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Plaintext-checkable encryption (PCE), first introduced by Canard et al., allows users searching on encrypted data through plaintexts. It provides a useful primitive in the cloud computing security. However, existing PCE schemes could not guarantee verifiability to prevent from malicious adversaries. To concentrate with, we provide the notion of plaintext-verifiably-checkable encryption (PVCE). Our PVCE scheme could check that if a ciphertext is valid, even though it could pass the check procedure, to avoid malicious ciphertexts. Furthermore, we extend the work in dual-server setting, called dual-server plaintext-verifiably-checkable encryption (DS-PVCE), to resist offline message recover attack. We also give the security definition IND-CCA-FS, IND-CCA-BS and IND-CCA-BS-II in standard model for DS-PVCE and prove that DS-PVCE is secure under these security notions. Then we give constructions of PVCE and DS-PVCE from pairing-friendly smooth projective hash function (PF-SPHF) and their instantiations based on k-MDDH assumption. At the end of this paper, we provide implementation to compare existing PCE schemes and our schemes, which shows that DS-PVCE has very high check efficiency compared with other PCE schemes.

With the higher rate of using cloud storage, protecting data privacy becomes an important issue. The most effective solution is to encrypt data before uploading to the cloud. However, how to efficiently search over data encrypted with different keys is still an open problem. To address this problem, we introduce a new notion of the identity-based encryption with equality test supporting flexible authorization (IBEET-FA). It supports the test of whether two ciphertexts encrypted under the different keys encapsulate the same message, and in the meanwhile supports fine-grained authorization of the test. Based on the equality test on ciphertexts, there is a direct way to support an authorized user to search over ciphertexts of different users, which accelerates secret data sharing among a group of users. Besides, IBEET-FA does not suffer from the complex key management problem of its counterpart in the traditional public key infrastructure. We propose a concrete construction of IBEET-FA and prove it to be securely based on simple mathematical assumptions. The experimental results show that our IBEET-FA scheme is efficient and can satisfy various types of search over encrypted data.