Lets say the server has corpus of ciphertext contains $enc(a),enc(b),enc(c), \dotsc enc(x)$. The encryption function is an additive homomorphic scheme (like Paillier). The server knows only the public key. The client holds both public and private keys.
Is there a way for client to identify if there is an encryption of $0$ in the corpus? i.e. is any of $a,b,c \dotsc x$ is a $0$. A trivial solution is for the client to iteratively retrieve each ciphertext and decrypt it to see if it is $0$. Can we do any better ? i.e. without iteratively retrieving each element or downloading the entire corpus of ciphertext ?
If it were a multiplicative homomorphic scheme, the server could multiply all the encryptions and give the result i.e. $enc(res) = enc(a)\times enc(b) \times \dotsc enc(x)$ , the client could just decrypt the result to see if the result is $0$ . i.e. $dec(enc(res)) = 0$ if at least one of $a,b,...,x = 0$.
Are there any such tricks possible in additive homomorphic scheme ?