  Encrypted control systems allow to evaluate feedback laws on external servers
without revealing private information about state and input data, the control
law, or the plant. While there are a number of encrypted control schemes
available for linear feedback laws, only few results exist for the evaluation
of more general control laws. Recently, an approach to encrypted polynomial
control was presented, relying on two-party secret sharing and an inter-server
communication protocol using homomorphic encryption. As homomorphic encryptions
are much more computationally demanding than secret sharing, they make up for a
tremendous amount of the overall computational demand of this scheme. For this
reason, in this paper, we investigate schemes for secure polynomial control
based solely on secret sharing. We introduce a novel secure three-party control
scheme based on three-party computation. Further, we propose a novel $n$-party
control scheme to securely evaluate polynomial feedback laws of arbitrary
degree without inter-server communication. The latter property makes it easier
to realize the necessary requirement regarding non-collusion of the servers,
with which perfect security can be guaranteed. Simulations suggest that the
presented control schemes are many times less computationally demanding than
the two-party scheme mentioned above.
