Reducible Rank Codes and Applications to Cryptography

Gabidulin, E.M.; Ourivski, A.; Ammar, Bassem; Honary, Bahram

doi:10.1007/978-1-4757-3585-7_8

Cited by 22 publications

(37 citation statements)

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“…We have obtained lower bounds on their generalized rank weights (GRWs) that extend the known lower bound on their minimum rank distance [9] and which give exact values for cartesian products, and we have obtained upper bounds that are always reached for the minimum rank distance and some reduction. We have obtained maximum rank distance (MRD) reducible codes with MRD main components for new parameters, extending the families of MRD codes for n > m considered in [9] and [23].…”

Section: Conclusion and Open Problemsmentioning

confidence: 80%

“…After some preliminaries in Section II, the paper is organized as follows: In Section III, we give lower and upper bounds on the GRWs of reducible codes, extending the lower bound on the minimum rank distance given in [9], and see that the given upper bound on the minimum rank distance can be reached by some reduction. In Section IV, we obtain new parameters for which reducible codes are MRD (or close to MRD) and with MRD components, and obtain explicit estimates on their GRWs, including those MRD codes found in [9] and considered for secure network coding in [23].…”

Section: B Organization Of the Papermentioning

confidence: 99%

“…In Section IV, we obtain new parameters for which reducible codes are MRD (or close to MRD) and with MRD components, and obtain explicit estimates on their GRWs, including those MRD codes found in [9] and considered for secure network coding in [23]. In Section V, we obtain all linear codes whose GRWs are all optimal, for all fixed packet and code sizes, up to rank equivalence.…”

Section: B Organization Of the Papermentioning

confidence: 99%

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Generalized rank weights of reducible codes, optimal cases and related properties

Martínez-Peñas

2016

2016 IEEE International Symposium on Information Theory (ISIT)

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Abstract-Reducible codes for the rank metric were introduced for cryptographic purposes. They have fast encoding and decoding algorithms, include maximum rank distance (MRD) codes and can correct many rank errors beyond half of their minimum rank distance, which makes them suitable for error-correction in network coding. In this paper, we study their security behaviour against information leakage on networks when applied as coset coding schemes, giving the following main results: 1) we give lower and upper bounds on their generalized rank weights (GRWs), which measure worst-case information leakage to the wire-tapper, 2) we find new parameters for which these codes are MRD (meaning that their first GRW is optimal), and use the previous bounds to estimate their higher GRWs, 3) we show that all linear (over the extension field) codes whose GRWs are all optimal for fixed packet and code sizes but varying length are reducible codes up to rank equivalence, and 4) we show that the information leaked to a wire-tapper when using reducible codes is often much less than the worst case given by their (optimal in some cases) GRWs. We conclude with some secondary related properties: Conditions to be rank equivalent to cartesian products of linear codes, conditions to be rank degenerate, duality properties and MRD ranks.

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Section: Conclusion and Open Problemsmentioning

confidence: 80%

Section: B Organization Of the Papermentioning

confidence: 99%

Section: B Organization Of the Papermentioning

confidence: 99%

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Generalized rank weights of reducible codes, optimal cases and related properties

Martínez-Peñas

2016

2016 IEEE International Symposium on Information Theory (ISIT)

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“…In Section 5, we apply Gabidulin codes [8,21] to obtain coding schemes with optimal information rates and communication overheads for n ≤ m, which can be seen as a rank-metric analog of the constructions in [2,13]. However, our method allows us to correct errors, and not only erasures as in the secret sharing case [2,13], and can be applied to other families of maximum rank distance codes, such as those in [7] for n > m. Finally, in Section 6, we discuss the applications in universal secure linear network coding and secure distributed storage with crisscross errors and erasures.…”

Section: Introductionmentioning

confidence: 99%

Universal secure rank-metric coding schemes with optimal communication overheads

Martínez-Peñas

2018

Cryptogr. Commun.

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We study the problem of reducing the communication overhead from a noisy wire-tap channel or storage system where data is encoded as a matrix, when more columns (or their linear combinations) are available. We present its applications to reducing communication overheads in universal secure linear network coding and secure distributed storage with crisscross errors and erasures and in the presence of a wire-tapper. Our main contribution is a method to transform coding schemes based on linear rank-metric codes, with certain properties, to schemes with lower communication overheads. By applying this method to pairs of Gabidulin codes, we obtain coding schemes with optimal information rate with respect to their security and rank error correction capability, and with universally optimal communication overheads, when n ≤ m, being n and m the number of columns and number of rows, respectively. Moreover, our method can be applied to other families of maximum rank distance codes when n > m. The downside of the method is generally expanding the packet length, but some practical instances come at no cost.

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“…Gibson was the first to prove the weakness of the system through a series of successful attacks [Gib95,Gib96]. Following these failures, the first works which modified the GPT scheme to avoid Gibson's attack were published in [GO01,GOHA03]. The idea is to hide further the structure of Gabidulin code by considering isometries for the rank metric.…”

Section: Introductionmentioning

confidence: 99%

Polynomial-time key recovery attack on the Faure–Loidreau scheme based on Gabidulin codes

Gaborit

Otmani

Kalachi

2017

Des. Codes Cryptogr.

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ABSTRACT. Encryption schemes based on the rank metric lead to small public key sizes of order of few thousands bytes which represents a very attractive feature compared to Hamming metric-based encryption schemes where public key sizes are of order of hundreds of thousands bytes even with additional structures like the cyclicity. The main tool for building public key encryption schemes in rank metric is the McEliece encryption setting used with the family of Gabidulin codes. Since the original scheme proposed in 1991 by Gabidulin, Paramonov and Tretjakov, many systems have been proposed based on different masking techniques for Gabidulin codes. Nevertheless, over the years most of these systems were attacked essentially by the use of an attack proposed by Overbeck.In 2005 Faure and Loidreau designed a rank-metric encryption scheme which was not in the McEliece setting. The scheme is very efficient, with small public keys of size a few kiloBytes and with security closely related to the linearized polynomial reconstruction problem which corresponds to the decoding problem of Gabidulin codes. The structure of the scheme differs considerably from the classical McEliece setting and until our work, the scheme had never been attacked. We show in this article that for a range of parameters, this scheme is also vulnerable to a polynomial-time attack that recovers the private key by applying Overbeck's attack on an appropriate public code. As an example we break in a few seconds parameters with 80-bit security claim. Our work also shows that some parameters are not affected by our attack but at the cost of a lost of efficiency for the underlying schemes.Post-quantum cryptography; Gabidulin code; Polynomial reconstruction; Faure-Loidreau scheme.

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Reducible Rank Codes and Applications to Cryptography

Cited by 22 publications

References 1 publication

Generalized rank weights of reducible codes, optimal cases and related properties

Generalized rank weights of reducible codes, optimal cases and related properties

Universal secure rank-metric coding schemes with optimal communication overheads

Polynomial-time key recovery attack on the Faure–Loidreau scheme based on Gabidulin codes

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