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In a multimedia cloud computing system, suppose all cloud users outsource their own data sets to the cloud in the encrypted form. Each outsourced set is associated with an access structure such that a valid data user, Bob, with the credentials satisfying the access structure is able to conduct computing over outsourced encrypted set (e.g., decryption or other kinds of computing function). Suppose Bob needs to compute the set intersection over a data owner Alice’s and his own outsourced encrypted sets. Bob’s simple solution is to download Alice’s and Bob’s outsourced encrypted sets, perform set intersection operation, and decrypt the set intersection ciphertexts. A better solution is for Bob to delegate the cloud to calculate the set intersection, without giving the cloud any ability in breaching the secrecy of the sets. To solve this problem, this work introduces a novel primitive called ciphertext-policy attribute-based encryption with outsourced set intersection for multimedia cloud computing. It is the first cryptographic algorithm supporting a fully outsourced encrypted storage, computation delegation, fine-grained authorization security for ciphertext-policy model, without relying on an online trusted authority or data owners, and multi-elements set, simultaneously. We construct a scheme that provably satisfies the desirable security properties, and analyze its efficiency.