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 linearly-coded network. Assuming no knowledge of the underlying linear network code (in particular, the network topology and underlying linear code may be random and 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 $\ell n^\prime - 2t - \rho - \mu$ packets for coherent communication, where $n^\prime$ is the number of outgoing links at the source, for any packet length $m \geq n^\prime$ (largest possible range). By lifting this construction, coding schemes for non-coherent communication are obtained with information rates close to optimal for practical instances. The required field size is $q^m$, where $q > \ell$, thus $q^m \approx \ell^{n^\prime}$, which is always smaller than that of a Gabidulin code tailored for $\ell$ shots, which would be at least $2^{\ell n^\prime}$. A Welch-Berlekamp sum-rank decoding algorithm for linearized Reed-Solomon codes is provided, having quadratic complexity in the total length $n = \ell n^\prime$, and which can be adapted to handle not only errors, but also erasures, wire-tap observations and non-coherent communication. Combined with the obtained field size, the given decoding complexity is of $\mathcal{O}(n^{\prime 4} \ell^2 \log(\ell)^2)$ operations in $\mathbb{F}_2$.