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 realarithmetic differentially private algorithms (Mironov, CCS 2012) and the potential for timing attacks on expected polynomialtime algorithms. We use a case study: 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 polynomialtime 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 "perbin 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 perbin 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.
BibTeX  Entry
@InProceedings{balcer_et_al:LIPIcs:2018:8353,
author = {Victor Balcer and Salil Vadhan},
title = {{Differential Privacy on Finite Computers}},
booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
pages = {43:143:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770606},
ISSN = {18688969},
year = {2018},
volume = {94},
editor = {Anna R. Karlin},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8353},
URN = {urn:nbn:de:0030drops83537},
doi = {10.4230/LIPIcs.ITCS.2018.43},
annote = {Keywords: Algorithms, Differential Privacy, Discrete Computation, Histograms}
}
Keywords: 

Algorithms, Differential Privacy, Discrete Computation, Histograms 
Seminar: 

9th Innovations in Theoretical Computer Science Conference (ITCS 2018) 
Issue Date: 

2018 
Date of publication: 

05.01.2018 