Low-Rank Parity-Check Codes Over Finite Commutative Rings

Kamche, Hermann Tchatchiem; Kalachi, Hervé Talé; Djomou, Franck Rivel Kamwa; Fouotsa, Emmanuel

doi:10.48550/arxiv.2106.08712

“…Puchinger et al (2021) studied the first decoding algorithm for Gabidulin codes over Galois rings that has a provable quadratic complexity in the code length. The papers (Renner et al, 2020b;Renner et al, 2021a;Djomou et al, 2021;Kamche et al, 2021) study low-rank parity-check codes over various finite rings.…”

Section: Discussionmentioning

confidence: 99%

Rank-Metric Codes and Their Applications

Bartz

¹

,

Holzbaur

²

,

Liu

³

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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“…Puchinger et al (2021) studied the first decoding algorithm for Gabidulin codes over Galois rings that has a provable quadratic complexity in the code length. The papers (Renner et al, 2020b;Renner et al, 2021a;Djomou et al, 2021;Kamche et al, 2021) study low-rank parity-check codes over various finite rings.…”

Section: Discussionmentioning

confidence: 99%

Rank-Metric Codes and Their Applications

Bartz¹,

Holzbaur²,

Liu³

et al. 2022

Preprint

0

View full text Add to dashboard Cite

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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“…Proposition 3.1. For a finitely generated R-module M , we have By [30,Lemma 9] we also have the following :…”

Section: 3mentioning

confidence: 99%

“…In [49], the recent and promising family of Low-Rank Parity-Check (LRPC) codes [20] was also generalized to the rings of integers modulo a prime power. This work was followed by the paper of Renner, Neri, and Puchinger [48] that defined LRPC codes over Galois rings, the paper of Kamwa, Tale, and Fouotsa [32] that generalized LRPC codes to the ring of integers modulo a positive integer and finally the work from [30] where the authors generalize LRPC codes to finite commutative rings. Note that LRPC codes is known as having a very poorer algebraic structure and, as a consequence, their use in code-based cryptography closes the door to structural attacks and in this case, a cryptanalysis must focus on the problem of solving the rank decoding problem.…”

mentioning

confidence: 99%

On the rank decoding problem over finite principal ideal rings

Kalachi¹,

Kamche²

2024

AMC

2

0

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The rank decoding problem has been the subject of much attention in this last decade. This problem, which is at the base of the security of publickey cryptosystems based on rank metric codes, is traditionally studied over finite fields. But the recent generalizations of certain classes of rank-metric codes from finite fields to finite rings have naturally created the interest to tackle the rank decoding problem in the case of finite rings. In this paper, we show that solving the rank decoding problem over finite principal ideal rings is at least as hard as the rank decoding problem over finite fields. We also show that computing the minimum rank distance for linear codes over finite principal ideal rings is equivalent to the same problem for linear codes over finite fields. Finally, we provide combinatorial type algorithms for solving the rank decoding problem over finite chain rings together with their average complexities.

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Low-Rank Parity-Check Codes Over Finite Commutative Rings

Cited by 4 publications

References 11 publications

Rank-Metric Codes and Their Applications

Rank-Metric Codes and Their Applications

Rank-Metric Codes and Their Applications

On the rank decoding problem over finite principal ideal rings

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