Homomorphic encryption allows computation over encrypted data, and can be used for delegating computation: data providers encrypt their data and send them to an aggregator, who can then perform computation over the encrypted data on behalf of a client, without the underlying data being exposed to the aggregator. However, since the aggregator is merely a third party, it may be malicious, and in particular, may submit an incorrect aggregation result to the receiver. Ohara et al. (APKC2014) studied secure aggregation of time-series data while enabling the correctness of aggregation to be verified. However, they only provided a concrete construction in the smart metering system and only gave an intuitive argument of security. In this paper, we define verifiable homomorphic encryption (VHE) which generalizes their scheme, and introduce formal security definitions. Further, we formally prove that Ohara et al.'s VHE scheme satisfies our proposed security definitions.

Replayable chosen ciphertext (RCCA) security was introduced by Canetti, Krawczyk, and Nielsen (CRYPTO'03) in order to handle an encryption scheme that is “non-malleable except tampering which preserves the plaintext.” RCCA security is a relaxation of CCA security and a useful security notion for many practical applications such as authentication and key exchange. Canetti et al. defined non-malleability against RCCA (NM-RCCA), indistinguishability against RCCA (IND-RCCA), and universal composability against RCCA (UC-RCCA). Moreover, they proved that these three security notions are equivalent when considering a PKE scheme whose plaintext space is super-polynomially large. Among these three security notions, NM-RCCA seems to play the central role since RCCA security was introduced in order to capture “non-malleability except tampering which preserves the plaintext.” However, their definition of NM-RCCA is not a natural extension of that of original non-malleability, and it is not clear whether their NM-RCCA captures the requirement of original non-malleability. In this paper, we propose definitions of indistinguishability-based and simulation-based non-malleability against RCCA by extending definitions of original non-malleability. We then prove that these two notions of non-malleability and IND-RCCA are equivalent regardless of the size of plaintext space of PKE schemes.