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.