InstaHide's Sample Complexity When Mixing Two Private Images
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Inspired by InstaHide challenge [Huang, Song, Li and Arora'20], [Chen, Song and Zhuo'20] recently provides one mathematical formulation of InstaHide attack problem under Gaussian images distribution. They show that it suffices to use O(n_𝗉𝗋𝗂𝗏^k_𝗉𝗋𝗂𝗏 - 2/(k_𝗉𝗋𝗂𝗏 + 1)) samples to recover one private image in n_𝗉𝗋𝗂𝗏^O(k_𝗉𝗋𝗂𝗏) + poly(n_𝗉𝗎𝖻) time for any integer k_𝗉𝗋𝗂𝗏, where n_𝗉𝗋𝗂𝗏 and n_𝗉𝗎𝖻 denote the number of images used in the private and the public dataset to generate a mixed image sample. Under the current setup for the InstaHide challenge of mixing two private images (k_𝗉𝗋𝗂𝗏 = 2), this means n_𝗉𝗋𝗂𝗏^4/3 samples are sufficient to recover a private image. In this work, we show that n_𝗉𝗋𝗂𝗏log ( n_𝗉𝗋𝗂𝗏 ) samples are sufficient (information-theoretically) for recovering all the private images.
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