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Image hash functions find extensive applications in content authentication, database search, and watermarking. This paper develops a novel algorithm for generating an image hash based on Fourier transform features and controlled randomization. We formulate the robustness of image hashing as a hypothesis testing problem and evaluate the performance under various image processing operations. We show that the proposed hash function is resilient to content-preserving modifications, such as moderate geometric and filtering distortions. We introduce a general framework to study and evaluate the security of image hashing systems. Under this new framework, we model the hash values as random variables and quantify its uncertainty in terms of differential entropy. Using this security framework, we analyze the security of the proposed schemes and several existing representative methods for image hashing. We then examine the security versus robustness tradeoff and show that the proposed hashing methods can provide excellent security and robustness.