  We consider an architecture of confidential cloud-based control synthesis
based on Homomorphic Encryption (HE). Our study is motivated by the recent
surge of data-driven control such as deep reinforcement learning, whose heavy
computational requirements often necessitate an outsourcing to the third party
server. To achieve more flexibility than Partially Homomorphic Encryption (PHE)
and less computational overhead than Fully Homomorphic Encryption (FHE), we
consider a Reinforcement Learning (RL) architecture over Leveled Homomorphic
Encryption (LHE). We first show that the impact of the encryption noise under
the Cheon-Kim-Kim-Song (CKKS) encryption scheme on the convergence of the
model-based tabular Value Iteration (VI) can be analytically bounded. We also
consider secure implementations of TD(0), SARSA(0) and Z-learning algorithms
over the CKKS scheme, where we numerically demonstrate that the effects of the
encryption noise on these algorithms are also minimal.
