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    Title: 多維度拉丁方陣及其臨界集之構造與應用
    The Construction and Applications of Latin k-hypercube and Its Critical Sets
    Authors: 陳淑美
    Chen, Shu-Mei
    Contributors: 左瑞麟
    Tso, Ray-Lin
    陳淑美
    Shu-Mei Chen
    Keywords: 拉丁方陣
    拉丁立體方陣
    多維度拉丁方陣
    臨界集
    秘密分享方案
    Latin squares
    Latin cubes
    Latin k-hypercubes
    Critical set
    Secret sharing schemes
    Date: 2018
    Issue Date: 2018-08-10 11:11:39 (UTC+8)
    Abstract: 「資訊安全」之相關研究如密碼學(Cryptography)、秘密分享方案(Secret sharing schemes)經常運用數學技術來設計與實踐,如1994年Cooper,Donovan和Seberry便舉出了如何運用拉丁方陣(Latin squares)來實踐秘密分享方案。
    拉丁方陣為組合設計(Combinatorial designs)中的一部分,於現代密碼學及編碼的設計,有相當多的貢獻。
    本論文將以組合設計中的拉丁方陣為基礎,進一步發展出拉丁立體方陣(Latin cubes)的相關方法論,包含如何建構拉丁立體方陣、如何找出拉丁立體方陣的臨界集(Critical sets)、如何由拉丁立體方陣的臨界集反推出拉丁立體方陣等,藉此增強拉丁方陣應用的複雜度及多元性,本論文亦依據拉丁立體方陣相關方法論發展出多維度拉丁方陣(稱之為Latin k-hypercubes)之相關方法論,也成功地將所提出之方法論運用至資訊隱藏領域。希望本論文所提出之方法論,後續可於資訊安全各類研究領域發展出更多相關應用。
    Researches related to "information security" such as Cryptography and Secret sharing schemes are usually designed and constructed using mathematical techniques. For example,in 1994 Cooper, Donovan and Seberry showed the method how to use the Latin square to design secret sharing schemes.
    The design of Latin squares are in the scope of the Combinatorial designs, and they have considerable contributions to Cryptography and coding theory.
    This thesis will develop the Latin cubes methodology based on the concepts of Latin squares and their critical sets. We will introduce how to construct a Latin cube,how to find the critical sets of the Latin cube,and how to rebuild the Latin cube using its critical sets and so on. The idea introduced here can be used to increase the complexity and diversity of the application of the Latin squares. Based on the methodology of Latin cubes,we will also develop the multi-dimensional Latin squares (called the Latin k-hypercubes) methodology,and show how it can be successfully applied to the areas of information hiding.
    We hope that the methodologies proposed in this thesis can be followed by more relevant applications in various fields of information security researches.
    Reference: [1] A. P. Street.(Math. 21 ,1992).Defining sets for t-designs and critical sets for Latin squares.New Zealand J.
    [2] A. P. Street and D. J. Street.(1987).Combinatorics of Experimental Design.Oxford University Press, Oxford.
    [3] B. Smetaniuk.(Math. 16 ,1979).On the minimal critical set of a Latin square.
    [4] Blakley,G. R.(1979).Safeguarding cryptographic keys . Proceedings of the National Computer Conference 48.
    [5] Chin-ChenChang ,Yung-ChenChou,The Duc Kieu.(2008).An Information Hiding Scheme Using Sudoku.The 3rd Intetnational .Conference on Innovative Computing Information and Control (ICICIC'08) 978-0-7695-3161-8/08 © 2008 IEEE
    [6] Cooper, J.,Donovan, D. , Seberry, J..( 4,1991) . Latin squares and critical sets of minimal size. Australas. J. Combin.
    [7] Donovan, D., Cooper, J., Nott, D.J., Seberry, J..( 1995 ) .Latin squares: critical sets and their lower bounds. Ars Combin. 39
    [8] Donovan, D., Cooper, J..( 1996 ) .Critical sets in back circulant Latin squares. Aequationes Math.
    [9] D. Curran and G. H. J. van Rees.( 1978) . Critical sets in Latin squares. Congr. Numer. 22
    [10] D. Raghavarao.(1988).Constructions and Combinatorial Problems in Design of Experiments. Dover Publications, New York
    [11] D. R. Stinson.(1996).Combinatorial Designs with Selected Applications. Lecture Notes, Dept. Comput. Sci., Univ. Manitoba, Winnipeg.
    [12] Daniel R. Droz.(2016).Orthogonal Sets of Latin Squares and Class-r Hypercubes Generated by Finite Algebraic Systems. Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor in Eberly College of Science of the Pennsylvania State University.
    [13] D. R. Stinson and G. H. J. van Rees.(1982).Some large critical sets. Congr. Numer. 34.
    [14] Hung-Li Fu.(2002).Combinatorial Designs ( Lecture Notes ).available at http://hlfu.math.nctu.edu.tw/course.php?YEAR=91.
    [15] J. Cooper, D. Donovan and J. Seberry.(Appl. 12 ,1994). Secret sharing schemes arising from Latin square. Bull. Inst. Combin.
    [16] Jerzy Wojdyło(Southeast Missouri State University).(2007). Latin Squares, Cubes and Hypercubes. available at https://www.slideserve.com/vangie/Latin-squares-cubes-and-hypercubes.
    [17] Pria Bharti, Roopali Soni.(November 2012).A New Approach of Data Hiding in Images using Cryptography and Steganography.International Journal of Computer Applications (0975 – 8887) Volume 58– No.18.
    [18] R. Tso, Y. Miao.(2017).A survey of secret sharing schemes based on Latin squares.(Conference Paper),13th International Conference on Intelligent Information Hiding and Multimedia Signal Processing, IIH-MSP 2017.
    [19] R. Mathon and A. Rosa.(1996) . 2-(v,k,λ) Designs of small order, in C. J. Colbourn and J. H. Dinitz, eds..The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton
    [20] Shamim Ahmed Laskar1 and Kattamanchi Hemachandran .(December 2012).High Capacity data hiding using LSB Steganography and Encryption. International Journal of Database Management Systems ( IJDMS ) Vol.4, No.6, December 2012.
    [21] Shamir, Adi,(1979).How to share a secret. Communications of the ACM 22 (11).
    [22] Shamir, A.(1979) . How to share a secret, Comm. ACM 22.
    [23] Smetaniuk, B.(1979) . On the minimal critical set of a Latin square. Util. Math. 16, 97–100
    [24] Steven T. Dougherty , Theresa A. Szczepanski.(2008).Latin k-hypercubes .Australasian Journal of Combinatorics Volume 40.
    [25] Stinson, D.R., van Rees, G.H.J. .(1982) . Some large critical sets. Congr. Numer. 34, 441–456.
    [26] Street, A.P.(Math. 21, 1992) . Defining sets for t -designs and critical sets for Latin squares, New Zealand J.
    [27] Tamara Gomez, Phoebe Coy.(2015).Latin Squares: Critical Sets.available at http://web.math.ucsb.edu/~padraic/ucsb_2014_15/ccs_problem_solving_w2015/Latin%20Squares%20Presentation%201.pdf.
    [28] Vaipuna Raass.(2016) .Critical Sets of Full Latin squares.A thesis submitted in fulfilment of the requirements for the DegreeOf Doctor of Philosophy at the University of Waikato.
    [29] 冷輝世,游孟霖 ,曾顯文.(2014).基於 LSB 的適性高負載資訊隱藏法. International Journal of Science and Engineering Vol.4 No.1:225-228
    [30] 維基百科https://en.wikipedia.org/wiki/Hypercube
    [31]維基百科拉丁方陣的數量https://zh.wikipedia.org/wiki/%E6%8B%89%E4%B8%81%E6%96%B9%E9%99%A3
    Description: 碩士
    國立政治大學
    資訊科學系碩士在職專班
    105971006
    Source URI: http://thesis.lib.nccu.edu.tw/record/#G0105971006
    Data Type: thesis
    DOI: 10.6814/THE.NCCU.EMCS.005.2018.B02
    Appears in Collections:[資訊科學系碩士在職專班] 學位論文

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