Assuming the existence of one-way functions, we show that there is no polynomial-time differentially private algorithm A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document} that takes a database D∈({0,1}d)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\in (\{0,1\}^d)^n$$\end{document} and outputs a “synthetic database” D^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{D}}$$\end{document} all of whose two-way marginals are approximately equal to those of D. (A two-way marginal is the fraction of database rows x∈{0,1}d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in \{0,1\}^d$$\end{document} with a given pair of values in a given pair of columns.) This answers a question of Barak et al. (PODS ‘07), who gave an algorithm running in time poly(n,2d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {poly}(n,2^d)$$\end{document}. Our proof combines a construction of hard-to-sanitize databases based on digital signatures (by Dwork et al., STOC ‘09) with encodings based on the PCP theorem. We also present both negative and positive results for generating “relaxed” synthetic data, where the fraction of rows in D satisfying a predicate c are estimated by applying c to each row of D^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{D}}$$\end{document} and aggregating the results in some way.

中文翻译：

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PCP 和生成合成数据的难度

1\}^d)^n$$\end{document} 并输出一个“合成数据库” D^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{ amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{D}}$$\end{document} 所有的其双向边际近似等于 D 的那些。（双向边际是数据库行的分数 x∈{0,1}d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym } \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in \{0, 1\}^d$$\end{document} 在给定的一对列中使用给定的一对值。）这回答了 Barak 等人的一个问题。(PODS '07), 谁给出了一个及时运行的算法 poly(n,2d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs } \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {poly}(n,2^d)$$\end{document}。我们的证明结合了基于数字签名（Dwork 等人，STOC '09）的难以清理的数据库结构与基于 PCP 定理的编码。我们还提供了生成“宽松”合成数据的负面和正面结果，