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Homomorphic encryption (HE) is public key encryption that enables computation over ciphertexts without decrypting them, while it is known that HE cannot achieve IND-CCA2 security. To overcome this iss...ue, the notion of keyed-homomorphic encryption (KH-PKE) was introduced, which has a separate homomorphic evaluation key and can achieve stronger security (Emura et al., PKC 2013). The contributions of this paper are twofold. First, the syntax of KH-PKE supposes that homomor-phic evaluation is performed for single operations, and its security notion called KH-CCA security was formulated based on this syntax. Consequently, if the homomorphic evaluation algorithm is enhanced in a way of gathering up sequential operations as a single evaluation, then it is not obvious whether or not KH-CCA security is preserved. In this paper, we show that KH-CCA security is in general not preserved under such modication, while KH-CCA security is preserved when the original scheme additionally satises circuit privacy. Secondly, Catalano and Fiore (ACM CCS 2015) proposed a conversion method from linearly HE schemes into two-level HE schemes, the latter admitting addition and a single multiplication for ci-phertexts. In this paper, we extend the conversion to the case of linearly KH-PKE schemes to obtain two-level KH-PKE schemes. Moreover, based on the generalized version of Catalano{Fiore conversion, we also construct a similar conversion from d-level KH-PKE schemes into 2d-level KH-PKE schemes.show more
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