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 <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> links, erase up to <inline-formula> <tex-math notation="LaTeX">$\rho $ </tex-math></inline-formula> packets, and wire-tap up to <inline-formula> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula> links, all throughout <inline-formula> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$\ell n^\prime - 2t - \rho - \mu $ </tex-math></inline-formula> packets for coherent communication, where <inline-formula> <tex-math notation="LaTeX">$n^\prime $ </tex-math></inline-formula> is the number of outgoing links at the source, for any packet length <inline-formula> <tex-math notation="LaTeX">$m \geq n^\prime $ </tex-math></inline-formula> (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 <inline-formula> <tex-math notation="LaTeX">$q^{m} $ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$q > \ell $ </tex-math></inline-formula>, thus <inline-formula> <tex-math notation="LaTeX">$q^{m} \approx \ell ^{n^\prime } $ </tex-math></inline-formula>, which is always smaller than that of a Gabidulin code tailored for <inline-formula> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> shots, which would be at least <inline-formula> <tex-math notation="LaTeX">$2^{\ell n^\prime } $ </tex-math></inline-formula>. A Welch-Berlekamp sum-rank decoding algorithm for linearized Reed-Solomon codes is provided, having quadratic complexity in the total length <inline-formula> <tex-math notation="LaTeX">$n = \ell n^\prime $ </tex-math></inline-formula>, 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 <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}(n^{\prime 4} \ell ^{2} \log (\ell)^{2}) $ </tex-math></inline-formula> operations in <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{2} $ </tex-math></inline-formula>, whereas the most efficient known decoding algorithm for a Gabidulin code has a complexity of <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}(n^{\prime 3.69} \ell ^{3.69} \log (\ell)^{2}) $ </tex-math></inline-formula> operations in <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{2} $ </tex-math></inline-formula>, assuming a multiplication in a finite field <inline-formula> <tex-math notation="LaTeX">$\mathbb {F} $ </tex-math></inline-formula> costs about <inline-formula> <tex-math notation="LaTeX">$\log (|\mathbb {F}|)^{2} $ </tex-math></inline-formula> operations in <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{2} $ </tex-math></inline-formula>.