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To this day, there are still some countries where the exchange of kidneys between multiple incompatible patient-donor pairs is restricted by law. Typically, legal regulations in this context are put in place to prohibit coercion and manipulation in order to prevent a market for organ trade. Yet, in countries where kidney exchange is practiced, existing platforms to facilitate such exchanges generally lack sufficient privacy mechanisms. In this paper, we propose a privacy-preserving protocol for kidney exchange that not only addresses the privacy problem of existing platforms but also is geared to lead the way in overcoming legal issues in those countries where kidney exchange is still not practiced. In our approach, we use the concept of secret sharing to distribute the medical data of patients and donors among a set of computing peers in a privacy-preserving fashion. These computing peers then execute our new Secure Multi-Party Computation (SMPC) protocol among each other to determine an optimal set of kidney exchanges. As part of our new protocol, we devise a privacy-preserving solution to the maximum matching problem on general graphs. We have implemented the protocol in the SMPC benchmarking framework MP-SPDZ and provide a comprehensive performance evaluation. Furthermore, we analyze the practicality of our protocol when used in a dynamic setting (where patients and donors arrive and depart over time) based on a data set from the United Network for Organ Sharing.