# PCPs and the hardness of generating synthetic data

### Citation:

Ullman, Jon, and Salil Vadhan. “PCPs and the hardness of generating synthetic data.” Journal of Cryptology 33 (2020): 2078-2112.
 TCC2011.pdf 300 KB ECCC2014.pdf 780 KB JoC 2020.pdf 451 KB

### Abstract:

Version HistoryFull version posted as ECCC TR10-017.

Published earlier in Yuval Ishai, ed., Proceedings of the 8th IACR Theory of Cryptography Conference (TCC ‘11), Lecture Notes on Computer Science. Springer-Verlag, Publishers: Vol. 5978, pp. 572-587. https://link.springer.com/chapter/10.1007/978-3-642-19571-6_24

Invited to J. Cryptology selected papers from TCC 2011.

Assuming the existence of one-way functions, we show that there is no polynomial-time, differentially private algorithm $$\mathcal{A}$$ that takes a database $$D ∈ ({0, 1}^d)^n$$ and outputs a “synthetic database” $$\hat{D}$$ all of whose two-way marginals are approximately equal to those of $$D$$. (A two-way marginal is the fraction of database rows $$x ∈ {0, 1}^d$$ 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 $$\mathrm{poly} (n, 2^d)$$.

Our proof combines a construction of hard-to-sanitize databases based on digital signatures (by Dwork et al., STOC ‘09) with encodings based on probabilistically checkable proofs.

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 $$\hat{D}$$ and aggregating the results in some way.

Publisher's Version

Last updated on 02/01/2021