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Sub-Quadratic Decoding of Gabidulin Codes [article]

Sven Puchinger, Antonia Wachter-Zeh
2016 arXiv   pre-print
This paper shows how to decode errors and erasures with Gabidulin codes in sub-quadratic time in the code length, improving previous algorithms which had at least quadratic complexity.  ...  In particular, we present fast algorithms for division, multi-point evaluation and interpolation of linearized polynomials and show how to efficiently compute minimal subspace polynomials.  ...  Sub-Quadratic Decoding of Gabidulin Codes Sven Puchinger1  ... 
arXiv:1601.06280v2 fatcat:skvl6rptjzgzdn3z7jk7zrmt4i

New Communication Models and Decoding of Maximum Rank Distance Codes [article]

Wrya K. Kadir
2021 arXiv   pre-print
Also the existing decoding algorithms for generalized twisted Gabidulin codes and additive generalized twisted Gabidulin codes are improved.  ...  In this paper an interpolation-based decoding algorithm to decode Gabidulin codes, transmitted through a finely restricted channel, is proposed.  ...  one can use any decoder of a Gabidulin code GG n,k+1 to recover the message.  ... 
arXiv:2109.02069v2 fatcat:dwx3bfvahnhrjfteaairtzhbwa

On decoding additive generalized twisted Gabidulin codes

Wrya K. Kadir, Chunlei Li
2020 Cryptography and Communications  
In this paper, we consider an interpolation-based decoding algorithm for a large family of maximum rank distance codes, known as the additive generalized twisted Gabidulin codes, over the finite field  ...  This paper extends the work of the conference paper Li and Kadir (2019) presented at the International Workshop on Coding and Cryptography 2019, which decoded these codes over finite fields in characteristic  ...  The work of C. Li Funding Information Open Access funding provided by University of Bergen.  ... 
doi:10.1007/s12095-020-00449-9 fatcat:hgs4etafhjcq7a3jskfqzcwr64

Decoding High-Order Interleaved Rank-Metric Codes [article]

Sven Puchinger, Julian Renner, Antonia Wachter-Zeh
2019 arXiv   pre-print
The relation of the new algorithm to existing interleaved decoders in the special case of Gabidulin codes is given.  ...  In contrast to previously-known decoding algorithms, the new decoder works for any rank-metric code, not only Gabidulin codes.  ...  quadratic in the code length and linear in the interleaving order.  ... 
arXiv:1904.08774v1 fatcat:zbedy7fyyzfzpbjyiihrqk7ql4

Fast Operations on Linearized Polynomials and their Applications in Coding Theory [article]

Sven Puchinger, Antonia Wachter-Zeh
2017 arXiv   pre-print
Finally, it is shown how the new fast operations on linearized polynomials lead to the first error and erasure decoding algorithm for Gabidulin codes with sub-quadratic complexity.  ...  We propose a new multiplication algorithm for skew polynomials (a generalization of linearized polynomials) which has sub-quadratic complexity in the polynomial degree s, independent of the underlying  ...  To give one example, one of the reviewer's suggestions significantly shortened the proof of Theorem 15.  ... 
arXiv:1512.06520v3 fatcat:ea54m3ztzff6ri7mjdt3hexkzq

Rank-Metric Codes and Their Applications [article]

Hannes Bartz, Lukas Holzbaur, Hedongliang Liu, Sven Puchinger, Julian Renner, Antonia Wachter-Zeh
2022 arXiv   pre-print
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.  ...  Acknowledgements The work of L. Holzbaur and A. Wachter  ... 
arXiv:2203.12384v1 fatcat:brysqya7ynem5i25adxfwf5vl4

DRANKULA: A McEliece-like Rank Metric based Cryptosystem Implementation

Ameera Salem Al Abdouli, Mohamed Al Ali, Emanuele Bellini, Florian Caullery, Alexandros Hasikos, Marc Manzano, Victor Mateu
2018 Proceedings of the 15th International Joint Conference on e-Business and Telecommunications  
Namely, we use the scheme proposed by Loidreau in PQCrypto 2017 using Gabidulin codes.  ...  We provide the pseudo-code of the core algorithms of the cryptosystem. In addition, we also show code optimization when special instructions like Carry-less multiplications are available.  ...  The algorithm takes advantage of the properties of linearized polynomials to achieve a sub-quadratic complexity.  ... 
doi:10.5220/0006838102300241 dblp:conf/icete/AbdouliABCHMM18 fatcat:kvohfmb4drfljkh6t3o7y47phq

A Novel Niederreiter-like cryptosystem based on the (u|u + υ)-construction codes

Roumaissa Mahdjoubi, Pierre Louis Cayrel, Sedat Akleylek, Guenda Kenza
2021 RAIRO - Theoretical Informatics and Applications  
and λ-Gabidulin codes.  ...  The security is based on the hardness of the Rank Syndrome Decoding (RSD) problem and it presents a (u|u + υ)-construction code using two different types of codes: Ideal Low Rank Parity Check (ILRPC) codes  ...  We denote by D 1 a decoding Algorithm 1 for the LRPC code C 1 which decodes up to t (1) errors and by D 2 the error-erasure decoding Algorithm 2 of λ-Gabidulin code C 2 which decodes up to t (2) errors  ... 
doi:10.1051/ita/2021010 fatcat:b7z4ngmt4nhvrcbmlie7xdiz2m

LIGA: A Cryptosystem Based on the Hardness of Rank-Metric List and Interleaved Decoding [article]

Julian Renner, Sven Puchinger, Antonia Wachter-Zeh
2020 arXiv   pre-print
We propose the new rank-metric code-based cryptosystem LIGA which is based on the hardness of list decoding and interleaved decoding of Gabidulin codes.  ...  formally defined problems related to list decoding and interleaved decoding of Gabidulin codes.  ...  Also, we are thankful to Michael Schelling for his observation that decryption of the FL system can be seen as error-erasure decoding.  ... 
arXiv:1812.04892v2 fatcat:mz3o2goz3zdnno6dcfgcy2fuye

Efficient Interpolation-Based Decoding of Interleaved Subspace and Gabidulin Codes [article]

Hannes Bartz, Antonia Wachter-Zeh
2014 arXiv   pre-print
These two efficient algorithms can also be applied for accelerating the decoding of interleaved Gabidulin codes.  ...  An interpolation-based decoding scheme for interleaved subspace codes is presented.  ...  Interleaved Gabidulin codes are a horizontal or vertical concatenation of s Gabidulin codes, for details see [14] - [16] .  ... 
arXiv:1408.1284v1 fatcat:55fnpfgn7bbpdoxcuu7bsad2ra

LIGA: a cryptosystem based on the hardness of rank-metric list and interleaved decoding

Julian Renner, Sven Puchinger, Antonia Wachter-Zeh
2021 Designs, Codes and Cryptography  
assumptions of formally defined problems related to list decoding and interleaved decoding of Gabidulin codes.  ...  AbstractWe propose the new rank-metric code-based cryptosystem which is based on the hardness of list decoding and interleaved decoding of Gabidulin codes. is an improved variant of the Faure–Loidreau  ...  Acknowledgements The work of J. Renner and A. Wachter-Zeh was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agree-  ... 
doi:10.1007/s10623-021-00861-z fatcat:ugo3dwngfrb7rpnqwiuckzpe5i

Application of rank metric codes in digital image watermarking

Pascal Lefèvre, Philippe Carré, Philippe Gaborit
2019 Signal processing. Image communication  
In particular, we describe a rank metric code family called Gabidulin codes analogous to the well known Reed-Solomon codes.  ...  One can decode the original message if the matrix rank of the detected codeword is small enough. We propose a study to validate the concept of rank metric in watermarking applications.  ...  Many algorithms to decode Gabidulin codes has been proposed in the literature such as [8, 9] . Reed-Solomon code decoding algorithm can be generalized for Gabidulin codes.  ... 
doi:10.1016/j.image.2018.12.015 fatcat:o47ai36h75dd7jjuawuuhio55u

Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric [article]

Hannes Bartz and Thomas Jerkovits and Sven Puchinger and Johan Rosenkilde
2021 arXiv   pre-print
We speed up existing decoding algorithms for three code classes in different metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved Gabidulin codes in the subspace metric, and linearized  ...  The new algorithms for the various decoding-related computational problems are interesting in their own and have further applications, in particular parts of decoders of several other codes and foundational  ...  The relation to decoding interleaved Gabidulin codes was given in [5] and lifted interleaved Gabidulin codes in [6] .  ... 
arXiv:2005.09916v2 fatcat:sqtkw73frraqhfiol7evekjcwy

Reliable and Secure Multishot Network Coding using Linearized Reed-Solomon Codes

Umberto Martinez-Penas, Frank R. Kschischang
2019 IEEE Transactions on Information Theory  
A Welch-Berlekamp sum-rank decoding algorithm for linearized Reed-Solomon codes is provided, having quadratic complexity in the total length n = ℓ n^' , and which can be adapted to handle not only errors  ...  The required field size is q^m , where q > ℓ , thus q^m ≈ℓ^n^' , which is always smaller than that of a Gabidulin code tailored for ℓ shots, which would be at least 2^ℓ n^' .  ...  With this point of view, the use of linearized Reed-Solomon codes instead of Gabidulin codes as MSRD codes extends the idea of using Reed-Solomon codes rather than Gabidulin codes as MDS codes (see Figs  ... 
doi:10.1109/tit.2019.2912165 fatcat:qziaieilgvevddundhpyqpg6ty

Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric

Hannes Bartz, Thomas Jerkovits, Sven Puchinger, Johan Rosenkilde
2021 IEEE Transactions on Information Theory  
that users recognise and abide by the legal requirements associated with these rights.  Users may download and print one copy of any publication from the public portal for the purpose of private study  ...  General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications  ...  The relation to decoding interleaved Gabidulin codes was given in [5] and lifted interleaved Gabidulin codes in [6] .  ... 
doi:10.1109/tit.2021.3067318 fatcat:sbfunjhzwfgkvhsbz47tcnlgoe
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