  The $k$-Nearest Neighbor Search ($k$-NNS) is the backbone of several
cloud-based services such as recommender systems, face recognition, and
database search on text and images. In these services, the client sends the
query to the cloud server and receives the response in which case the query and
response are revealed to the service provider. Such data disclosures are
unacceptable in several scenarios due to the sensitivity of data and/or privacy
laws.
  In this paper, we introduce SANNS, a system for secure $k$-NNS that keeps
client's query and the search result confidential. SANNS comprises two
protocols: an optimized linear scan and a protocol based on a novel sublinear
time clustering-based algorithm. We prove the security of both protocols in the
standard semi-honest model. The protocols are built upon several
state-of-the-art cryptographic primitives such as lattice-based additively
homomorphic encryption, distributed oblivious RAM, and garbled circuits. We
provide several contributions to each of these primitives which are applicable
to other secure computation tasks. Both of our protocols rely on a new circuit
for the approximate top-$k$ selection from $n$ numbers that is built from $O(n
+ k^2)$ comparators.
  We have implemented our proposed system and performed extensive experimental
results on four datasets in two different computation environments,
demonstrating more than $18-31\times$ faster response time compared to
optimally implemented protocols from the prior work. Moreover, SANNS is the
first work that scales to the database of 10 million entries, pushing the limit
by more than two orders of magnitude.
