International Association
for Cryptologic Research

We study the space complexity of the two related fields of {\em differential privacy} and {\em adaptive data analysis}. Specifically, \begin{enumerate} \item Under standard cryptographic assumptions, we show that there exists a problem $P$ that requires exponentially more space to be solved efficiently with differential privacy, compared to the space needed without privacy. To the best of our knowledge, this is the first separation between the space complexity of private and non-private algorithms. \item The line of work on adaptive data analysis focuses on understanding the number of {\em samples} needed for answering a sequence of adaptive queries. We revisit previous lower bounds at a foundational level, and show that they are a consequence of a space bottleneck rather than a sampling bottleneck. \end{enumerate} To obtain our results, we define and construct an encryption scheme with multiple keys that is built to withstand a limited amount of key leakage in a very particular way.