Application of rank metric codes in digital image watermarking

Lefèvre, Pascal; Carré, Philippe; Gaborit, Philippe

doi:10.1016/j.image.2018.12.015

Cited by 14 publications

(9 citation statements)

References 17 publications

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“…For instance, rank-metric codes can be used in combinatorics (Lewis and Morales, 2020), to construct linear authentication codes (Wang et al, 2003) and space-time codes (Gabidulin et al, 2000;Lusina et al, 2003;Lu and Kumar, 2004;Augot et al, 2013;Kamche and Mouaha, 2019) (see (Martínez-Peñas et al, 2021) for a recent survey). They have also been used for digital image watermarking (Lefèvre et al, 2019), low-rank matrix recovery (Forbes and Shpilka, 2012;Müelich et al, 2017) (using rank-metric codes over infinite fields (Roth, 1991;Roth, 1996;Augot et al, 2018)), and private information retrieval over networks (Tajeddine et al, 2019).…”

Section: Further Applications Of Rank-metric Codesmentioning

confidence: 99%

Rank-Metric Codes and Their Applications

Bartz¹,

Holzbaur²,

Liu³

et al. 2022

Preprint

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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: Further Applications Of Rank-metric Codesmentioning

confidence: 99%

Rank-Metric Codes and Their Applications

Bartz¹,

Holzbaur²,

Liu³

et al. 2022

Preprint

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“…having rank t and therefore rk(S) = t according to (9). Table I shows the simulation results, including the different scenarios for comparison.…”

Section: ) 2-interleaved Gabidulin Code: Simulation Of a 2-interleavedmentioning

confidence: 99%

“…The rank metric measures the distance between two codewords, represented as matrices, as the rank of their differences. Gabidulin codes are of interest for many applications related to communication, cryptography, space-time coding, network coding, distributed storage systems and digital watermarking [4]- [9].…”

Section: Introductionmentioning

confidence: 99%

Decoding of Space-Symmetric Rank Errors

Jerkovits¹,

Sidorenko²,

Wachter-Zeh³

2021

Preprint

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This paper investigates the decoding of certain Gabidulin codes that were transmitted over a channel with space-symmetric errors. Space-symmetric errors are additive error matrices that have the property that their column and row spaces are equal. We show that for channels restricted to space-symmetric errors, with high probability errors of rank up to 2(n − k)/3 can be decoded with a Gabidulin code of length n and dimension k, using a weak-self orthogonal basis as code locators.

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“…For example, Baudry et al discuss in [6] the coding strategies and robustness with BSC or AWGN watermarking channels. In [7], Lefèvre et al completes the study of robust watermarking by applying rank metric codes in other error structures. However, the use of codes in semi-fragile watermarking slightly differs from robust watermarking.…”

Section: Introductionmentioning

confidence: 99%