Efficient Decoding of Interleaved Low-Rank Parity-Check Codes

Renner, Julian; Jerkovits, Thomas; Bartz, Hannes

doi:10.1109/redundancy48165.2019.9003356

Cited by 10 publications

(4 citation statements)

References 12 publications

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“…In (Renner et al, 2019b), interleaved low-rank parity-check codes were defined (similar as vertically and heterogeneously interleaved Gabidulin codes), an efficient decoding algorithm for this code class was devised, and upper bounds on the decoding failure rate of this algorithm were derived.…”

Section: Discussionmentioning

confidence: 99%

Rank-Metric Codes and Their Applications

Bartz

Holzbaur

Liu

et al. 2022

FNT in Communications and Information Theory

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The rank metric measures the distance between two matrices by the rank of their difference. Codes designed for the rank metric have attracted considerable attention in recent years, reinforced by network coding and further motivated by a variety of applications. In code-based cryptography, the hardness of the corresponding generic decoding problem can lead to systems with reduced public-key size. In distributed data storage, codes in the rank metric have been used repeatedly to construct codes with locality, and in coded caching, they have been employed for the placement of coded symbols. This survey gives a general introduction to rank-metric codes, explains their most important applications, and highlights their relevance to these areas of research.Codes in the rank metric have drawn increasing interest in recent years due to their application to cryptography, distributed storage and network coding. The survey by Gorla and Ravagnani (2018) mainly focuses on combinatorial properties of rank-metric codes, while the one by Sheekey (2019a) considers MRD codes and their properties. In particular, Gorla and Ravagnani (2018) and Gorla ( 2019) treat (amongst others): isometries, anticodes, duality, MacWilliams identities, generalized weights in the rank metric. We will therefore not considers these topics in this survey. An English version of the textbook by Gabidulin (2021) has been published recently which contains a collection of Gabidulin's results in the area of rank-metric codes. Our survey deals shortly with properties of rank-metric codes, but the main focus is on their decoding and their applications to code-based cryptography, storage, and network coding.In this chapter, we give an introduction to rank-metric codes, their properties and decoding. In Section 2.1, we provide the basic notation used in this survey. Section 2.2 defines linearized polynomials and recalls some basic properties. In Section 2.3, we define the rank metric and state bounds on the cardinality of rank-metric codes, i.e., spherepacking, Gilbert-Varshamov, and Singleton-like bounds. Section 2.4, 2.5 and 2.6 provide the weight distribution of rank metric codes, constant-rank codes and covering property of rank-metric codes, respectively. Section 2.7 defines Gabidulin codes, proves their minimum rank distance and gives generator and parity-check matrices. In Section 2.8, we describe syndrome-based decoding of Gabidulin codes, i.e., we prove the key equation, show how to solve it and how to reconstruct the final error. We also outline error-erasure decoding and summarize known fast decoders. In Section 2.9, we discuss results on list decoding of Gabidulin codes. Further, we introduce interleaved Gabidulin codes (Section 2.10), folded Gabidulin codes (Section 2.11) and summarize further classes of MRD codes (Section 2.13).

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Section: Discussionmentioning

confidence: 99%

Rank-Metric Codes and Their Applications

Bartz

Holzbaur

Liu

et al. 2022

FNT in Communications and Information Theory

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“…It is, however, not very restrictive: if the parity-check matrix entries H i,j are chosen uniformly at random from F, this property is fulfilled with the probability that a random λ(n − k) × n matrix has full (free) rank n. This probability is arbitrarily close to 1 for increasing difference of λ(n − k) and n (cf. [19] for the field and Lemma 7 in Section 5.2 for the ring case).…”

Section: Unity Property If Every Entrymentioning

confidence: 95%

Low-Rank Parity-Check Codes over Galois Rings

Renner,

Neri,

Puchinger

2020

Preprint

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Low-rank parity-check (LRPC) are rank-metric codes over finite fields, which have been proposed by Gaborit et al. (2013) for cryptographic applications. Inspired by a recent adaption of Gabidulin codes to certain finite rings by Kamche et al. (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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“…Thus, horizontal interleaving appears to be better suited for cryptographic purposes. This is also reflected in recent proposals as for example in the KEM LowMS [6] that is based on horizontally interleaved Gabidulin codes, in the signature scheme Durandal [5] based on the closely related rank-support-learning (RSL) problem [10], and in the cryptosystem [2] that makes use of horizontally interleaved low-rank parity-check (LRPC) codes [39].…”

Section: Introductionmentioning

confidence: 93%

Interpolation-based decoding of folded variants of linearized and skew Reed–Solomon codes

Hörmann

Bartz

2023

Des. Codes Cryptogr.

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The sum-rank metric is a hybrid between the Hamming metric and the rank metric and suitable for error correction in multishot network coding and distributed storage as well as for the design of quantum-resistant cryptosystems. In this work, we consider the construction and decoding of folded linearized Reed–Solomon (FLRS) codes, which are shown to be maximum sum-rank distance (MSRD) for appropriate parameter choices. We derive an efficient interpolation-based decoding algorithm for FLRS codes that can be used as a list decoder or as a probabilistic unique decoder. The proposed decoding scheme can correct sum-rank errors beyond the unique decoding radius with a computational complexity that is quadratic in the length of the unfolded code. We show how the error-correction capability can be optimized for high-rate codes by an alternative choice of interpolation points. We derive a heuristic upper bound on the decoding failure probability of the probabilistic unique decoder and verify its tightness by Monte Carlo simulations. Further, we study the construction and decoding of folded skew Reed-Solomon codes in the skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with different block sizes that come along with an efficient decoding algorithm.

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Efficient Decoding of Interleaved Low-Rank Parity-Check Codes

Cited by 10 publications

References 12 publications

Rank-Metric Codes and Their Applications

Rank-Metric Codes and Their Applications

Low-Rank Parity-Check Codes over Galois Rings

Interpolation-based decoding of folded variants of linearized and skew Reed–Solomon codes

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