  Oblivious RAM (ORAM) and private information retrieval (PIR) are classic
cryptographic primitives used to hide the access pattern to data whose storage
has been outsourced to an untrusted server. Unfortunately, both primitives
require considerable overhead compared to plaintext access. For large-scale
storage infrastructure with highly frequent access requests, the degradation in
response time and the exorbitant increase in resource costs incurred by either
ORAM or PIR prevent their usage. In an ideal scenario, a privacy-preserving
storage protocols with small overhead would be implemented for these heavily
trafficked storage systems to avoid negatively impacting either performance
and/or costs. In this work, we study the problem of the best $\mathit{storage\
access\ privacy}$ that is achievable with only $\mathit{small\ overhead}$ over
plaintext access.
  To answer this question, we consider $\mathit{differential\ privacy\ access}$
which is a generalization of the $\mathit{oblivious\ access}$ security notion
that are considered by ORAM and PIR. Quite surprisingly, we present strong
evidence that constant overhead storage schemes may only be achieved with
privacy budgets of $\epsilon = \Omega(\log n)$. We present asymptotically
optimal constructions for differentially private variants of both ORAM and PIR
with privacy budgets $\epsilon = \Theta(\log n)$ with only $O(1)$ overhead. In
addition, we consider a more complex storage primitive called key-value storage
in which data is indexed by keys from a large universe (as opposed to
consecutive integers in ORAM and PIR). We present a differentially private
key-value storage scheme with $\epsilon = \Theta(\log n)$ and $O(\log\log n)$
overhead. This construction uses a new oblivious, two-choice hashing scheme
that may be of independent interest.
