Is the following possible with homomorphic encryption (or some other technique)?
Suppose the cloud server keeps a key-value list for a user with each value field encrypted (prv key with user). Now, the cloud server needs to include some of its own key-value pairs to this list and sign the whole list homomorphically and return the list to the user. The major objective is to keep the user values hidden from a cloud server.
$$\{ K_1: \{V_1\}_{U}, K_2: \{V_2\}_{U} \}$$ Or
$$\{ \{K_1: V_1\}_{U}, \{K_2: V_2\}_{U} \}$$
is turned to
$$\{ \{ K_1: V_1, K_2: V_2 \}_{CS} \}_{U}$$
$$ $$ Let me rephrase...
Party 1 (user) encrypts few items $V_1, V_2, V_3$ using homomorphic encryption key $U$ to yield $\{V_1\}_{U}, \{V_2\}_{U}, \{V_3\}_{U}$ and sends them to party 2 (cloud server).
Since, it is homomorphic encryption, party 2 can multiply ciphertexts $\{V_1\}_{U} × \{V_2\}_{U} ×\{V_3\}_{U}$ to obtain $\{V_1, V_2, V_3\}_{U}$. Right?
My question is, can it do something more with the available ciphertexts?
Can party 2 compute $\{\ \{V_1, V_2, V_3\}_{P2}\ \}_{U}$ ?