Homomorphic encryption (HE) protects data in-use, but can be computationally expensive. To avoid the costly bootstrapping procedure that refreshes ciphertexts, some works have explored client-aided outsourcing protocols, where the client intermittently refreshes ciphertexts for a server that is performing homomorphic computations. But is this approach secure against malicious servers? We present a CPA-secure encryption scheme that is completely insecure in this setting. We define a new notion

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... security, called funcCPA, that we prove is sufficient. Additionally, we show: -Homomorphic encryption schemes that have a certain type of circuit privacy -for example, schemes in which ciphertexts can be "sanitized" -are funcCPA-secure. -In particular, assuming certain existing HE schemes are CPA-secure, they are also funcCPA-secure. -For certain encryption schemes, like Brakerski-Vaikuntanathan, that have a property that we call oblivious secret key extraction, funcCPAsecurity implies circular security -i.e., that it is secure to provide an encryption of the secret key in a form usable for bootstrapping (to construct fully homomorphic encryption). In summary, funcCPA-security lies strictly between CPA-security and CCA2-security (under reasonable assumptions), and has an interesting relationship with circular security, though it is not known to be equivalent.