Answering n∧{2+o(1)} Counting Queries with Differential Privacy is Hard

A central problem in differentially private data analysis is how to design efficient algorithms capable of answering large numbers of counting queries on a sensitive database. Counting queries are of the form "What fraction of individual records in the database satisfy the property q?" We prove that if one-way functions exist, then there is no algorithm that takes as input a database D is an element of ({0, 1}(d))(n), and k = (Theta) over tilde (n(2)) arbitrary efficiently computable counting queries, runs in time poly(d, n), and returns an approximate answer to each query, while satisfying differential privacy. We also consider the complexity of answering "simple" counting queries, and make some progress in this direction by showing that the above result holds even when we require that the queries are computable by constant -depth (AC(0)) circuits. Our result is almost tight because it is known that (Omega) over tilde (n(2)) counting queries can be answered efficiently while satisfying differential privacy. Moreover, many more than n(2) queries (even exponential in n) can be answered in exponential time. We prove our results by extending the connection between differentially private query release and cryptographic traitor-tracing schemes to the setting where the queries are given to the sanitizer as input, and by constructing a traitor-tracing scheme that is secure in this setting.