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Near-Optimal Private Approximation Protocols via a Black Box Transformation
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BibTeX
@MISC{Woodruff_near-optimalprivate,
author = {David P. Woodruff},
title = {Near-Optimal Private Approximation Protocols via a Black Box Transformation},
year = {}
}
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Abstract
We show the following transformation: any two-party pro-tocol for outputting a (1 + ε)-approximation to f(x, y) = n j=1 g(xj, yj) with probability at least 2/3, for any nonnegative efficienty computable function g, can be transformed into a two-party private approximation protocol with only a polylogarithmic factor loss in communication, computation, and round complexity. In general it is insufficient to use secure function evaluation or fully homomorphic encryption on a standard, non-private protocol for approximating f. This is because the approximation may reveal information about x and y that does not follow from f(x, y). Applying our transformation and variations of it, we obtain near-optimal private approximation protocols for a wide range of problems in the data stream literature for which previously nothing was known. We give near-optimal private approximation protocols for the ℓp-distance for every p ≥ 0, for the heavy hitters and importance sampling problems with respect to any ℓp-norm, for the max-dominance and other dominant ℓp-norms, for the distinct summation problem, for entropy, for cascaded frequency moments, for subspace approximation and block sampling, and for measuring independence of datasets. Using a result for data streams, we obtain private approximation protocols with polylogarithmic communication for every non-decreasing and symmetric function g(xj, yj) = h(xj − yj) with at most quadratic growth. If the original (non-private) protocol is a simultaneous protocol, e.g., a sketching algorithm, then our only cryptographic assumption is efficient symmetric computationally-private information retrieval; otherwise it is fully homomorphic encryption. For all but one of these problems, the original protocol is a sketching algorithm. Our protocols generalize straightforwardly to more than two parties.
Keyphrases
near-optimal private approximation protocol black box transformation homomorphic encryption dominant p-norms two-party pro-tocol following transformation private approximation protocol heavy hitter two-party private approximation protocol original protocol subspace approximation symmetric function cryptographic assumption non-private protocol importance sampling problem cascaded frequency moment nonnegative efficienty computable function data stream literature distinct summation problem sketching algorithm quadratic growth block sampling polylogarithmic communication polylogarithmic factor loss secure function evaluation round complexity data stream simultaneous protocol wide range efficient symmetric computationally-private information retrieval
