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Proceedings ArticleDOI

On a class of three-weight codes with cryptographic applications

01 Jul 2012-pp 2551-2555
TL;DR: Making use of the finite projective geometry, a sufficient and necessary condition for a linear code to be a three-weight code is given and the geometric approach is established, which provides a convenient method to construct three- Weight codes.
Abstract: Linear codes with good algebraic structures have been used in a number of cryptographic or information-security applications, such as wire-tap channels of type II and secret sharing schemes. For a code-based secret sharing scheme, the problem of determining the minimal access sets is reduced to finding the minimal codewords of the dual code. It is well known that the latter problem is a hard problem for an arbitrary linear code. Constant weight codes and two-weight codes have been studied in the literature, for their applications to secret sharing schemes. In this paper, we study a class of three-weight codes. Making use of the finite projective geometry, we will give a sufficient and necessary condition for a linear code to be a three-weight code. The geometric approach that we will establish also provides a convenient method to construct three-weight codes. More importantly, we will determine the minimal codewords of a three-weight code, making use of the geometric approach.
Citations
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TL;DR: In this paper, a class of three-weight cyclic codes whose duals have two zeros is presented, where $p$ is an odd prime, and the weight distribution of this class of cyclic code is settled.
Abstract: Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a class of three-weight cyclic codes over $\gf(p)$ whose duals have two zeros is presented, where $p$ is an odd prime. The weight distribution of this class of cyclic codes is settled. Some of the cyclic codes are optimal. The duals of a subclass of the cyclic codes are also studied and proved to be optimal.

113 citations

Journal ArticleDOI
TL;DR: This article studies the efficacy of the Metropolis algorithm for the minimum-weight codeword problem and provides both theoretical and experimental justification to show why the generator space is a worthwhile search space for this problem.
Abstract: This article studies the efficacy of the Metropolis algorithm for the minimum-weight codeword problem . The input is a linear code $C$ given by its generator matrix and our task is to compute a nonzero codeword in the code $C$ of least weight. In particular, we study the Metropolis algorithm on two possible search spaces for the problem: 1) the codeword space and 2) the generator space . The former is the space of all codewords of the input code and is the most natural one to use and hence has been used in previous work on this problem. The latter is the space of all generator matrices of the input code and is studied for the first time in this article. In this article, we show that for an appropriately chosen temperature parameter the Metropolis algorithm mixes rapidly when either of the search spaces mentioned above are used. Experimentally, we demonstrate that the Metropolis algorithm performs favorably when compared to previous attempts. When using the generator space, the Metropolis algorithm is able to outperform the previous algorithms in most of the cases. We have also provided both theoretical and experimental justification to show why the generator space is a worthwhile search space to use for this problem.

9 citations

Proceedings ArticleDOI
12 Jul 2014
TL;DR: It is proved that the Markov chains associated with the Metropolis algorithm mix rapidly for suitable choices of the temperature parameter T, and performed very well in comparison to previously known experimental results.
Abstract: We study the performance of the Metropolis algorithm for the problem of finding a code word of weight less than or equal to M, given a generator matrix of an [n,k]-binary linear code. The algorithm uses the set Sk of all kxk invertible matrices as its search space where two elements are considered adjacent if one can be obtained from the other via an elementary row operation (i.e by adding one row to another or by swapping two rows.) We prove that the Markov chains associated with the Metropolis algorithm mix rapidly for suitable choices of the temperature parameter T. We ran the Metropolis algorithm for a number of codes and found that the algorithm performed very well in comparison to previously known experimental results.

3 citations


Cites background from "On a class of three-weight codes wi..."

  • ...This problem is important for several reasons: a minimum weight code word is a measure of the error correction capability of the code [9], also, codes with large minimum weight code words have applications in diverse areas such as cryptography [18, 17, 16], pseudorandom generators [1, 11]....

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References
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01 Jan 1999
TL;DR: It is shown that the minimal codewords in the dual code completely specify the access structure of the secret-sharing scheme, and conversely, the apparently new notion of minimal codEWords in a linear code.
Abstract: The use of a linear code to "split" secrets into equal-size shares is considered. The determination of which sets of shares can be used to obtain the secret leads to the apparently new notion of minimal codewords in a linear code. It is shown that the minimal codewords in the dual code completely specify the access structure of the secret-sharing scheme, and conversely.

314 citations


"On a class of three-weight codes wi..." refers methods in this paper

  • ...I. INTRODUCTION Linear codes with good Hamming weight properties have been used in many cryptographic or information-security areas, for examples, wire-tap channels of type II [7] and secret sharing schemes [4], [8], [10]....

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Journal ArticleDOI
TL;DR: It is shown that for even codes the set of zero neighbors is strictly optimal in this class of algorithms, which implies that general asymptotic improvements of the zero-neighbors algorithm in the frame of gradient-like approach are impossible.
Abstract: Minimal vectors in linear codes arise in numerous applications, particularly, in constructing decoding algorithms and studying linear secret sharing schemes. However, properties and structure of minimal vectors have been largely unknown. We prove basic properties of minimal vectors in general linear codes. Then we characterize minimal vectors of a given weight and compute their number in several classes of codes, including the Hamming codes and second-order Reed-Muller codes. Further, we extend the concept of minimal vectors to codes over rings and compute them for several examples. Turning to applications, we introduce a general gradient-like decoding algorithm of which minimal-vectors decoding is an example. The complexity of minimal-vectors decoding for long codes is determined by the size of the set of minimal vectors. Therefore, we compute this size for long randomly chosen codes. Another example of algorithms in this class is given by zero-neighbors decoding. We discuss relations between the two decoding methods. In particular, we show that for even codes the set of zero neighbors is strictly optimal in this class of algorithms. This also implies that general asymptotic improvements of the zero-neighbors algorithm in the frame of gradient-like approach are impossible. We also discuss a link to secret-sharing schemes.

313 citations


"On a class of three-weight codes wi..." refers background in this paper

  • ...However, determining minimal codewords is a hard problem for an arbitrary linear code (see [1], for example)....

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Journal ArticleDOI
TL;DR: A sufficient condition for all nonzero codewords of a linear code to be minimal is derived from exponential sums, and then used to construct secret sharing schemes with nice access structures.
Abstract: Secret sharing has been a subject of study for over 20 years, and has had a number of real-world applications. There are several approaches to the construction of secret sharing schemes. One of them is based on coding theory. In principle, every linear code can be used to construct secret sharing schemes. But determining the access structure is very hard as this requires the complete characterization of the minimal codewords of the underlying linear code, which is a difficult problem in general. In this paper, a sufficient condition for all nonzero codewords of a linear code to be minimal is derived from exponential sums. Some linear codes whose covering structure can be determined are constructed, and then used to construct secret sharing schemes with nice access structures.

298 citations


"On a class of three-weight codes wi..." refers methods in this paper

  • ...I. INTRODUCTION Linear codes with good Hamming weight properties have been used in many cryptographic or information-security areas, for examples, wire-tap channels of type II [7] and secret sharing schemes [4], [8], [10]....

    [...]

Journal ArticleDOI
TL;DR: The notion of higher (or generalized) weights of codes is just as natural as that of the classical Hamming weight and the authors adopt the geometric point of view.
Abstract: The notion of higher (or generalized) weights of codes is just as natural as that of the classical Hamming weight. The authors adopt the geometric point of view and always treat the q-ary case. Some results and proofs being new, the main goal is to present a clear picture of what is known on the subject.

217 citations

Journal ArticleDOI
TL;DR: The generalized Hamming weight of Wei and the dimension/length profile (DLP) of Forney are extended to two-code formats and are useful to design a perfect secrecy coding scheme for the coordinated multiparty model.
Abstract: The noiseless wire-tap channel of type II with coset coding scheme was provided by Ozarow and Wyner. In this correspondence, the user is split into multiple parties who are coordinated in coding their data symbols by using the same encoder. The adversary can tap not only partial transmitted symbols but also partial data symbols. We are interested in the equivocation of the data symbols to this adversary who has more power than that of Ozarow and Wyner. The generalized Hamming weight of Wei and the dimension/length profile (DLP) of Forney are extended to two-code formats: relative generalized Hamming weight and relative dimension/length profile (RDLP). Upper and lower bounds of the new concepts are investigated. They are useful to design a perfect secrecy coding scheme for the coordinated multiparty model. Under a general secrecy standard, the coordinated model can provide a higher transmission rate than an uncoordinated (time-sharing) model.

127 citations


"On a class of three-weight codes wi..." refers methods in this paper

  • ...I. INTRODUCTION Linear codes with good Hamming weight properties have been used in many cryptographic or information-security areas, for examples, wire-tap channels of type II [7] and secret sharing schemes [4], [8], [10]....

    [...]