  The database community, at least for the last decade, has been grappling with
querying encrypted data, which would enable secure database as a service
solutions. A recent breakthrough in the cryptographic community (in 2009)
related to fully homomorphic encryption (FHE) showed that arbitrary computation
on encrypted data is possible. Successful adoption of FHE for query processing
is, however, still a distant dream, and numerous challenges have to be
addressed. One challenge is how to perform algebraic query processing of
encrypted data, where we produce encrypted intermediate results and operations
on encrypted data can be composed. In this paper, we describe our solution for
algebraic query processing of encrypted data, and also outline several other
challenges that need to be addressed, while also describing the lessons that
can be learnt from a decade of work by the database community in querying
encrypted data.
