  In a private information retrieval (PIR) system, the user needs to retrieve
one of the possible messages from a set of storage servers, but wishes to keep
the identity of requested message private from any given server. Existing
efforts in this area have made it clear that the efficiency of the retrieval
will be impacted significantly by the amount of the storage space allowed at
the servers. In this work, we consider the tradeoff between the storage cost
and the retrieval cost. We first present three fundamental results: 1) a
regime-wise 2-approximate characterization of the optimal tradeoff, 2) a cyclic
permutation lemma that can produce more sophisticated codes from simpler ones,
and 3) a relaxed entropic linear program (LP) lower bound that has a polynomial
complexity. Equipped with the cyclic permutation lemma, we then propose two
novel code constructions, and by applying the lemma, obtain new
storage-retrieval points. Furthermore, we derive more explicit lower bounds by
utilizing only a subset of the constraints in the relaxed entropic LP in a
systematic manner. Though the new upper bound and lower bound do not lead to a
more precise approximate characterization in general, they are significantly
tighter than the existing art.
