有限计算机上的差分隐私
Differential Privacy on Finite Computers
摘要：
我们考虑了设计和分析差分私有算法的问题，这些算法可以在{\em strict}多项式时间中在{\em离散}计算模型上实现，其动机是对实数算术差分私有算法的浮点实现的已知攻击（Mironov，CCS 2012）以及对预期的多项式时间算法的潜在定时攻击。作为一个案例研究，我们研究的基本问题，近似的直方图的分类数据集在一个可能的大数据宇宙$\mathcal{X}$。经典拉普拉斯机制（Dwork，McSherry，Nissim，Smith，TCC 2006和J. Privacy \& Confidence 2017）不满足我们的要求，因为它基于实数运算和自然离散模拟，例如几何机制（Ghosh，Roughgarden，Sundarajan，STOC 2009和SICOMP 2012），在$|\mathcal{X}|$中花费至少线性的时间，其在输入的比特长度中可以是指数的。在本文中，我们提供严格的多项式时间离散算法近似直方图的同时精度（所有箱的最大误差）匹配拉普拉斯机制的常数因子，同时保留相同的（纯）差分隐私保证。我们的一个算法产生一个稀疏的直方图作为输出。它的“每箱精度”（个别箱的错误）是由$\log|\mathcal{X}|$的一个因素比拉普拉斯机制，但我们证明了一个下界，表明这是必要的任何算法，产生一个稀疏的直方图。第二种算法通过产生密集直方图的紧凑且有效的可计算表示来避免该下限，并且匹配拉普拉斯机制的每箱精度，其基于离散几何机制的适当箝位版本的$(n+1)$方式独立实现。
英文摘要：
We consider the problem of designing and analyzing differentially private algorithms that can be implemented on {\em discrete} models of computation in {\em strict} polynomial time, motivated by known attacks on floating point implementations of real-arithmetic differentially private algorithms (Mironov, CCS 2012) and the potential for timing attacks on expected polynomial-time algorithms. As a case study, we examine the basic problem of approximating the histogram of a categorical dataset over a possibly large data universe $\mathcal{X}$. The classic Laplace Mechanism (Dwork, McSherry, Nissim, Smith, TCC 2006 and J. Privacy \& Confidentiality 2017) does not satisfy our requirements, as it is based on real arithmetic, and natural discrete analogues, such as the Geometric Mechanism (Ghosh, Roughgarden, Sundarajan, STOC 2009 and SICOMP 2012), take time at least linear in $|\mathcal{X}|$, which can be exponential in the bit length of the input. In this paper, we provide strict polynomial-time discrete algorithms for approximate histograms whose simultaneous accuracy (the maximum error over all bins) matches that of the Laplace Mechanism up to constant factors, while retaining the same (pure) differential privacy guarantee. One of our algorithms produces a sparse histogram as output. Its "per-bin accuracy" (the error on individual bins) is worse than that of the Laplace Mechanism by a factor of $\log|\mathcal{X}|$, but we prove a lower bound showing that this is necessary for any algorithm that produces a sparse histogram. A second algorithm avoids this lower bound, and matches the per-bin accuracy of the Laplace Mechanism, by producing a compact and efficiently computable representation of a dense histogram, it is based on an $(n+1)$-wise independent implementation of an appropriately clamped version of the Discrete Geometric Mechanism.
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