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,...

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Vydané v:IEEE transactions on information theory Ročník 65; číslo 8; s. 4785 - 4803
Hlavní autori: Martinez-Penas, Umberto, Kschischang, Frank R.
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York IEEE 01.08.2019
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Predmet:
Algorithms
Codes
Coding
Coherence
Communication
Complexity
Complexity theory
Decoding
Encoding
Fields (mathematics)
Knowledge engineering
Knowledge management
Linearization
Linearized Reed-Solomon codes
Links
Multiplication
multishot network coding
Network coding
network error-correction
Network topologies
Packets (communication)
Reed-Solomon codes
Reliability
sum-rank metric
sum-subspace codes
Wire
wire-tap channel
Wiretapping
ISSN:0018-9448, 1557-9654
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Shrnutí: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>.
Bibliografia:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2019.2912165