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In this paper, a new image hashing scheme satisfying robustness and security is proposed. We exploit the property of dimensionality reduction inherent in compressive sensing/sampling (CS) for image hash design. The gained benefits include (1) the hash size can be kept small and (2) the CS-based hash is computationally secure. We study the use of visual information fidelity (VIF) for hash comparison under Stirmark attacks. We further derive the relationships between the hash of an image and both of its MSE distortion and visual quality measured by VIF, respectively. Hence, based on hash comparisons, both the distortion and visual quality of a query image can be approximately estimated without accessing its original version. We also derive the minimum distortion for manipulating an image to be unauthentic to measure the security of our scheme.