  We study the space complexity of the two related fields of differential
privacy and adaptive data analysis. Specifically,
  (1) 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.
  (2) The line of work on adaptive data analysis focuses on understanding the
number of 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.
  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.
