  This paper studies how a system operator and a set of agents securely execute
a distributed projected gradient-based algorithm. In particular, each
participant holds a set of problem coefficients and/or states whose values are
private to the data owner. The concerned problem raises two questions: how to
securely compute given functions; and which functions should be computed in the
first place. For the first question, by using the techniques of homomorphic
encryption, we propose novel algorithms which can achieve secure multiparty
computation with perfect correctness. For the second question, we identify a
class of functions which can be securely computed. The correctness and
computational efficiency of the proposed algorithms are verified by two case
studies of power systems, one on a demand response problem and the other on an
optimal power flow problem.
