对有限域上遍历矩阵的性质进行了分析,给出了有限域上遍历矩阵的计数定理,并对遍历矩阵序对(A,B)关于矩阵M的双侧幂乘集〈A〉M〈B〉的秩及基数进行了全面分析.给出了Rk(A,B)集的构成及其基数的有关定理,所得到的结论对利用遍历矩阵实现有关的公钥密码具有理论上的指导意义.
Abstract
We analyzed the properties of the ergodic matrix over finite field, And deduced the theorem on the number of the ergodic matrices over finite fields. We also comprehensively analyzes the rank and cardinal number of the two-side exponentiation set about the given matrix M and the ergodic matrix pair (A,B). And given the related theorems about the structure and cardinal number of the set. The results have the important theoretical significance for constructing the public key cryptography based on the ergodic matrices.
关键词
有限域 /
遍历矩阵 /
遍历矩阵的双侧幂乘集
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Key words
finite field /
ergodic matrix /
two-side exponentiation set of ergodic matrix pair
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参考文献
[1] Lin D. D., Algebra Basis & Finite Field, Higher Education Publishing Company, Beijing, 2006, 7 (in Chinese).
[2] Darafsheh M. R., Order of elements in the groups related to the general linear group, Finite Fields and TheirApplications, 2005, 11(4): 738-747.
[3] Darafsheh M. R., The maximum element order in the groups related to the linear groups which is a multipleof the defining characteristic, Finite Fields and Their Applications, 2008, 14(4): 992-1001.
[4] Lidl R., Niederreiter H., Introduction to Fintie Fields and Their Applications, Cambridge University Press,Cambridge, 1986.
[5] Green J. A., The characters of the finite general linear groups, Trans. Amer. Math. Soc., 1955, 80: 402-447.
[6] Shi W., Xiao Y., Quotient of the Group and the Number of Conjugacy, Southeast Asian Bulletin of Mathematics,1998, 22: 301-305 (in Chinese).
[7] Zhao Y. Z., Zhao B., Pei S. H., Jiang Z. H., Design and implement on the HFEM public key scheme, Journalon Communications, 2011, 32(6): 24-31 (in Chinese).
[8] Pei S. H., Zhao Y. Z., Zhao H. W., Public key encryption scheme based on the ergodic matrices, ActaElectronica Sinica, 2010, 38(8): 1908-1913 (in Chinese).
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脚注
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基金
"十一五"国家密码发展基金资助项目(2006L014J00002)
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