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The development of large-scale distributed control systems has led to theoutsourcing of costly computations to cloud-computing platforms, as well as toconcerns about privacy of the collected sensitive data. This paper develops acloud-based protocol for a quadratic optimization problem involving multipleparties, each holding information it seeks to maintain private. The protocol isbased on the projected gradient ascent on the Lagrange dual problem andexploits partially homomorphic encryption and secure multi-party computationtechniques. Using formal cryptographic definitions of indistinguishability, theprotocol is shown to achieve computational privacy, i.e., there is nocomputationally efficient algorithm that any involved party can employ toobtain private information beyond what can be inferred from the party's inputsand outputs only. In order to reduce the communication complexity of theproposed protocol, we introduced a variant that achieves this objective at theexpense of weaker privacy guarantees. We discuss in detail the computationaland communication complexity properties of both algorithms theoretically andalso through implementations. We conclude the paper with a discussion oncomputational privacy and other notions of privacy such as the non-uniqueretrieval of the private information from the protocol outputs.

Cloud-based Quadratic Optimization with Partially Homomorphic Encryption