach will be constructed. The decentralized (n, n)-threshold HE scheme is fisrt modified to reduce the error contained in the common evaluation key which affects to the entire parameter size. Then by applying (t, n)-threshold secret sharing scheme, (n, n)-threshold HE scheme is converted to (t, n)-threshold HE scheme. Moreover, proactive secret sharing scheme is applied to update secret key share of the constructed (t, n)-threshold HE scheme whenever needed.