We consider the problem of efficiently evaluating a secret polynomial at a
given public point, when the polynomial is stored on an untrusted server. The
server performs the evaluation and returns a certificate, and the client can
efficiently check that the evaluation is correct using some pre-computed keys.
Our protocols support two important features: the polynomial itself can be
encrypted on the server, and it can be dynamically updated by changing
individual coefficients cheaply without redoing the entire setup. As an
important application, we show how these new techniques can be used to
instantiate a Dynamic Proof of Retrievability (DPoR) for arbitrary outsourced
data storage that achieves low server storage size and audit complexity. Our
methods rely only on linearly homomorphic encryption and pairings, and
preliminary timing results indicate reasonable performance for polynomials with
millions of coefficients, and efficient DPoR with for instance 1TB size
databases.

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