  Contextual bandit is a general framework for online learning in sequential
decision-making problems that has found application in a wide range of domains,
including recommendation systems, online advertising, and clinical trials.
  A critical aspect of bandit methods is that they require to observe the
contexts --i.e., individual or group-level data-- and rewards in order to solve
the sequential problem. The large deployment in industrial applications has
increased interest in methods that preserve the users' privacy. In this paper,
we introduce a privacy-preserving bandit framework based on homomorphic
encryption{\color{violet} which allows computations using encrypted data}. The
algorithm \textit{only} observes encrypted information (contexts and rewards)
and has no ability to decrypt it. Leveraging the properties of homomorphic
encryption, we show that despite the complexity of the setting, it is possible
to solve linear contextual bandits over encrypted data with a
$\widetilde{O}(d\sqrt{T})$ regret bound in any linear contextual bandit
problem, while keeping data encrypted.
