  Shannon showed that to achieve perfect secrecy in point-to-point
communication, the message rate cannot exceed the shared secret key rate giving
rise to the simple one-time pad encryption scheme. In this paper, we extend
this work from point-to-point to networks. We consider a connected network with
pairwise communication between the nodes. We assume that each node is provided
with a certain amount of secret bits before communication commences. An
eavesdropper with unlimited computing power has access to all communication and
can hack a subset of the nodes not known to the rest of the nodes. We
investigate the limits on information-theoretic secure communication for this
network. We establish a tradeoff between the secure channel rate (for a node
pair) and the secure network rate (sum over all node pair rates) and show that
perfect secrecy can be achieved if and only if the sum rate of any subset of
unhacked channels does not exceed the shared unhacked-secret-bit rate of these
channels. We also propose two practical and efficient schemes that achieve a
good balance of network and channel rates with perfect secrecy guarantee. This
work has a wide range of potential applications for which perfect secrecy is
desired, such as cyber-physical systems, distributed-control systems, and
ad-hoc networks.
