  Measurement-device-independent quantum key distribution (MDI-QKD) can provide
enhanced security, as compared to traditional QKD, and it constitutes an
important framework for a quantum network with an untrusted network server.
Still, a key assumption in MDI-QKD is that the sources are trusted. We propose
here a MDI quantum network with a single untrusted source. We have derived a
complete proof of the unconditional security of MDI-QKD with an untrusted
source. Using simulations, we have considered various real-life imperfections
in its implementation, and the simulation results show that MDI-QKD with an
untrusted source provides a key generation rate that is close to the rate of
initial MDI-QKD in the asymptotic setting. Our work proves the feasibility of
the realization of a quantum network. The network users need only low-cost
modulation devices, and they can share both an expensive detector and a
complicated laser provided by an untrusted network server.
