International Association
for Cryptologic Research

We initiate a study of the composition properties of interactive differentially private mechanisms. An interactive differentially private mechanism is an algorithm that allows an analyst to adaptively ask queries about a sensitive dataset, with the property that an adversarial analyst's view of the interaction is approximately the same regardless of whether or not any individual's data is in the dataset. Previous studies of composition of differential privacy have focused on non-interactive algorithms, but interactive mechanisms are needed to capture many of the intended applications of differential privacy and a number of the important differentially private primitives. We focus on concurrent composition, where an adversary can arbitrarily interleave its queries to several differentially private mechanisms, which may be feasible when differentially private query systems are deployed in practice. We prove that when the interactive mechanisms being composed are pure differentially private, their concurrent composition achieves privacy parameters (with respect to pure or approximate differential privacy) that match the (optimal) composition theorem for noninteractive differential privacy. We also prove a composition theorem for interactive mechanisms that satisfy approximate differential privacy. That bound is weaker than even the basic (suboptimal) composition theorem for noninteractive differential privacy, and we leave closing the gap as a direction for future research, along with understanding concurrent composition for other variants of differential privacy.

Assuming the existence of one-way functions, we show that there is no polynomial-time differentially private algorithm $${\mathcal {A}}$$ A that takes a database $$D\in (\{0,1\}^d)^n$$ D ∈ ( { 0 , 1 } d ) n and outputs a “synthetic database” $${\hat{D}}$$ D ^ all of whose two-way marginals are approximately equal to those of D . (A two-way marginal is the fraction of database rows $$x\in \{0,1\}^d$$ x ∈ { 0 , 1 } d with a given pair of values in a given pair of columns.) This answers a question of Barak et al. (PODS ‘07), who gave an algorithm running in time $$\mathrm {poly}(n,2^d)$$ poly ( n , 2 d ) . Our proof combines a construction of hard-to-sanitize databases based on digital signatures (by Dwork et al., STOC ‘09) with encodings based on the PCP theorem. We also present both negative and positive results for generating “relaxed” synthetic data, where the fraction of rows in D satisfying a predicate c are estimated by applying c to each row of $${\hat{D}}$$ D ^ and aggregating the results in some way.

We introduce hardness in relative entropy, a new notion of hardness for search problems which on the one hand is satisfied by all one-way functions and on the other hand implies both next-block pseudoentropy and inaccessible entropy, two forms of computational entropy used in recent constructions of pseudorandom generators and statistically hiding commitment schemes, respectively. Thus, hardness in relative entropy unifies the latter two notions of computational entropy and sheds light on the apparent “duality” between them. Additionally, it yields a more modular and illuminating proof that one-way functions imply next-block inaccessible entropy, similar in structure to the proof that one-way functions imply next-block pseudoentropy (Vadhan and Zheng, STOC ‘12).