The Asymptotic Capacity of Private Search

The private search problem is introduced, where a dataset comprised of $L$ i.i.d. records is replicated across $N$ non-colluding servers, each record takes values uniformly from an alphabet of size $K$, and a user wishes to search for all records that match a privately chosen value, without revealing any information about the chosen value to any individual server. The capacity of private search is the maximum number of bits of desired information that can be retrieved per bit of download. The asymptotic (large $K$) capacity of private search is shown to be $1-1/N$, even as the scope of private search is further generalized to allow approximate (OR) search over a number of realizations that grows with $K$. The results are based on the asymptotic behavior of a new converse bound for private information retrieval with arbitrarily dependent messages.