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Detect Zero by Using Symmetric Homomorphic Encryption

First, we prove that a homomorphic encryption with a function of detecting zero, detecting equality, comparing the value or detecting overflow on cipher texts is not secure if there is no restriction to limit the times of operating these functions. Then we analyze that a homomorphic encryption scheme can still detect zero with some restrictions. If we allow all the people to detect zero on cipher texts, then there is no method better than the key-owner decrypts the cipher text and announces the result. However, if we allow only few people to detect zero when key owner agrees, then we can construct a symmetric homomorphic encryption scheme with our new version of ring learn with error assumption.