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Online social networking applications have become more and more popular in the advance of the technological age. Much of our personal information has been disclosed in social networking activities and privacy-preserving still remains a research challenge in social network. Public key encryption scheme with equality test(PKEET), which is an extension of public key encryption with keyword search (PEKS), seems to be a solution. PKEET enables the tester to check whether two given ciphertexts are derived from the same plaintext. Recently, Zhu et al. proposed a pairing-free public key encryption scheme with equality test based on the traditional public key cryptosystem. However, it suffers from certificates management issue. In this paper, we propose a pairing-free identity-based encryption scheme with authorized equality test(PF-IBEAET). The PF-IBEAET scheme also provides fine-grained authorizations. We prove that the scheme is one way secure against chosen identity and chosen ciphertext attack (OW-ID-CCA) and indistinguishable against chosen-identity and chosen-ciphertext attack (IND-ID-CCA) in the random oracle model (ROM). Performance analysis shows that the scheme achieves a better performance than similar schemes.

The "if–then–else" construction is one of the most elementary programming commands, and its abstract laws have been widely studied, starting with McCarthy. Possibly, the most obvious extension of this is to include the operation of composition of programs, which gives a semigroup of functions (total, partial, or possibly general binary relations) that can be recombined using if–then–else. We show that this particular extension admits no finite complete axiomatization and instead focus on the case where composition of functions with predicates is also allowed (and we argue there is good reason to take this approach). In the case of total functions — modeling halting programs — we give a complete axiomatization for the theory in terms of a finite system of equations. We obtain a similar result when an operation of equality test and/or fixed point test is included.