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

Umberto Martinez-Penas, Frank R. Kschischang
<span title="">2019</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/niovmjummbcwdg4qshgzykkpfu" style="color: black;">IEEE Transactions on Information Theory</a> </i> &nbsp;
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 ρ packets, and wire-tap up to μ links, all throughout ℓ 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
more &raquo; ... nication 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 for coherent communication, where n^' is the number of outgoing links at the source, for any packet length m ≥ n^' (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 > ℓ , thus q^m ≈ℓ^n^' , which is always smaller than that of a Gabidulin code tailored for ℓ shots, which would be at least 2^ℓ n^' . 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. Combined with the obtained field size, the given decoding complexity is of O(n^' 4ℓ^2 (ℓ)^2) operations in F_2 .
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