Cloud-based control is prevalent in many modern control applications. Such applications require security for the sake of data secrecy and system safety. The presented research proposes an encrypted adaptive control framework that can be secured for cloud computing with encryption and without issues caused by encryption overflow and large execution delays. This objective is accomplished by implementing a somewhat homomorphic encryption (SHE) scheme on a modified model reference adaptive controller with accompanying encryption parameter tuning rules. Additionally, this paper proposes a virtual false data injection attack (FDIA) trap based on the SHE scheme. The trap guarantees a probability of attack detection by the adjustment of encryption parameters, thus protecting the system from malicious third parties. The formulated algorithm is then simulated, verifying that after tuning encryption parameters, the encrypted controller produces desired plant outputs while guaranteeing detection or compensation of FDIAs.