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Probl. Peredachi Inf., 1985, Volume 21, Issue 1, Pages 3–16 (Mi ppi967)  

This article is cited in 53 scientific papers (total in 53 papers)

Information Theory and Coding Theory

Theory of Codes with Maximum Rank Distance

È. M. Gabidulin


Abstract: The article considers codes over $GF(q^N)$. A new metric, called the rank metric, is introduced; the maximum number of coordinates of vector $\mathbf{x}=(x_1,…,x_n)$ that are linearly dependent over $GF(q)$ is called its norm. For this metric a theory analogous to the theory of MDS codes is formulated. Codes with maximum rank distance are described; their spectrum is obtained; and encoding and decoding algorithms are given.

Full text: PDF file (1379 kB)

English version:
Problems of Information Transmission, 1985, 21:1, 1–12

Bibliographic databases:

UDC: 621.391.15
Received: 17.09.1984

Citation: È. M. Gabidulin, “Theory of Codes with Maximum Rank Distance”, Probl. Peredachi Inf., 21:1 (1985), 3–16; Problems Inform. Transmission, 21:1 (1985), 1–12

Citation in format AMSBIB
\Bibitem{Gab85}
\by \`E.~M.~Gabidulin
\paper Theory of Codes with Maximum Rank Distance
\jour Probl. Peredachi Inf.
\yr 1985
\vol 21
\issue 1
\pages 3--16
\mathnet{http://mi.mathnet.ru/ppi967}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=791529}
\zmath{https://zbmath.org/?q=an:0585.94013}
\transl
\jour Problems Inform. Transmission
\yr 1985
\vol 21
\issue 1
\pages 1--12


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. V. Urivskii, T. Johansson, “New Technique for Decoding Codes in the Rank Metric and Its Cryptography Applications”, Problems Inform. Transmission, 38:3 (2002), 237–246  mathnet  crossref  mathscinet  zmath
    2. Manoj K.N., Rajan B.S., “Full rank distance codes and optimal STBC for BPSK modulation”, ISIT: 2002 IEEE International Symposium on Information Theory, Proceedings, 2002, 276–276  crossref  isi
    3. Gabidulin, EM, “Reducible rank codes and their applications to cryptography”, IEEE Transactions on Information Theory, 49:12 (2003), 3289  crossref  mathscinet  isi
    4. È. M. Gabidulin, V. A. Obernikhin, “Codes in the Vandermonde $\mathcal F$-Metric and Their Applications”, Problems Inform. Transmission, 39:2 (2003), 159–169  mathnet  crossref  mathscinet  zmath
    5. Berger T., Loidreau Pierre., “Designing an efficient and secure public-key cryptosystem based on reducible rank codes”, Lecture Notes in Comput. Sci., Progress in cryptology (INDOCRYPT 2004), 3348, Springer, Berlin, 2004, 218–229  crossref  mathscinet  zmath  isi
    6. È. M. Gabidulin, N. I. Pilipchuk, “Symmetric Rank Codes”, Problems Inform. Transmission, 40:2 (2004), 103–117  mathnet  crossref  mathscinet  zmath
    7. Pilipchuk, NI, “On codes correcting symmetric rank errors”, Coding and Cryptography, 3969 (2006), 14  crossref  mathscinet  zmath  isi
    8. Loidreau, P, “A Welch-Berlekamp like algorithm for decoding Gabidulin codes”, Coding and Cryptography, 3969 (2006), 36  crossref  mathscinet  zmath  isi
    9. Overbeck, R, “Extending Gibson's attacks on the GPT cryptosystem”, Coding and Cryptography, 3969 (2006), 178  crossref  mathscinet  zmath  isi
    10. Faure, C, “A new public-key cryptosystem based on the problem of reconstructing p-polynomials”, Coding and Cryptography, 3969 (2006), 304  crossref  mathscinet  zmath  isi
    11. Gabidulin, EM, “Symmetric matrices and codes correcting rank errors beyond the [(d-1)/2] bound”, Discrete Applied Mathematics, 154:2 (2006), 305  crossref  mathscinet  zmath  isi  elib
    12. Gadouleau M., Yan Zh., “Decoder error probability of MRD codes”, Proceedings of 2006 IEEE Information Theory Workshop, 2006, 264–268  crossref  isi
    13. Gadouleau M., Yan Zh., “Security of the GPT-type cryptosystems”, 2006 IEEE International Symposium on Information Theory, 2006, 724–728  crossref  isi
    14. Gabidulin, EM, “Error and erasure correcting algorithms for rank codes”, Designs Codes and Cryptography, 49:1–3 (2008), 105  crossref  mathscinet  isi
    15. Gabidulin, EM, “Properties of subspace subcodes of Gabidulin codes”, Advances in Mathematics of Communications, 2:2 (2008), 147  crossref  mathscinet  zmath  isi
    16. Gabidulin E.M., Rashwan H., Honary B., “On improving security of GPT cryptosystems”, 2009 IEEE International Symposium on Information Theory, 2009, 1110–1114  crossref  isi
    17. Bossert M., Gabidulin E.M., “One family of Algebraic Codes for Network Coding”, 2009 IEEE International Symposium on Information Theory, 2009, 2863–2866  crossref  isi
    18. E. M. Gabidulin, N. I. Pilipchuk, M. Bossert, “Decoding of random network codes”, Problems Inform. Transmission, 46:4 (2010), 300–320  mathnet  crossref  mathscinet  isi
    19. Rashwan H., Gabidulin E.M., Honary B., “A Smart Approach for GPT Cryptosystem Based on Rank Codes”, 2010 IEEE International Symposium on Information Theory, IEEE International Symposium on Information Theory, 2010, 2463–2467  isi
    20. Loidreau P., “Designing a Rank Metric Based McEliece Cryptosystem”, Post-Quantum Cryptography, Proceedings, Lecture Notes in Computer Science, 6061, 2010, 142–152  crossref  isi
    21. A. Wachter, V. R. Sidorenko, M. Bossert, V. V. Zyablov, “On (partial) unit memory codes based on Gabidulin codes”, Problems Inform. Transmission, 47:2 (2011), 117–129  mathnet  crossref  mathscinet  isi
    22. B. A. Pogorelov, M. A. Pudovkina, “Naturalnye metriki i ikh svoistva. Ch. 1. Podmetriki i nadmetriki”, Matem. vopr. kriptogr., 2:4 (2011), 49–74  mathnet  crossref
    23. Sidorenko V., Richter G., Bossert M., “Linearized Shift-Register Synthesis”, IEEE Trans Inform Theory, 57:9 (2011), 6025–6032  crossref  isi  elib
    24. Rashwan H., Gabidulin E.M., Honary B., “Security of the GPT cryptosystem and its applications to cryptography”, Security and Communication Networks, 4:8 (2011), 937–946  crossref  isi
    25. Etzion T., “Sequence Folding, Lattice Tiling, and Multidimensional Coding”, IEEE Trans Inform Theory, 57:7 (2011), 4383–4400  crossref  isi  elib
    26. Gaborit Ph., Schrek J., Zemor G., “Full Cryptanalysis of the Chen Identification Protocol”, Post-Quantum Cryptography, Lecture Notes in Computer Science, 7071, ed. Yang B., Springer-Verlag Berlin, 2011, 35–50  crossref  isi
    27. Yang Sh., Honold T., “Good Random Matrices Over Finite Fields”, Adv. Math. Commun., 6:2 (2012), 203–227  crossref  isi  elib
    28. Tan V.Y.F., Balzano L., Draper S.C., “Rank Minimization Over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations”, IEEE Trans. Inf. Theory, 58:4 (2012), 2018–2039  crossref  isi  elib
    29. E. M. Gabidulin, N. I. Pilipchuk, “Rank subcodes in multicomponent network coding”, Problems Inform. Transmission, 49:1 (2013), 40–53  mathnet  crossref  isi
    30. E. M. Gabidulin, N. I. Pilipchuk, B. Honary, H. Rashwan, “Information security in a random network coding network”, Problems Inform. Transmission, 49:2 (2013), 179–191  mathnet  crossref  isi
    31. Gaborit Ph., Ruatta O., Schrek J., Zemor G., “Ranksign: An Efficient Signature Algorithm Based on the Rank Metric”, Post-Quantum Cryptography, Pqcrypto 2014, Lecture Notes in Computer Science, 8772, ed. Mosca M., Springer-Verlag Berlin, 2014, 88–107  crossref  isi
    32. Gaborit Ph., Ruatta O., Schrek J., Zemor G., “New Results For Rank-Based Cryptography”, Progress in Cryptology - Africacrypt 2014, Lecture Notes in Computer Science, 8469, eds. Pointcheval D., Vergnaud D., Springer-Verlag Berlin, 2014, 1–12  crossref  isi
    33. O. V. Trushina, E. M. Gabidulin, “A new method for ensuring anonymity and security in network coding”, Problems Inform. Transmission, 51:1 (2015), 75–81  mathnet  crossref  isi  elib
    34. E. M. Gabidulin, N. I. Pilipchuk, “Multicomponent codes with maximum code distance”, Problems Inform. Transmission, 52:3 (2016), 276–283  mathnet  crossref  isi  elib
    35. Pilipchuk N.I., “Cardinality of Subspace Multicomponent Codes”, 2017 Fourth International Conference on Engineering and Telecommunication (En&T), ed. Pavlyukova E. Uzhinskaya L. Suroegina Z., IEEE, 2017, 11–14  crossref  isi
    36. Puchinger S., Nielsen Johan Rosenkilde Ne, Li W., Sidorenko V., “Row Reduction Applied to Decoding of Rank-Metric and Subspace Codes”, Designs Codes Cryptogr., 82:1-2, SI (2017), 389–409  crossref  isi
    37. Horlemann-Trautmann A.-L., Marshall K., Rosenthal J., “Extension of Overbeck'S Attack For Gabidulin-Based Cryptosystems”, Designs Codes Cryptogr., 86:2, SI (2018), 319–340  crossref  mathscinet  zmath  isi  scopus
    38. Lau Terry Shue Chien, Tan Ch.H., “Key Recovery Attack on Mcnie Based on Low Rank Parity Check Codes and Its Reparation”, Advances in Information and Computer Security, Iwsec 2018, Lecture Notes in Computer Science, 11049, eds. Inomata A., Yasuda K., Springer International Publishing Ag, 2018, 19–34  crossref  zmath  isi  scopus
    39. Kim J.-L., Galvez L., Kim Y.-S., Lee N., “A New Lrpc-Kronecker Product Codes Based Public-Key Cryptography”, Apkc'18: Proceedings of the 5Th Acm Asia Public-Key Cryptography Workshop, Assoc Computing Machinery, 2018, 25–33  crossref  isi  scopus
    40. Martinez-Penas U., Kschischang F.R., “Reliable and Secure Multishot Network Coding Using Linearized Reed-Solomon Codes”, 2018 56Th Annual Allerton Conference on Communication, Control, and Computing (Allerton), Annual Allerton Conference on Communication Control and Computing, IEEE, 2018, 702–709  isi
    41. Aragon N., Gaborit Ph., Hauteville A., Tillich J.-P., “A New Algorithm For Solving the Rank Syndrome Decoding Problem”, 2018 IEEE International Symposium on Information Theory (Isit), IEEE International Symposium on Information Theory, IEEE, 2018, 2421–2425  isi
    42. Martinez-Penas U., “Skew and Linearized Reed-Solomon Codes and Maximum Sum Rank Distance Codes Over Any Division Ring”, J. Algebra, 504 (2018), 587–612  crossref  mathscinet  zmath  isi
    43. Aragon N., Gaborit Ph., Hauteville A., Ruatta O., Zemor G., “Low Rank Parity Check Codes: New Decoding Algorithms and Applications to Cryptography”, IEEE Trans. Inf. Theory, 65:12 (2019), 7697–7717  crossref  mathscinet  zmath  isi  scopus
    44. Kamche H.T. Mouaha Ch., “Rank-Metric Codes Over Finite Principal Ideal Rings and Applications”, IEEE Trans. Inf. Theory, 65:12 (2019), 7718–7735  crossref  mathscinet  zmath  isi  scopus
    45. Lau Terry Shue Chien, Tan Ch.H., “New Rank Codes Based Encryption Scheme Using Partial Circulant Matrices”, Designs Codes Cryptogr., 87:12 (2019), 2979–2999  crossref  mathscinet  zmath  isi  scopus
    46. Antrobus J. Gluesing-Luerssen H., “Maximal Ferrers Diagram Codes: Constructions and Genericity Considerations”, IEEE Trans. Inf. Theory, 65:10 (2019), 6204–6223  crossref  mathscinet  zmath  isi  scopus
    47. Liu X., Liu H., “Rank-Metric Complementary Dual Codes”, J. Appl. Math. Comput., 61:1-2 (2019), 281–295  crossref  mathscinet  zmath  isi  scopus
    48. Martinez-Penas U., “Theory of Supports For Linear Codes Endowed With the Sum-Rank Metric”, Designs Codes Cryptogr., 87:10 (2019), 2295–2320  crossref  mathscinet  zmath  isi  scopus
    49. Kadhe S., El Rouayheb S., Duursma I., Sprintson A., “Codes With Locality in the Rank and Subspace Metrics”, IEEE Trans. Inf. Theory, 65:9 (2019), 5454–5468  crossref  mathscinet  zmath  isi
    50. Martinez-Penas U., “Universal Secure Rank-Metric Coding Schemes With Optimal Communication Overheads”, Cryptogr. Commun., 11:2 (2019), 147–166  crossref  mathscinet  zmath  isi  scopus
    51. Khan E., Khalid A., Ali A., Atif M., Khan A.S., “Geo Security Using Gpt Cryptosystem”, Int. J. Adv. Comput. Sci. Appl., 11:2 (2020), 514–521  isi
    52. Xu Ch., Zhang Y., Mei L., Xu L., Zuo C., “Attacking the Niederreiter-Type Cryptosystem Based on Rank Metric”, Int. J. Embed. Syst., 13:4 (2020), 398–404  crossref  isi
    53. Renner J., Neri A., Puchinger S., “Low-Rank Parity-Check Codes Over Galois Rings”, Designs Codes Cryptogr., 89:2 (2021), 351–386  crossref  mathscinet  zmath  isi  scopus
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