Ernst M. Gabidulin. Theory of codes with maximum rank distance. Problems of Information Transmission, 21:1-12, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....linear code over F q m . For any pair of codewords c; c 2 C, the rank distance between them is de ned to be the rank over F q of the m n matrix corresponding to c c obtained by expanding each entry of c c as an m tuple along a basis of F q m over F q and is denoted by Rank q (c c ) [2]. It is easily veri ed that this rank distance satis es the following requirements of a metric: For all c; c 2 F m ; A part of the content of this report has been accepted for presentation in IEEE International Symposium on Information Theory, Yokohama, Japan to be held during June ....

....c over all possible pairs of distinct codewords. Clearly, the rank distance between two codewords c 6= c 2 C is at most the Hamming distance between them. Combining this with the Singleton bound one gets, Rank q (C) minfm; n k 1g: The case where Rank q (C) n k 1 has been studied in [2, 3, 6] and are called MRD (Maximum Rank Distance) codes. In [10, 11] codes with Rank q (C) m calling them Full Rank Distance (FRD) Codes have been studied. Notice that the cases Rank q (C) n k 1 corresponds to considering only those cases where m n k 1 and the cases Rank q (C) m ....

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E. M. Gabidulin, \Theory of Codes with Maximum Rank Distance," Problemy Peredachi Informatsii, 21, 99.3-14, Jan.-Mar.1985.

....Block codes. 1 Introduction m . For any pair of codewords c; c 2 C, the rank distance between them is de ned to be the rank over F q of the m n matrix corresponding to c c obtained by expanding each entry of c c as an m tuple along a basis of F q m over F q and is denoted by r q (c c ) [4]. The rank of C, denoted by d R;min is de ned as the minimum of r q (c c ) over all possible pairs of distinct codewords. Clearly, the rank distance between two codewords c 6= c 2 C is at most the Hamming distance between them. Combining this with the Singleton bound one gets, d R;min ....

....q (c c ) over all possible pairs of distinct codewords. Clearly, the rank distance between two codewords c 6= c 2 C is at most the Hamming distance between them. Combining this with the Singleton bound one gets, d R;min minfm; n k 1g: 1) The case where d R;min = n k 1 has been studied in [4, 5, 8] and are called MRD (Maximum Rank Distance) codes. In this paper we study codes with d R;min = m calling them Full Rank Distance (FRD) Codes . Notice that the cases d R;min = n k 1 corresponds to considering only those cases where m n k 1 and the cases d R;min = m corresponds to studying ....

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E. M. Gabidulin, \Theory of Codes with Maximum Rank Distance," Problemy Peredachi Informatsii, 21, 99.3-14, Jan.-Mar.1985.

....over Fq m . For any pair of codewords c1 ; c2 2 C, the rank distance between them is de ned to be the rank over Fq of the m n matrix corresponding to c1 c2 obtained by expanding each entry of c1 c2 as an m tuple along a basis of Fq m over Fq and is denoted by Rankq ( c1 c2 ) [2]. The rank of C, denoted by Rankq (C) is de ned as the minimum of Rankq ( c 1 c2 ) over all possible pairs of distinct codewords. Rank distance codes over nite elds have been studied by several authors for applications in storage devices and more recently for applications in SpaceTime coding. ....

E. M. Gabidulin, \Theory of Codes with Maximum Rank Distance," Problemy Peredachi Informatsii, 21, 99.3-14, Jan.- Mar.1985.

....Ste90, Ste94, V er95b] have been proposed . They have low computational requirements and high speed. The counterpart is that the communication complexity is significant. In an attempt to improve the performances of the above systems, Kefei Chen has suggested the idea of using rank metric codes [Gab85] instead of Hamming metric codes in cryptographic schemes. He has designed two authentication schemes [Che94, Che96] with claimed better performances than the above systems. The security of these protocols relies on the following informal assumption: The syndrome decoding problem for rank ....

....resulting from underestimating the necessary sizes, it appears that rank distance codes are not better than usual codes. The authentication scheme [Har89] was broken by P. V eron [V er95a] 2 Background 2.1 Rank distance codes The rank distance codes were introduced by E.M. Gabidulin [Gab85] and rely on the following observation. Let x = x 1 ; x n ) be a n dimensional vector over GF (q ) where q is the power of a prime. Let b 1 ; b m be a basis of GF (q ) Write each element x j 2 ) as x j = fi 1;j b 1 Delta Delta Delta fi m;j b m , where fi i;j 2 ....

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E.M. Gabidulin. Theory of codes with maximum rank distance. Problems of Information Transmission, 21:1--12, 1985.

.... generalized version of one among many NP complete rank problems studied in [21] and [9] In our scheme R will be a finite field GF (q) MinRank over a field can be defined in terms of codes: it is a decoding problem for a subfield subcode of Gabidulin s linear rank distance code over GF (q # ) [12, 10, 34]. Moreover the best known attacks known to decode rank distance codes are currently based on MinRank [10] and therefore MinRank is essential to the security of Chen and GPT public key schemes [13, 4, 10] MinRank also appears in attacks known on the HFE [29, 8, 9] TTM cryptosystem [17] and ....

Ernst M. Gabidulin. Theory of codes with maximum rank distance. Problems of Information Transmission, 21:1-12, 1985.

....codes that have been studied in the literature so far consider burst errors of a given rectangular shape, say t 1 Theta t 2 [1, 3, 4, 7, 10, 11, 12] However, there are also papers that study other shapes as well. For instance, in [2] the authors study circular type of bursts. In [6, 9, 15], the authors consider metrics given by the rank of the array: a particular case, is the correction of criss cross type of errors. Metrics for different channels, including 2 dimensional clusters, are presented in [8] A recent application of correction of 2 dimensional clusters appeared in the ....

E. M. Gabidulin, Theory of Codes with Maximum Rank Distance, Probl. Inform. Transm., vol. 21, No. 1, pp. 3--16, Jan.-Mar. 1985.

....a row or a column in error (without apriori knowledge of which one occurred) There exist codes that can do so. Moreover, the known codes are stronger in the sense that they can correct the rank of an array. The idea of using the rank as a metric comes from Delsarte [4] See also Gabidulin [6] and Roth [12] However, these constructions are based on finite field arithmetic, as Reed Solomon codes. Therefore, for very large arrays, they may become impractical, since they may need a very large look up table. In this paper, we will present array codes that have the same error correcting ....

....Therefore, for very large arrays, they may become impractical, since they may need a very large look up table. In this paper, we will present array codes that have the same error correcting capability in terms of rows and columns (although sometimes they cannot correct the rank) as the ones in [4][6][12] but they have less complexity. The new codes are based on simple parity along lines of different slopes, in the spirit of [3] There are applications in which information bits are stored in n Theta n bit arrays. The error patterns are such that all corrupted bits are confined to at most ....

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E.M. Gabidulin, Theory of codes with maximum rank distance, Prob. Info. Trans., Vol. 21, No. 1, (1985), 3--16.

....Ste90, Ste94, V er95b] have been proposed 3 . They have low computational requirements and high speed. The counterpart is that the communication complexity is significant. In an attempt to improve the performances of the above systems, Kefei Chen has suggested the idea of using rank metric codes [Gab85] instead of Hamming metric codes in cryptographic schemes. He has designed two authentication schemes [Che94, Che96] with claimed better performances than the above systems. The security of these protocols relies on the following informal assumption: The syndrome decoding problem for rank ....

....resulting from underestimating the necessary sizes, it appears that rank distance codes are not better than usual codes. 3 The authentication scheme [Har89] was broken by P. V eron [V er95a] 2 2 Background 2.1 Rank distance codes The rank distance codes were introduced by E.M. Gabidulin [Gab85] and rely on the following observation. Let x = x 1 ; x n ) be a n dimensional vector over GF (q m ) where q is the power of a prime. Let b 1 ; b m be a basis of GF (q m ) Write each element x j 2 GF (q m ) as x j = fi 1;j b 1 Delta Delta Delta fi m;j b m , where ....

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E.M. Gabidulin. Theory of codes with maximum rank distance. Problems of Information Transmission, 21:1--12, 1985.

....= 2. A Singleton type bound on the minimum rank states that the minimum rank and the redundancy of any [n Theta Delta ; k] tensor code over a field F satisfy the relation n Delta Gamma k ( Gamma 1) n : 2) This bound was stated by Delsarte in [7] for the case Delta = 2 (see also [10] and the generalization for larger Delta in [26] Furthermore, Delsarte obtained a construction of [n Theta n; k] array codes over GF (q) that attains this bound for every n (see also [10] and [26] We describe next this optimal construction, which we denote by C(n; 2; q) the ....

.... k ( Gamma 1) n : 2) This bound was stated by Delsarte in [7] for the case Delta = 2 (see also [10] and the generalization for larger Delta in [26] Furthermore, Delsarte obtained a construction of [n Theta n; k] array codes over GF (q) that attains this bound for every n (see also [10] and [26] We describe next this optimal construction, which we denote by C(n; 2; q) the parameter 2 stands for Delta = 2) Let fi = fi j ] n j=1 and = n =1 be two vectors in GF (q n ) each with entries that are linearly independent over GF (q) The array code C(n; 2; q) ....

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E.M. Gabidulin, Theory of codes with maximum rank distance, Probl. Peredach. Inform., 21 (1985), 3--16 (in Russian; pp. 1--12 in the English translation).

....such that is the smallest rank of any nonzero matrix in C. The parameter is referred to as the minimum rank of C. The Singleton bound on the minimum rank takes the form mn Gamma k ( Gamma 1) n ; 2) where we assume that m n. This bound was stated by Delsarte in [4] see also Gabidulin [8] and Roth [19] Furthermore, those references contain a construction of [n Theta n; k] array codes over the field F q = GF (q) that attain this bound for every n. We describe next this optimal construction, which we denote by C q (n; s) where = s 1. Let ff = ff i ] n i=1 be a row ....

....entries that are linearly independent over F q . The array code C q (n; s) consists of all n Theta n matrices Gamma = Gamma i;j ] n i;j=1 over F q such that n X i;j=1 Gamma i;j ff q i j = 0 ; 0 s : 3) Two polynomial time decoding algorithms for C q (n; s) are presented in [8] and [19] for recovering any error pattern of rank s=2. The construction C q (n; s) can be generalized to obtain optimal (s 1) m Theta n; k] array codes by means of code shortening. Namely, to form an (s 1) m Theta n; m Gammas)n] array code for m n out of an (s 1) n Theta n; n Gammas)n] ....

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E.M. Gabidulin, Theory of codes with maximum rank distance, Probl. Peredach. Inform., 21 (1985), 3--16 (in Russian; pp. 1--12 in the English translation).

....nonzero elements of S are linearly independent over GF (p) Proof. We present a decoding algorithm that uniquely recovers the error vector e provided that the number of errors is t or less. The algorithm is based upon the one described in [21] for decoding maximum rank array codes (see also [11]) Let y 2 F n be the received word and let E 1 ; E 2 ; E be the given error values, which we regard as vectors in the linear space F = GF (p m ) over GF (p) Also, let ffi = ffi 1 ffi 2 : ffi ae ] be a vector over F whose entries form a basis of size ae t of the linear ....

E.M. Gabidulin, Theory of codes with maximum rank distance, Probl. Peredach. Inform., 21 (1985), 3--16 (in Russian; pp. 1--12 in the English translation).

....erasures were constructed in [20] Another well studied model of transmission assumes that symbols of a codeword are arranged in a matrix and errors are confined to a prescribed number of its rows or columns. Polynomial time codes for this problem were suggested by Delsarte [49] Gabidulin [68], and Roth [134] see Chapter xx (Blaum, Farrell, van Tilborg) Asymptotically good codes correcting insertions, deletions, and transpositions were constructed in [142] 3 Difficult Problems In this section, we group algorithmic problems whose solutions involve a large amount of backtracking ....

E. M. Gabidulin, "Theory of codes with maximum rank distance," Problems of Info. Trans., 21 (1) (1985), 3--16 and 1--12.

....an m vector over GF (2 m ) whose coordinates are linearly independent over GF (2) and let G be the k Theta m Gabidulin matrix with generating vector g. The Gabidulin code C with dimension k and generating vector g has generator matrix G, and corrects e = m Gamma k) div 2 errors. Gabidulin in [3] and [5] gives fast decoding algorithms for C for which g acts as a key. Errors for Gabidulin codes are not counted using the usual Hamming metric, but with the rank metric induced by the rank norm jvj of a vector v over GF (2 m ) which is defined to be the number of coordinates of v that are ....

GABIDULIN E.M. "Theory of Codes with Maximum Rank Distance." Problems of Information Transmission, Vol 21 no. 1, 1985.

....= SH of a GRS code. This is a Knapsack Type cryptosystem. Therefore, one might hope that this PKC is secure. But in fact, many knapsack type PKC including the Niederreiter PKC and some of its modifications have recently be shown to be insecure. See [9, 10] The PKC based on a family of rank codes [11] was proposed in [12] It looks like a McEliece type PKC. An important difference is that an open key G cr = SG X is a sum of a scrambled generator matrix SG and a hiding matrix X. In [12] a hiding matrix X of rank 1 was used. Recently, Gibson showed that for such hiding matrices a PKC can be ....

....of Algebraic Coding All the public key cryptosystems under consideration are based on codes over large alphabets. In fact, these codes are generalized Reed Solomon codes and rank codes. In this section, background information on these codes is given (for more detailed information, see [17] and [11]) Let GF (q) be a finite field with q elements. The nth Cartesian power GF (q) n of GF (q) is a metric space with respect to the Hamming distance function d(x; y) jfi j 1 i n; x i 6= y i gj ; 1) where x = x) 2 GF (q) n and y = y 1 ; y 2 ; y n ) 2 GF (q) n The Hamming weight ....

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E. M. Gabidulin, "Theory of Codes with Maximum Rank Distance," Probl. Inform. Transm., vol.21, No. 1, pp.1-12, July 1985.

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Ernst M. Gabidulin. Theory of codes with maximum rank distance. Problems of Information Transmission, 21:1-12, 1985.

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E.M. Gabidulin, Theory of codes with maximum rank distance, Probl. Peredach. Inform., 21 (1985), 3--16 (in Russian; pp. 1--12 in the English translation).

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E. M. Gabidulin, Theory of Codes with Maximum Rank Distance, Probl. Inform. Transm., vol. 21, No. 1, pp. 3--16, Jan.-Mar. 1985.

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