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An image hash is a randomized compact representation of image content and finds applications in image authentication, image and video watermarking, and image similarity comparison. Usually, an image-hashing scheme is required to be robust and secure, and the security issue is particularly important in applications, such as multimedia authentication, watermarking, and fingerprinting. In this paper, we investigate the security of image hashing from the perspective of unicity distance, a concept pioneered by Shannon in one of his seminal papers. Using two recently proposed image-hashing schemes as representatives, we show that the concept of unicity distance can be adapted to evaluate the security of image hashing. Our analysis shows that the secret hashing key, or its equivalent form, can be estimated with high accuracy when the key is reused several dozen times. The estimated unicity distance determines the maximum number of key reuses in the investigated hashing schemes. A countermeasure of randomized key initialization is discussed to avoid key reuse and strengthen the security of robust image hashing.