Let's suppose we have a asymmetric crypto-system $H$ which is homomorphic with respect to some function $F$.
- Alice encrypts a message $m$ with her private key $e$ in the crypto-system $H$ and obtains the ciphertext $C = H(M)$
- Alice sends $C$ to Bob
- Bob computes $F$ over $C$ and obtains $C' = F(C)$
- Bob sends $C'$ to Alice
- Alice wants to check whether $C'$ is equal to some number $n$.
Let's suppose the decryption operation is costly, much more costly than the encryption. Can Alice check the equality by comparing the encryption of $n (H(n))$ and $C'$? Is that possible? With which homomorphic crypto-system(s)?