Differential Privacy on Finite Computers

Abstract

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 polynomialtime 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. * A condensed version of this paper appeared in ITCS 2018 [BV18] †

Optimum Noise Mechanism for Differentially Private Queries in Discrete Finite Sets

“…In the literature, some contributions are made in providing the noise mechanisms for discrete queries such as geometric distribution [5], [15] or a quantized Gaussian distribution, as proposed in [11]. First, we find the ( , δ)− differential privacy trade offs for the addition methodologies based on these mechanisms and compare them to highlight the effect of clamping.…”

Differential Privacy for Deep and Federated Learning: A Survey

“…The probability P (∆ = δ) can be interpreted as the probability of adding discrete noise δ to the result of the query function f . The Geometric mechanism is a discretized version of the Laplace mechanism [57]. 4) Exponential mechanism [58]: is most suited when we have to select a noisy (i.e., random) response from the set of all possible outputs, instead of adding a noise to the result of the query function [53].…”

An Uncertainty Principle is a Price of Privacy-Preserving Microdata

“…Balcer and Vadhan [3] recently showed a statistical price of privacy-preserving release of the top-k counts in a histogram. They proved an analogous O(log 2 (d/k)) penalty for point queries under ǫ-DP (and also results for approximate DP).…”

“…Thus, given a set Q of counting queries, for each q ∈ Q, the data collector computes a noisy answer a q by adding noise with distribution F q to the true answer and then must postprocess them to create microdata. 3 We assume the data collector chooses the noise distributions to achieve their desired privacy definition (e.g., ǫ-DP, ρ-zCDP).…”