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Plaintext checkable encryption with check delegation revisited

Plaintext checkable encryption (PCE) is a new notion which provides an add-on checkability on traditional public key encryption. In general, PCE allows that given a target plaintext, a ciphertext and a public key, the specific check algorithm can verify whether the target plaintext is identical to the underlying plaintext of the ciphertext with the public key. With the new functionality of checkability, PCE cannot achieve the CPA security undoubtedly. To achieve the CPA security, we can change the system framework and then restrict that only the specific and authenticated checker can perform such the check algorithm. In addition, the sender who encrypts a message does not need to have any information or identity of any checker. In this paper, we revisit PCE with designated checker. We formally present two new definitions, PCEAC and PCEDC, and, we propose two constructions. Finally, an extension of the definition is formalised to capture checkability only for a set of plaintexts.