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Fragile watermarking is a technology for detecting manipulations in an image, such as the digital signature in cryptography. However, the classification of attacks is not fixed in the security evaluation of existing fragile watermarking schemes, although many schemes have been proposed. In this paper, typical fragile watermarking schemes using a public key cryptosystem are evaluated from the viewpoint of a practical watermarking service. In this evaluation, nine attacks on watermarked images are defined and are classified as: watermarked image attacks, similar to cipher text attacks in cryptography; known original image attacks, similar to known plain text attacks; chosen original image attacks, similar to chosen plain text attacks. We show that the existing schemes are not secure against all attacks and propose a new scheme that is secure against such attacks and useful for a practical watermarking service.