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 $O(n_{\mathsf{priv}}^{k_{\mathsf{priv}} - 2/(k_{\mathsf{priv}} + 1)})$ samples to recover one private image in $n_{\mathsf{priv}}^{O(k_{\mathsf{priv}})} + \mathrm{poly}(n_{\mathsf{pub}})$ time for any integer $k_{\mathsf{priv}}$, where $n_{\mathsf{priv}}$ and $n_{\mathsf{pub}}$ 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 ($k_{\mathsf{priv}} = 2$), this means $n_{\mathsf{priv}}^{4/3}$ samples are sufficient to recover a private image. In this work, we show that $n_{\mathsf{priv}} \log ( n_{\mathsf{priv}} )$ samples are sufficient (information-theoretically) for recovering all the private images.

中文翻译：

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混合两个私有图像时InstaHide的样本复杂度

受InstaHide挑战[Huang，Song，Li和Arora'20]的启发，[Chen，Song和Zhuo'20]最近提供了一种在高斯图像分布下InstaHide攻击问题的数学公式。他们表明使用$ O（n _ {\ mathsf {priv}} ^ {k _ {\ mathsf {priv}}-2 /（k _ {\ mathsf {priv}} + 1）}）$个样本即可恢复一个$ n _ {\ mathsf {priv}} ^ {O（k _ {\ mathsf {priv}}}} + \ mathrm {poly}（n _ {\ mathsf {pub}}）$时间中的私有图像\ mathsf {priv}} $，其中$ n _ {\ mathsf {priv}} $和$ n _ {\ mathsf {pub}} $表示私有和公共数据集中用于生成混合图像样本的图像数量。在当前的InstaHide挑战混合两个私有图像（$ k _ {\ mathsf {priv}} = 2 $）的设置下，这意味着$ n _ {\ mathsf {priv}} ^ {4/3} $个样本足以恢复私有映像。在这项工作中