The Complexity of Verifying Circuits as Differentially Private
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We study the problem of verifying differential privacy for straight line programs with probabilistic choice. Programs in this class can be seen as randomized Boolean circuits. We focus on 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.
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