Reliable and Secure Multishot Network Coding using Linearized Reed-Solomon Codes
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Multishot network coding is considered in a worst-case adversarial setting in which an omniscient adversary with limitless computational resources may inject erroneous packets in up to t links, erase up to ρ packets, and wire-tap up to μ links, all throughout ℓ 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 ℓ n^' - 2t - ρ - μ packets, where n^' is the number of outgoing links, for any packet length m ≥ n^' (largest possible range), with only the restriction that ℓ < 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 = ℓ n^' , and which can be adapted to handle not only errors, but also erasures, wire-tap observations and non-coherent communication.
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