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The wireless medium contains domain-specific information that can be used to complement and enhance traditional security mechanisms. In this paper we propose ways to exploit the spatial variability of the radio channel response in a rich scattering environment, as is typical of indoor environments. Specifically, we describe a physical-layer authentication algorithm that utilizes channel probing and hypothesis testing to determine whether current and prior communication attempts are made by the same transmit terminal. In this way, legitimate users can be reliably authenticated and false users can be reliably detected. We analyze the ability of a receiver to discriminate between transmitters (users) according to their channel frequency responses. This work is based on a generalized channel response with both spatial and temporal variability, and considers correlations among the time, frequency and spatial domains. Simulation results, using the ray-tracing tool WiSE to generate the time-averaged response, verify the efficacy of the approach under realistic channel conditions, as well as its capability to work under unknown channel variations.