Author

# Ernst M. Gabidulin

Bio: Ernst M. Gabidulin is an academic researcher from Moscow Institute of Physics and Technology. The author has contributed to research in topics: Block code & Linear code. The author has an hindex of 22, co-authored 76 publications receiving 1609 citations.

##### Papers published on a yearly basis

##### Papers

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08 Apr 1991TL;DR: A new modification of the McEliece public-key cryptosystem is proposed that employs the so-called maximum-rank-distance codes in place of Goppa codes and that hides the generator matrix of the MRD code by addition of a randomly-chosen matrix.

Abstract: A new modification of the McEliece public-key cryptosystem is proposed that employs the so-called maximum-rank-distance (MRD) codes in place of Goppa codes and that hides the generator matrix of the MRD code by addition of a randomly-chosen matrix. A short review of the mathematical background required for the construction of MRD codes is given. The cryptanalytic work function for the modified McEliece system is shown to be much greater than that of the original system. Extensions of the rank metric are also considered.

265 citations

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31 Oct 2005TL;DR: A new construction of rank codes is presented, which defines new codes and includes known codes, and it is argued that these are different codes.

Abstract: The only known construction of error-correcting codes in rank metric was proposed in 1985. These were codes with fast decoding algorithm. We present a new construction of rank codes, which defines new codes and includes known codes. This is a generalization of E.M. Gabidulin, 1985. Though the new codes seem to be very similar to subcodes of known rank codes, we argue that these are different codes. A fast decoding algorithm is described

155 citations

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TL;DR: This work provides construction methods of full-rank codes over different complex signal constellations, for arbitrary numbers of antennas, and codeword periods, and derives a Singleton-type bound on the rate of a code for the rank metric.

Abstract: The critical design criterion for space-time codes in asymptotically good channels is the minimum rank between codeword pairs. Rank codes are a two-dimensional matrix code construction where by the rank is the metric of merit. We look at the application of rank codes to space-time code design. In particular, we provide construction methods of full-rank codes over different complex signal constellations, for arbitrary numbers of antennas, and codeword periods. We also derive a Singleton-type bound on the rate of a code for the rank metric, and we show that rank codes satisfy this bound with equality.

140 citations

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TL;DR: A new family of so-called reducible rank codes which are a generalization of rank product codes is presented, which includes maximal rank distance (MRD) codes for lengths n>N in the field F/sub N/.

Abstract: We present a new family of so-called reducible rank codes which are a generalization of rank product codes . This family includes maximal rank distance (MRD) codes for lengths n>N in the field F/sub N/. We give methods for encoding and decoding reducible rank codes. A public key cryptosystem based on these codes and on the idea of a column scrambler is proposed. The column scrambler "mixes" columns of a generator (parity-check) matrix of a code. It makes the system more resistant to structural attacks such as Gibson's attacks. Possible attacks on the system are thoroughly studied. The system is found to be secure against known attacks for public keys of about 16 kbits and greater.

77 citations

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TL;DR: In this paper, transmitted signals are considered as square matrices of the Maximum rank distance (MRD) (n, k, d)-codes and a new composed decoding algorithm is proposed to correct simultaneously rank errors and rank erasures.

Abstract: In this paper, transmitted signals are considered as square matrices of the Maximum rank distance (MRD) (n, k, d)-codes. A new composed decoding algorithm is proposed to correct simultaneously rank errors and rank erasures. If the rank of errors and erasures is not greater than the Singleton bound, then the algorithm gives always the correct decision. If it is not a case, then the algorithm gives still the correct solution in many cases but some times the unique solution may not exist.

75 citations

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01 Jan 1996TL;DR: A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols.

Abstract: From the Publisher:
A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols; more than 200 tables and figures; more than 1,000 numbered definitions, facts, examples, notes, and remarks; and over 1,250 significant references, including brief comments on each paper.

13,597 citations

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TL;DR: In this paper, the problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of Rotter and Kschischang.

Abstract: The problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of Rotter and Kschischang. A large class of constant-dimension subspace codes is investigated. It is shown that codes in this class can be easily constructed from rank-metric codes, while preserving their distance properties. Moreover, it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank-metric codes where partial information about the error is available. This partial information may be in the form of erasures (knowledge of an error location but not its value) and deviations (knowledge of an error value but not its location). Taking erasures and deviations into account (when they occur) strictly increases the error correction capability of a code: if mu erasures and delta deviations occur, then errors of rank t can always be corrected provided that 2t les d - 1 + mu + delta, where d is the minimum rank distance of the code. For Gabidulin codes, an important family of maximum rank distance codes, an efficient decoding algorithm is proposed that can properly exploit erasures and deviations. In a network coding application, where n packets of length M over F(q) are transmitted, the complexity of the decoding algorithm is given by O(dM) operations in an extension field F(qn).

668 citations

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TL;DR: The problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of Rotter and Kschischang and an efficient decoding algorithm is proposed that can properly exploit erasures and deviations.

Abstract: The problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of K\"otter and Kschischang. A large class of constant-dimension subspace codes is investigated. It is shown that codes in this class can be easily constructed from rank-metric codes, while preserving their distance properties. Moreover, it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank-metric codes where partial information about the error is available. This partial information may be in the form of erasures (knowledge of an error location but not its value) and deviations (knowledge of an error value but not its location). Taking erasures and deviations into account (when they occur) strictly increases the error correction capability of a code: if $\mu$ erasures and $\delta$ deviations occur, then errors of rank $t$ can always be corrected provided that $2t \leq d - 1 + \mu + \delta$, where $d$ is the minimum rank distance of the code. For Gabidulin codes, an important family of maximum rank distance codes, an efficient decoding algorithm is proposed that can properly exploit erasures and deviations. In a network coding application where $n$ packets of length $M$ over $F_q$ are transmitted, the complexity of the decoding algorithm is given by $O(dM)$ operations in an extension field $F_{q^n}$.

563 citations

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TL;DR: In this paper, the trade-offs between resilience, security, and local repairability in distributed storage systems are investigated. But the authors focus on coding schemes that enable optimal local repairs and further bring these two concepts together to develop locally repairable coding schemes for DSS that are secure against eavesdroppers.

Abstract: This paper aims to go beyond resilience into the study of security and local-repairability for distributed storage systems (DSSs). Security and local-repairability are both important as features of an efficient storage system, and this paper aims to understand the trade-offs between resilience, security, and local-repairability in these systems. In particular, this paper first investigates security in the presence of colluding eavesdroppers, where eavesdroppers are assumed to work together in decoding the stored information. Second, this paper focuses on coding schemes that enable optimal local repairs. It further brings these two concepts together to develop locally repairable coding schemes for DSS that are secure against eavesdroppers. The main results of this paper include: 1) an improved bound on the secrecy capacity for minimum storage regenerating codes; 2) secure coding schemes that achieve the bound for some special cases; 3) a new bound on minimum distance for locally repairable codes; 4) code construction for locally repairable codes that attain the minimum distance bound; and 5) repair-bandwidth-efficient locally repairable codes with and without security constraints.

298 citations

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01 Jan 2003

TL;DR: In this article, the authors present a survey of 4-regular graphs with large girth, including Xp,q and PSL2(q) graphs, with a focus on the number of vertices.

Abstract: An overview 1. Graph theory 2. Number theory 3. PSL2(q) 4. The graphs Xp,q Appendix A. 4-regular graphs with large girth Index Bibliography.

239 citations