Low-rank parity-check codes over Galois rings

Low-rank parity-check (LRPC) codes are rank-metric codes over finite fields, which have been proposed by Gaborit et al. (Proceedings of the workshop on coding and cryptography WCC, vol 2013, 2013) for cryptographic applications. Inspired by a recent adaption of Gabidulin codes to certain finite ring...

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Published in:Designs, codes, and cryptography Vol. 89; no. 2; pp. 351 - 386
Main Authors: Renner, Julian, Neri, Alessandro, Puchinger, Sven
Format: Journal Article
Language:English
Published: New York Springer US 01.02.2021
Springer Nature B.V
Subjects:
Algorithms
Binary system
Codes
Coding and Information Theory
Computer Science
Cryptography
Cryptology
Decoding
Discrete Mathematics in Computer Science
Fields (mathematics)
Parity
Rings (mathematics)
Upper bounds
Rank-metric codes
Algebraic coding theory
11T71
Low-rank parity-check codes
Galois rings
ISSN:0925-1022, 1573-7586, 1573-7586
Online Access:Get full text
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Summary:Low-rank parity-check (LRPC) codes are rank-metric codes over finite fields, which have been proposed by Gaborit et al. (Proceedings of the workshop on coding and cryptography WCC, vol 2013, 2013) for cryptographic applications. Inspired by a recent adaption of Gabidulin codes to certain finite rings by Kamche et al. (IEEE Trans Inf Theory 65(12):7718–7735, 2019), we define and study LRPC codes over Galois rings—a wide class of finite commutative rings. We give a decoding algorithm similar to Gaborit et al.’s decoder, based on simple linear-algebraic operations. We derive an upper bound on the failure probability of the decoder, which is significantly more involved than in the case of finite fields. The bound depends only on the rank of an error, i.e., is independent of its free rank. Further, we analyze the complexity of the decoder. We obtain that there is a class of LRPC codes over a Galois ring that can decode roughly the same number of errors as a Gabidulin code with the same code parameters, but faster than the currently best decoder for Gabidulin codes. However, the price that one needs to pay is a small failure probability, which we can bound from above.
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Communicated by I. Landjev.
ISSN:0925-1022
1573-7586
1573-7586
DOI:10.1007/s10623-020-00825-9