Message authentication is a process that allows sender Alice to transmit a source state to receiver Bob such that the latter is assured of the authenticity. We study this problem in the physical-layer where the channel is degraded by a Rayleigh fading and a Gaussian noise. We propose an efficient construction for our problem. Our idea is to amplify a basic codeword of the source state with a small but dynamic scaling factor. This scaling factor is asymptotically 1. It thus essentially incurs no power expansion. Our authentication is achieved through a statistical test. Our scheme is provably secure and performs better than the related existing schemes. Further, our error analysis allows it to be realized with very practical parameters.