We present the concept of key-rotatable and security-updatable homomorphic encryption (KR-SU-HE) scheme, which is defined as a class of public-key homomorphic encryption in which the keys and the security of any ciphertext can be rotated and updated while still keeping the underlying plaintext intact and unrevealed. After formalising the syntax and security notions for KR-SU-HE schemes, we build a concrete scheme based on the Learning With Errors assumption. We then perform several careful implementations and optimizations to show that our proposed scheme is efficiently practical.