One of the basic motivations for solving DCOPs is maintaining agents' privacy. Thus, researchers have evaluated the privacy loss of DCOP algorithms and defined corresponding notions of privacy preservation for secured DCOP algorithms. However, no secured protocol was proposed for Max-Sum, which is among the most studied DCOP algorithms. As part of the ongoing effort of designing secure DCOP algorithms, we propose P-Max-Sum, the first private algorithm that is based on Max-Sum. The proposed algorithm has multiple agents preforming the role of each node in the factor graph, on which the Max-Sum algorithm operates. P-Max-Sum preserves three types of privacy: topology privacy, constraint privacy, and assignment/decision privacy. By allowing a single call to a trusted coordinator, P-Max-Sum also preserves agent privacy. The two main cryptographic means that enable this privacy preservation are secret sharing and homomorphic encryption. In addition, we design privacy-preserving implementations of four variants of Max-Sum. We conclude by analyzing the price of privacy in terns of runtime overhead, both theoretically and by extensive experimentation.