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Unmanned Aerial Vehicles (UAVs), or drones, are increasingly being utilized for public safety circumstances including post-disaster recovery of destroyed communication infrastructure. For instance, drones are temporarily positioned within an affected area to create a wireless mesh network among public safety personnel. To serve the need for high-rate video-based damage assessment, drone-assisted communication can utilize high- bandwidth millimeter wave (mmWave) technologies such as IEEE 802.11ad. However, short-range mmWave communication makes it hard for optimally- positioned drones to be authenticated with a centralized network control center. Therefore and assuming that there are potential imposters, we propose two lightweight and fast authentication mechanisms that take into account the physical limitations of mmWave communication. First, we propose a drone-to-drone authentication mechanism, which is based on proxy signatures from a control center. Accordingly, any newly joining drone can authenticate itself to an exist one rather than attempting to authenticate to the outof-reach control center. Second, we propose a drone-to- ground authentication mechanism, to enable each drone to authenticate itself to its associated ground users. Such authentication approach is based on challenge-response broadcast type, and it is still utilizing fast proxy signature approach. The evaluation of the proposed authentication mechanisms, conducted using NS-3 implementation of IEEE 802.11ad protocol, show their efficiency and practicality.