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 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 communication and perfect secrecy is obtained based on linearized Reed-Solomon codes. The scheme achieves the maximum possible secret message size of ln' - 2t - rho - mu 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) approximate to l(n'), which is always smaller than that of a Gabidulin code tailored for l shots, which would be at least 2(ln)'. A Welch-Berlekamp sum-rank decoding algorithm for linearized Reed-Solomon codes is provided, having quadratic complexity in the total length n = ln', 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('4)l(2)log(l)(2)) operations in F-2, whereas the most efficient known decoding algorithm for a Gabidulin code has a complexity of O(n('3.69) l(3.69) log(l)(2)) operations in F-2, assuming a multiplication in a finite field F costs about log(|F|)(2) operations in F-2.