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In some applications, such as surveillance cameras, it is essential that images acquired with the same device be deemed distinct. To fulfil this requirement, many fragile watermarking methods need to employ a different key for every single image. Keeping track of the correct key associated with each image may become a challenging task as the number of images increases. An existing image index-based scheme as a suitable solution for this problem is revisited, and its security limitations in applications are highlighted where higher localisation accuracy is required. Then, a new fragile watermarking scheme that enhances the localisation accuracy is proposed, while hindering brute force attacks. Experiments demonstrate that the proposed scheme outperforms the state-of-the-art private fragile watermarking methods.