  We consider the problem of communicating information over a network secretly
and reliably in the presence of a hidden adversary who can eavesdrop and inject
malicious errors. We provide polynomial-time, rate-optimal distributed network
codes for this scenario, improving on the rates achievable in previous work.
Our main contribution shows that as long as the sum of the adversary's jamming
rate Zo and his eavesdropping rate Zi is less than the network capacity C,
(i.e., Zo+Zi<C), our codes can communicate (with vanishingly small error
probability) a single bit correctly and without leaking any information to the
adversary. We then use this to design codes that allow communication at the
optimal source rate of C-Zo-Zi, while keeping the communicated message secret
from the adversary. Interior nodes are oblivious to the presence of adversaries
and perform random linear network coding; only the source and destination need
to be tweaked. In proving our results we correct an error in prior work by a
subset of the authors in this work.
