Reliable and Secure Multishot Network Coding using Linearized Reed-Solomon Codes

Multishot network coding is considered in a worst-case adversarial setting in which an omniscient adversary with unbounded computational resources may inject erroneous packets in up to t links, erase up to $\rho$ packets, and wire-tap up to $\mu$ links, all throughout $\ell$ shots of a (random) linearly-coded network. Assuming no knowledge of the underlying linear network code (in particular, the network topology and underlying linear code may change with time), a coding scheme achieving zero-error communication and perfect secrecy is obtained based on linearized Reed-Solomon codes. The scheme achieves the maximum possible secret message size of $\ell n'-2t-\rho-\mu$ packets, where $n'$ is the number of outgoing links at the source, for any packet length $m \geq n'$ (largest possible range), with only the restriction that $\ell \lt q$ (size of the base field). By lifting this construction, coding schemes for non-coherent communication are obtained with information rates close to optimal for practical instances. A Welch-Berlekamp sum-rank decoding algorithm for linearized Reed-Solomon codes is provided, having quadratic complexity in the total length $n = \ell n'$, and which can be adapted to handle not only errors, but also erasures, wire-tap observations and non-coherent communication.