| Summary: | In multi-client functional encryption (MC-FE) for predicate queries, clients generate ciphertexts of plaintexts <inline-formula> <tex-math notation="LaTeX">$x_{1}, \ldots, x_{n}$ </tex-math></inline-formula> binding with a time period <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> and store them on a cloud server, and the cloud server receives a function key corresponding to a predicate <inline-formula> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> from a trusted center and learns whether <inline-formula> <tex-math notation="LaTeX">$f(x_{1}, \ldots, x_{n}) = 1$ </tex-math></inline-formula> or not by running the decryption algorithm on the multiple ciphertexts of the same time period. MC-FE for predicates can be used for a network event or medical data monitoring system based on time series data gathered by multiple clients. In this paper, we propose efficient MC-FE schemes that support conjunctive equality or range queries on encrypted data in the multi-client settings. First, we propose an efficient multi-client hidden vector encryption (MC-HVE) scheme in bilinear groups and prove the selective security with static corruptions. Our MC-HVE scheme is very efficient since a function key is composed of four group elements, a ciphertext consists of <inline-formula> <tex-math notation="LaTeX">$O(\ell)$ </tex-math></inline-formula> group elements where <inline-formula> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> is the size of a plaintext, and the decryption algorithm only requires four pairing operations. Second, we propose an efficient multi-client range query encryption (MC-RQE) scheme and prove the selective weak security with static corruptions. Our MC-RQE scheme which uses a binary tree is efficient since a ciphertext consists of <inline-formula> <tex-math notation="LaTeX">$O(\log D)$ </tex-math></inline-formula> group elements and a function key consists of <inline-formula> <tex-math notation="LaTeX">$O(n \log D)$ </tex-math></inline-formula> group elements where <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> is the maximum value of a range.
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