Arbitrarily varying networks: Capacity-achieving computationally efficient codes

We consider the problem of communication over a network containing a hidden and malicious adversary that can control a subset of network resources, and aims to disrupt communications. We focus on omniscient node-based adversary, i.e., the adversary can control a subset of nodes, and knows the message, network code and packets on all links. Characterizing information-theoretically optimal communication rates as a function of network parameters and bounds on the adversarially controlled network is in general open, even for unicast (single source, single destination) problems. In this work we characterize the information-theoretically optimal randomized capacity of such problems, i.e., under the assumption that the source node shares (an asymptotically negligible amount of) independent common randomness with each network node a priori. We propose a novel computationally-efficient communication scheme whose rate matches a natural information-theoretically 'erasure outer bound' on the optimal rate. Our schemes require no prior knowledge of network topology, and can be implemented in a distributed manner as an overlay on top of classical distributed linear network coding.