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Characterization of some minimal codes for secret sharing

ASIAN-EUROPEAN JOURNAL OF MATHEMATICS 2019年第2期12卷

作者： Makhlouf, Sassia Noui, Lemnouar Univ Batna1 Fac Econ Commercial & Management Sci Batna Algeria Univ Batna2 Fac Math & Comp Sci Dept Math Batna Algeria

Recently, several authors used linear codes to construct secret sharing schemes. It is known that if each nonzero codeword of a code C is minimal, then the dual code C-perpendicular to is suitable for secret sharing. To seek such codes Ashikhmin-Barg give a sufficient condition from weights;in [n, k, d](q) code C, let w(min) and w(max) be the minimum and maximum nonzero weights, respectively. If w(min)/w(max) > q - 1/q (*) then all nonzero codewords of C are minimal. In this paper, a necessary and sufficient condition is given for self-dual codes and for MDS codes to verify the inequality (*). Special codes are examined and applied for secret sharing schemes.

关键词： Secret sharing scheme self-dual codes MDS codes minimal codeword dual code

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Linear codes from quadratic forms

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APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING 2017年第6期28卷 535-547页

作者： Du, Xiaoni Wan, Yunqi Northwest Normal Univ Coll Math & Stat Lanzhou 730070 Gansu Peoples R China

Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for an odd prime power q, we present a class of linear codes over finite fields with quadratic forms via a general construction and then determine the explicit complete weight enumerators of these linear codes. Our construction covers some related ones via quadratic form functions and the linear codes may have applications in cryptography and secret sharing schemes.

关键词： Linear code Weight distribution Complete weight enumerators Quadratic form function minimal codeword

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On the Intersection of Binary Linear Codes

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Journal of Systems Science & Complexity 2016年第3期29卷 814-824页

作者： LIAO Dajian LIU Zihui College of Science Huaihai Institute of Technology Department of Mathematics Beijing Institute of Technology

For a binary linear code,a new relation between the intersection and(2,2)-separating property is addressed,and a relation between the intersection and the trellis complexity is also *** above relations,the authors will apply several classes of binary codes to secret sharing scheme and determine their trellis complexity and separating *** authors also present the properties of the intersection of certain kinds of two-weight binary *** using the concept of value function,the intersecting properties of general binary codes are described.

关键词： minimal access structure minimal codeword separating property trellis complexity two- weight code value function.

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The fullrank value function

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DESIGNS CODES AND CRYPTOGRAPHY 2016年第2期81卷 297-315页

作者： Liu, Zihui Wu, Xin-Wen Beijing Inst Technol Dept Math Beijing 100081 Peoples R China Griffith Univ Sch Informat & Commun Technol Gold Coast Qld 4222 Australia

Motivated by applications to the wiretap channel of type II with the coset coding scheme as well as secret sharing, the concept of the fullrank value function is introduced in this paper. Several constructing methods for fullrank value functions are presented by using the finite projective geometry. The relationship between fullrank value functions and secret sharing schemes, and the relationship between fullrank value functions and separation of linear codes are given as well. In particular, new (2,2)-separating codes are constructed by using our methods.

关键词： Value function minimal codeword Separating property Feasible set Support weight

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On relative constant-weight codes

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DESIGNS CODES AND CRYPTOGRAPHY 2015年第1期75卷 127-144页

作者： Liu, Zihui Wu, Xin-Wen Beijing Inst Technol Dept Math Beijing 100081 Peoples R China Griffith Univ Sch Informat & Commun Technol Gold Coast Qld 4222 Australia

In this paper, relative two-weight and three-weight codes are studied, which are both called relative constant-weight codes. A geometric approach is introduced to construct and characterize relative constant-weight codes, using the finite projective geometry. A sufficient and necessary condition is derived for linear codes to be relative constant-weight codes, based on the geometric approach. A family of infinite number of relative constant-weight codes are constructed, which includes dual Hamming codes and subcodes of punctured Reed-Muller codes as special instances. It is well known that determining all the minimal codewords is a hard problem for an arbitrary linear code. For relative constant-weight codes, minimal codewords are completely determined in this paper. Based on the above-mentioned results, applications of relative constant-weight codes to wire-tap channel of type II and secret sharing are discussed. A comparative study shows that relative constant-weight codes form a new family. They are not covered by the previously well-known three-weight codes or linear codes for which minimal codewords can be determined.

关键词： Relative two-weight code Relative three-weight code minimal codeword Finite projective geometry Wire-tap channel of type II Secret sharing

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A New Method to Construct Secret Sharing Schemes Based on Linear Codes

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Computer Technology and Application 2015年第2期6卷 89-94页

作者： Selda Calkavur Department of Mathematics Kocaeli University Kocaeli Turkey

Secret sharing is an important topic in cryptography and has applications in information security. The coding theory has been an important role in the constructing of secret sharing schemes. It is known that every linear code can be used to construct secret sharing schemes. So, we use the parity-check matrix of a linear code to construct secret sharing schemes based on linear codes. We also describe some techniques to recover the secret and determine the access structure of the new scheme. In this paper, we use the Massey＇s secret sharing scheme.

关键词： Linear code parity-check matrix secret sharing scheme minimal codeword minimal access set.

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Binary Cyclic Codes and minimal codewords

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Computer Technology and Application 2013年第9期4卷 486-489页

作者： Selda Calkavur Department of Computer Programming Koeaeli University Kocael 41135 Turkey

Cyclic codes form an important class of codes. They have very interesting algebraic structure. Furthermore, they are equivalent to many important codes, such as binary Hamming codes, Golay codes and BCH codes. minimal codewords in linear codes are widely used in constructing decoding algorithms and studying linear secret sharing scheme. In this paper, we show that in the binary cyclic code all of the codewords are minimal, except 0 and 1. Then, we obtain a result about the number of minimal codewords in the binary cyclic codes.

关键词： Linear code cyclic code binary cyclic code generator polynomial minimal codeword secret sharing.

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Local weight distribution of the (256,93) third-order binary Reed-Muller code

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IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES 2007年第3期E90A卷 698-701页

作者： Yasunaga, Kenji Fujiwara, Toru Kasami, Tadao Osaka Univ Grad Sch Informat Sci & Technol Suita Osaka 5650871 Japan Nara Inst Sci & Technol Nara 6300192 Japan

Local weight distribution is the weight distribution of minimal codewords in a linear code. We give the local weight distribution of the (256,93) third-order binary Reed-Muller code. For the computation, a coset partitioning algorithm is modified by using a binary shift invariance property. This reduces the time complexity by about 1/256 for the code. A necessary and sufficient condition for minimality in Reed-Muller codes is also presented.

关键词： local weight distribution minimal codeword Reed-Muller code binary shift

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