# PCPs and the Hardness of Generating Synthetic Data

### Abstract

Assuming the existence of one-way functions, we show that there is no polynomial-time, differentially private algorithm A that takes a database D\in ({0,1}^d)^n and outputs a synthetic database'' 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 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 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 PCP-based Levin-reductions from NP search problems to finding approximate solutions to CSPs.