Chen, Yiling, Or Sheffet, and Salil Vadhan. “Privacy games.” ACM Transactions on Economics and Computation 8, no. 2 (2020): Article 9. Publisher's VersionAbstract

Version History: 

Previously published as: Yiling Chen, Or Sheffet, and Salil Vadhan. Privacy games. In Proceedings of the 10th International Conference on Web and Internet Economics (WINE ‘14), volume 8877 of Lecture Notes in Computer Science, pages 371–385. Springer-Verlag, 14–17 December 2014. (WINE Publisher's Version linked here:; PDF attached as WINE2014.

The problem of analyzing the effect of privacy concerns on the behavior of selfish utility-maximizing agents has received much attention lately. Privacy concerns are often modeled by altering the utility functions of agents to consider also their privacy loss. Such privacy aware agents prefer to take a randomized strategy even in very simple games in which non-privacy aware agents play pure strategies. In some cases, the behavior of privacy aware agents follows the framework of Randomized Response, a well-known mechanism that preserves differential privacy. 

Our work is aimed at better understanding the behavior of agents in settings where their privacy concerns are explicitly given. We consider a toy setting where agent A, in an attempt to discover the secret type of agent B, offers B a gift that one type of B agent likes and the other type dislikes. As opposed to previous works, B's incentive to keep her type a secret isn't the result of "hardwiring" B's utility function to consider privacy, but rather takes the form of a payment between B and A. We investigate three different types of payment functions and analyze B's behavior in each of the resulting games. As we show, under some payments, B's behavior is very different than the behavior of agents with hardwired privacy concerns and might even be deterministic. Under a different payment we show that B's BNE strategy does fall into the framework of Randomized Response.

Balcer, Victor, and Salil Vadhan. “Differential privacy on finite computers.” Journal of Privacy and Confidentiality 9, no. 2 (2019). Publisher's VersionAbstract

Version History: 

Also presented at TPDP 2017; preliminary version posted as arXiv:1709.05396 [cs.DS].

2018: Published in Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018), volume 94 of Leibniz International Proceedings in Informatics (LIPIcs), pp 43:1-43:21.

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.

Bun, Mark, Jonathan Ullman, and Salil Vadhan. “Fingerprinting codes and the price of approximate differential privacy.” SIAM Journal on Computing, Special Issue on STOC '14 47, no. 5 (2018): 1888-1938. Publisher's VersionAbstract

Version HistorySpecial Issue on STOC ‘14. Preliminary versions in STOC ‘14 and arXiv:1311.3158 [cs.CR].

We show new information-theoretic lower bounds on the sample complexity of (ε, δ)- differentially private algorithms that accurately answer large sets of counting queries. A counting query on a database \(D ∈ (\{0, 1\}^d)^n\) has the form “What fraction of the individual records in the database satisfy the property \(q\)?” We show that in order to answer an arbitrary set \(Q\) of \(\gg d/ \alpha^2\) counting queries on \(D\) to within error \(±α\) it is necessary that \(n ≥ \tilde{Ω}(\sqrt{d} \log |Q|/α^2ε)\). This bound is optimal up to polylogarithmic factors, as demonstrated by the private multiplicative weights algorithm (Hardt and Rothblum, FOCS’10). In particular, our lower bound is the first to show that the sample complexity required for accuracy and (ε, δ)-differential privacy is asymptotically larger than what is required merely for accuracy, which is \(O(\log |Q|/α^2 )\). In addition, we show that our lower bound holds for the specific case of \(k\)-way marginal queries (where \(|Q| = 2^k \binom{d}{k}\) ) when \(\alpha\) is not too small compared to d (e.g., when \(\alpha\) is any fixed constant). Our results rely on the existence of short fingerprinting codes (Boneh and Shaw, CRYPTO’95; Tardos, STOC’03), which we show are closely connected to the sample complexity of differentially private data release. We also give a new method for combining certain types of sample-complexity lower bounds into stronger lower bounds.

Murtagh, Jack, and Salil Vadhan. “The complexity of computing the optimal composition of differential privacy.” Theory of Computing 14 (2018): 1-35. Publisher's VersionAbstract

Version History: Full version posted on CoRR, abs/1507.03113, July 2015Additional version published in Proceedings of the 13th IACR Theory of Cryptography Conference (TCC '16-A)

In the study of differential privacy, composition theorems (starting with the original paper of Dwork, McSherry, Nissim, and Smith (TCC '06)) bound the degradation of privacy when composing several differentially private algorithms. Kairouz, Oh, and Viswanath (ICML '15) showed how to compute the optimal bound for composing \(k\) arbitrary (\(\epsilon\),\(\delta\))- differentially private algorithms. We characterize the optimal composition for the more general case of \(k\) arbitrary (\(\epsilon_1\) , \(\delta_1\) ), . . . , (\(\epsilon_k\) , \(\delta_k\) )-differentially private algorithms where the privacy parameters may differ for each algorithm in the composition. We show that computing the optimal composition in general is \(\#\)P-complete. Since computing optimal composition exactly is infeasible (unless FP\(=\)\(\#\)P), we give an approximation algorithm that computes the composition to arbitrary accuracy in polynomial time. The algorithm is a modification of Dyer’s dynamic programming approach to approximately counting solutions to knapsack problems (STOC '03).

Karwa, Vishesh, and Salil Vadhan. “Finite sample differentially private confidence intervals.” In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018), volume 94 of Leibniz International Proceedings in Informatics (LIPIcs), 44:1-44:9. Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. ITCS, 2018. Publisher's VersionAbstract

Version History: Also presented at TPDP 2017. Preliminary version posted as arXiv:1711.03908 [cs.CR].

We study the problem of estimating finite sample confidence intervals of the mean of a normal population under the constraint of differential privacy. We consider both the known and unknown variance cases and construct differentially private algorithms to estimate confidence intervals. Crucially, our algorithms guarantee a finite sample coverage, as opposed to an asymptotic coverage. Unlike most previous differentially private algorithms, we do not require the domain of the samples to be bounded. We also prove lower bounds on the expected size of any differentially private confidence set showing that our the parameters are optimal up to polylogarithmic factors.

Murtagh, Jack, Kathryn Taylor, George Kellaris, and Salil P. Vadhan. “Usable differential privacy: A case study with PSI.” arXiv, 2018, 1809.04103 [cs.CR]. ArXiv VersionAbstract

Version History: v1, 11 September 2018

Differential privacy is a promising framework for addressing the privacy concerns in sharing sensitive datasets for others to analyze. However differential privacy is a highly technical area and current deployments often require experts to write code, tune parameters, and optimize the trade-off between the privacy and accuracy of statistical releases. For differential privacy to achieve its potential for wide impact, it is important to design usable systems that enable differential privacy to be used by ordinary data owners and analysts. PSI is a tool that was designed for this purpose, allowing researchers to release useful differentially private statistical information about their datasets without being experts in computer science, statistics, or privacy. We conducted a thorough usability study of PSI to test whether it accomplishes its goal of usability by non-experts. The usability test illuminated which features of PSI are most user-friendly and prompted us to improve aspects of the tool that caused confusion. The test also highlighted some general principles and lessons for designing usable systems for differential privacy, which we discuss in depth.

Wood, Alexandra, Micah Altman, Aaron Bembenek, Mark Bun, Marco Gaboardi, James Honaker, Kobbi Nissim, David R. OBrien, Thomas Steinke, and Salil Vadhan. “Differential privacy: A primer for a non-technical audience.” Vanderbilt Journal of Entertainment & Technology Law 21, no. 1 (2018): 209-275. Publisher's VersionAbstract

Version History: Preliminary version workshopped at PLSC 2017.

Differential privacy is a formal mathematical framework for quantifying and managing privacy risks. It provides provable privacy protection against a wide range of potential attacks, including those currently unforeseen. Differential privacy is primarily studied in the context of the collection, analysis, and release of aggregate statistics. These range from simple statistical estimations, such as averages, to machine learning. Tools for differentially private analysis are now in early stages of implementation and use across a variety of academic, industry, and government settings. Interest in the concept is growing among potential users of the tools, as well as within legal and policy communities, as it holds promise as a potential approach to satisfying legal requirements for privacy protection when handling personal information. In particular, differential privacy may be seen as a technical solution for analyzing and sharing data while protecting the privacy of individuals in accordance with existing legal or policy requirements for de-identification or disclosure limitation.

This primer seeks to introduce the concept of differential privacy and its privacy implications to non-technical audiences. It provides a simplified and informal, but mathematically accurate, description of differential privacy. Using intuitive illustrations and limited mathematical formalism, it discusses the definition of differential privacy, how differential privacy addresses privacy risks, how differentially private analyses are constructed, and how such analyses can be used in practice. A series of illustrations is used to show how practitioners and policymakers can conceptualize the guarantees provided by differential privacy. These illustrations are also used to explain related concepts, such as composition (the accumulation of risk across multiple analyses), privacy loss parameters, and privacy budgets. This primer aims to provide a foundation that can guide future decisions when analyzing and sharing statistical data about individuals, informing individuals about the privacy protection they will be afforded, and designing policies and regulations for robust privacy protection.

Wood, Alexandra, Micah Altman, Suso Baleato, and Salil Vadhan. Comments on the City of Seattle Open Data Risk Assessment, 2017. Publisher's VersionAbstract
The transparency goals of the open data movement serve important social, economic, and democratic functions in cities like Seattle. At the same time, some municipal datasets about the city and its citizens’ activities carry inherent risks to individual privacy when shared publicly. In 2016, the City of Seattle declared in its Open Data Policy that the city’s data would be “open by preference,” except when doing so may affect individual privacy. To ensure its Open Data program effectively protects individuals, Seattle committed to performing an annual risk assessment and tasked the Future of Privacy Forum (FPF) with creating and deploying an initial privacy risk assessment methodology for open data.This Draft Report provides tools and guidance to the City of Seattle and other municipalities navigating the complex policy, operational, technical, organizational, and ethical standards that support privacyprotective open data programs. Although there is a growing body of research on open data privacy, open data managers and departmental data owners need to be able to employ a standardized methodology for assessing the privacy risks and benefits of particular datasets internally, without a bevy of expert statisticians, privacy lawyers, or philosophers. By following a flexible, risk-based assessment process, the City of Seattle – and other municipal open data programs – can maximize the utility and openness of civic data while minimizing privacy risks to individuals and community concerns about ethical challenges, fairness, and equity.This Draft Report first describes inherent privacy risks in an open data landscape, with an emphasis on potential harms related to re-identification, data quality, and fairness. Accompanying this, the Draft Report includes a Model Open Data Benefit Risk Analysis (MODBRA). The model template evaluates the types of data contained in a proposed open dataset, the potential benefits – and concomitant risks – of releasing the dataset publicly, and strategies for effective de-identification and risk mitigation. This holistic assessment guides city officials to determine whether to release the dataset openly, in a limited access environment, or to withhold it from publication (absent countervailing public policy considerations). The Draft Report methodology builds on extensive work done in this field by experts at the National Institute of Standards and Technology, the University of Washington, the Berkman Klein Center for Internet & Society at Harvard University, and others, and adapts existing frameworks to the unique challenges faced by cities as local governments, technological system integrators, and consumer facing service providers.
Vadhan, Salil. “The Complexity of Differential Privacy.” In Tutorials on the Foundations of Cryptography, 347-450. Springer, Yehuda Lindell, ed. 2017. Publisher's VersionAbstract

Version History: 

August 2016: Manuscript v1 (see files attached)

March 2017: Manuscript v2 (see files attached); Errata

April 2017: Published Version (in Tutorials on the Foundations of Cryptography; see Publisher's Version link and also SPRINGER 2017.PDF, below) 


Differential privacy is a theoretical framework for ensuring the privacy of individual-level data when performing statistical analysis of privacy-sensitive datasets. This tutorial provides an introduction to and overview of differential privacy, with the goal of conveying its deep connections to a variety of other topics in computational complexity, cryptography, and theoretical computer science at large. This tutorial is written in celebration of Oded Goldreich’s 60th birthday, starting from notes taken during a minicourse given by the author and Kunal Talwar at the 26th McGill Invitational Workshop on Computational Complexity [1].


Nissim, Kobbi, Aaron Bembenek, Alexandra Wood, Mark Bun, Marco Gaboardi, Urs Gasser, David O'Brien, Thomas Steinke, and Salil Vadhan. “Bridging the gap between computer science and legal approaches to privacy.” Harvard Journal of Law & Technology 31, no. 2 (2017). Publisher's VersionAbstract

Version History: Workshopped at PLSC (Privacy Law Scholars Conference) ‘16.


The analysis and release of statistical data about individuals and groups of individuals carries inherent privacy risks, and these risks have been conceptualized in different ways within the fields of law and computer science. For instance, many information privacy laws adopt notions of privacy risk that are sector- or context-specific, such as in the case of laws that protect from disclosure certain types of information contained within health, educational, or financial records. In addition, many privacy laws refer to specific techniques, such as deidentification, that are designed to address a subset of possible attacks on privacy. In doing so, many legal standards for privacy protection rely on individual organizations to make case-by-case determinations regarding concepts such as the identifiability of the types of information they hold. These regulatory approaches are intended to be flexible, allowing organizations to (1) implement a variety of specific privacy measures that are appropriate given their varying institutional policies and needs, (2) adapt to evolving best practices, and (3) address a range of privacy-related harms. However, in the absence of clear thresholds and detailed guidance on making case-specific determinations, flexibility in the interpretation and application of such standards also creates uncertainty for practitioners and often results in ad hoc, heuristic processes. This uncertainty may pose a barrier to the adoption of new technologies that depend on unambiguous privacy requirements. It can also lead organizations to implement measures that fall short of protecting against the full range of data privacy risks.

Chen, Yiling, Stephen Chong, Ian A. Kash, Tal Moran, and Salil P. Vadhan. “Truthful mechanisms for agents that value privacy.” ACM Transactions on Economics and Computation 4, no. 3 (2016). Publisher's VersionAbstract

Version History: Special issue on EC ‘13. Preliminary version at arXiv:1111.5472 [cs.GT] (Nov. 2011).

Recent work has constructed economic mechanisms that are both truthful and differentially private. In these mechanisms, privacy is treated separately from truthfulness; it is not incorporated in players’ utility functions (and doing so has been shown to lead to nontruthfulness in some cases). In this work, we propose a new, general way of modeling privacy in players’ utility functions. Specifically, we only assume that if an outcome \({o}\) has the property that any report of player \({i}\) would have led to \({o}\) with approximately the same probability, then \({o}\) has a small privacy cost to player \({i}\). We give three mechanisms that are truthful with respect to our modeling of privacy: for an election between two candidates, for a discrete version of the facility location problem, and for a general social choice problem with discrete utilities (via a VCG-like mechanism). As the number \({n}\) of players increases, the social welfare achieved by our mechanisms approaches optimal (as a fraction of \({n}\)).

Altman, Micah, Alexandra Wood, David R. O'Brien, Salil Vadhan, and Urs Gasser. “Towards a modern approach to a privacy-aware government data releases.” Berkeley Technology Law Journal 30, no. 3 (2016): 1967-2072. Publisher's VersionAbstract
Governments are under increasing pressure to publicly release collected data in order to promote transparency, accountability, and innovation. Because much of the data they release pertains to individuals, agencies rely on various standards and interventions to protect privacy interests while supporting a range of beneficial uses of the data. However, there are growing concerns among privacy scholars, policymakers, and the public that these approaches are incomplete, inconsistent, and difficult to navigate. To identify gaps in current practice, this Article reviews data released in response to freedom of information and Privacy Act requests, traditional public and vital records, official statistics, and e-government and open government initiatives. It finds that agencies lack formal guidance for implementing privacy interventions in specific cases. Most agencies address privacy by withholding or redacting records that contain directly or indirectly identifying information based on an ad hoc balancing of interests, and different government actors sometimes treat similar privacy risks vastly differently. These observations demonstrate the need for a more systematic approach to privacy analysis and also suggest a new way forward. In response to these concerns, this Article proposes a framework for a modern privacy analysis informed by recent advances in data privacy from disciplines such as computer science, statistics, and law. Modeled on an information security approach, this framework characterizes and distinguishes between privacy controls, threats, vulnerabilities, and utility. When developing a data release mechanism, policymakers should specify the desired data uses and expected benefits, examine each stage of the data lifecycle to identify privacy threats and vulnerabilities, and select controls for each lifecycle stage that are consistent with the uses, threats, and vulnerabilities at that stage. This Article sketches the contours of this analytical framework, populates selected portions of its contents, and illustrates how it can inform the selection of privacy controls by discussing its application to two real-world examples of government data releases.
Nissim, Kobbi, Uri Stemmer, and Salil Vadhan. “Locating a small cluster privately.” In Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems (PODS ‘16), 413-427. ACM, 2016. Publisher's VersionAbstract

Version HistoryFull version posted as arXiv:1604.05590 [cs.DS].

We present a new algorithm for locating a small cluster of points with differential privacy [Dwork, McSherry, Nissim, and Smith, 2006]. Our algorithm has implications to private data exploration, clustering, and removal of outliers. Furthermore, we use it to significantly relax the requirements of the sample and aggregate technique [Nissim, Raskhodnikova, and Smith, 2007], which allows compiling of “off the shelf” (non-private) analyses into analyses that preserve differential privacy.

Gaboardi, Marco, Hyun Woo Lim, Ryan Rogers, and Salil Vadhan. “Differentially private chi-squared hypothesis testing: Goodness of fit and independence testing.” In M. Balcan and K. Weinberger, editors, Proceedings of the 33rd International Conference on Machine Learning (ICML ‘16). 2111-2120, 2016. Publisher's VersionAbstract

Version History: Preliminary version posted as arXiv:1602.03090.

Hypothesis testing is a useful statistical tool in determining whether a given model should be rejected based on a sample from the population. Sample data may contain sensitive information about individuals, such as medical information. Thus it is important to design statistical tests that guarantee the privacy of subjects in the data. In this work, we study hypothesis testing subject to differential privacy, specifically chi-squared tests for goodness of fit for multinomial data and independence between two categorical variables.

We propose new tests for goodness of fit and independence testing that like the classical versions can be used to determine whether a given model should be rejected or not, and that additionally can ensure differential privacy. We give both Monte Carlo based hypothesis tests as well as hypothesis tests that more closely follow the classical chi-squared goodness of fit test and the Pearson chi-squared test for independence. Crucially, our tests account for the distribution of the noise that is injected to ensure privacy in determining significance.

We show that these tests can be used to achieve desired significance levels, in sharp contrast to direct applications of classical tests to differentially private contingency tables which can result in wildly varying significance levels. Moreover, we study the statistical power of these tests. We empirically show that to achieve the same level of power as the classical non-private tests our new tests need only a relatively modest increase in sample size.

Gaboardi, Marco, James Honaker, Gary King, Jack Murtagh, Kobbi Nissim, Jonathan Ullman, and Salil Vadhan. “PSI (Ψ): a private data-sharing interface.” In Poster presentation at the 2nd Workshop on the Theory and Practice of Differential Privacy (TPDP ‘16), 2016. ArXiv VersionAbstract

Version History: Paper posted as arXiv:1609.04340 [cs.CR].

We provide an overview of the design of PSI (“a Private data Sharing Interface”), a system we are developing to enable researchers in the social sciences and other fields to share and explore privacy-sensitive datasets with the strong privacy protections of differential privacy.