The complexity of verifying loop-free programs as differentially private
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Accepted manuscript
Date
2020-06-29
Authors
Gaboardi, Marco
Nissim, K.
Purser, D.
Version
OA Version
Citation
M. Gaboardi, K. Nissim, D. Purser. 2020. "The Complexity of Verifying Loop-Free Programs as Differentially Private." 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). https://doi.org/10.4230/LIPIcs.ICALP.2020.129
Abstract
We study the problem of verifying differential privacy for loop-free programs with probabilistic choice. Programs in this class can be seen as randomized Boolean circuits, which we will use as a formal model to answer two different questions: first, deciding whether a program satisfies a prescribed level of privacy; second, approximating the privacy parameters a program realizes. We show that the problem of deciding whether a program satisfies ε-differential privacy is coNP^#P-complete. In fact, this is the case when either the input domain or the output range of the program is large. Further, we show that deciding whether a program is (ε,δ)-differentially private is coNP^#P-hard, and in coNP^#P for small output domains, but always in coNP^{#P^#P}. Finally, we show that the problem of approximating the level of differential privacy is both NP-hard and coNP-hard. These results complement previous results by Murtagh and Vadhan [Jack Murtagh and Salil P. Vadhan, 2016] showing that deciding the optimal composition of differentially private components is #P-complete, and that approximating the optimal composition of differentially private components is in P.
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License
© Marco Gaboardi, Kobbi Nissim, and David Purser; licensed under Creative Commons License CC-BY.