  We investigate the problem of information theoretically secure communication
in a line network with erasure channels and state feedback. We consider a
spectrum of cases for the private randomness that intermediate nodes can
generate, ranging from having intermediate nodes generate unlimited private
randomness, to having intermediate nodes generate no private randomness, and
all cases in between. We characterize the secret message capacity when either
only one of the channels is eavesdropped or all of the channels are
eavesdropped, and we develop polynomial time algorithms that achieve these
capacities. We also give an outer bound for the case where an arbitrary number
of channels is eavesdropped. Our work is the first to characterize the secrecy
capacity of a network of arbitrary size, with imperfect channels and feedback.
As a side result, we derive the secret key and secret message capacity of an
one-hop network, when the source has limited randomness.
