Copyright Notice: This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. These works may not be reposted without the explicit permission of the copyright holder. Also, some of these works have been submitted for publication. Copyright may be transferred without further notice and this version may no longer be accessible.
We initiate a study of the composition properties of interactive differentially private mechanisms. An interactive differentially private mechanism is an algorithm that allows an analyst to adaptively ask queries about a sensitive dataset, with the property that an adversarial analyst's view of the interaction is approximately the same regardless of whether or not any individual's data is in the dataset. Previous studies of composition of differential privacy have focused on non-interactive algorithms, but interactive mechanisms are needed to capture many of the intended applications of differential privacy and a number of the important differentially private primitives.
We focus on concurrent composition, where an adversary can arbitrarily interleave its queries to several differentially private mechanisms, which may be feasible when differentially private query systems are deployed in practice. We prove that when the interactive mechanisms being composed are pure differentially private, their concurrent composition achieves privacy parameters (with respect to pure or approximate differential privacy) that match the (optimal) composition theorem for noninteractive differential privacy. We also prove a composition theorem for interactive mechanisms that satisfy approximate differential privacy. That bound is weaker than even the basic (suboptimal) composition theorem for noninteractive differential privacy, and we leave closing the gap as a direction for future research, along with understanding concurrent composition for other variants of differential privacy.
We consider the problem of designing and analyzing differentially private algorithms that can be implemented on discrete models of computation in 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 \(X\). 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 \(|X|\), 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 \(\log |X|\), 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.