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本文构造了一类具有Markov性质的分段扩张线性映射, 计算表明其具有正的李雅普诺夫指数. 通过理论分析, 证明了其极限分布是均匀分布, 并给出了其最小周期的计算公式. 通过分析符号熵测度应用于此系统时的具体涵义, 指出该系统的复杂度与系统的最小周期之间存在着矛盾关系, 说明了此系统具有可控的复杂度并给出了其复杂度随系统参数变化时的理论极限公式. 通过统计测试和复杂度测试表明, 此系统产生的混沌序列服从均匀分布, 而通过参数的调整该系统的符号熵可以接近其理论极限, 而与Logistic映射和斜帐篷映射的基于近似熵和符号熵的对比实验可知, 本文所提系统具有更高的复杂度, 并有更长的最小周期. 这表明此系统比较适合用于构造保密通信系统.In this article, a kind of piecewise expanding linear system is constructed. It has a positive Lyapunov exponent as calculated. It is proved that the system has a uniform limit distribution The formula of the least period of the system is also presented. It is indicated that there is a contradictory relationship between the complexity and the least period of the system when the symbol entropy is applied to the system. The theoretical limit of the complexity of the system with changing parameters is presented. Simulation of the system shows that the sequence generated by the chaos is uniformly distributed. It also tells that the system can have higher complexity but longer least-period than the logistic system and the Tent-Map system. Experiments show that the system is suitable for constructing the cipher.
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Keywords:
- chaos /
- Markov property /
- complexity /
- uniform distribution
[1] Matthews R 1989 Cryptologia 13 29
[2] Baptista M S 1998 Phys. Lett. A 240 50
[3] Xiao D, Liao X F, Deng S J 2008 Phys. Lett. A 372 4682
[4] Li P Y, Gu L, Sui Y X, Yang H J 2010 Optics and Precision Engineering 18 2102 (in Chinese) [李佩玥, 古力, 隋永新, 杨怀江 2010 光学 精密工程 18 2102]
[5] Kanso A, Ghebleh M 2012 Commun Nonlinear Sci Numer Simulat 17 2943
[6] Hu H P, Liu S H, Wang Z X, Wu X G 2004 Chinese Journal of Computers 27 408 (in Chinese) [胡汉平, 刘双红, 王祖喜, 吴晓刚 2004 计算机学报 27 408]
[7] Ye R S 2011 Optics Communications 284 5290
[8] Tong X J 2013 Commun Nonlinear Sci. Numer Simulat 18 1725
[9] Wang X M, Zhang J S, Zhang W F 2003 Acta Phys. Sin. 52 2737 (in Chinese) [王小敏, 张家树, 张文芳 2003 52 2737]
[10] Li Z, Cai J P, Chen X J, Lu X F 2009 Wireless Communications and Networking Conference 2009 IEEE 4 p1
[11] Li Z, Cai J P, Lu X F, Si J B 2009 Communications 2009 ICC 09. IEEE International Conference on 6 p1
[12] Li Z, Cai J P, Chang Y L 2009 IEEE Trans Commun 57 812
[13] Xiao G Z, Wei S M, Lam K Y, Imamura K 2000 IEEE Trans Inform Theory 46 2203
[14] Kolokotronis N, Kalouptsidis N 2003 IEEE Trans Inform Theory 49 3047
[15] Kurosawa K, Sato F, Sakata T, Kishimoto W 2000 IEEE Trans Inform Theory 46 694
[16] Lauder A G B, Paterson K G 2003 IEEE Trans Inform Theory 49 273
[17] Cai J P, Li Z, Song W T 2003 Acta Phys. Sin. 52 1871 (in Chinese) [蔡觉平, 李赞, 宋文涛 2003 52 1871]
[18] Pincus S M 1995 Chaos 5 110
[19] Xiao F H, Yan G R, Han Y H 2004 Acta Phys. Sin. 53 2877 (in Chinese) [肖方红, 阎桂荣, 韩宇航 2004 53 2877]
[20] Azad R K, Rao J S, Ramaswamy R 2002 Chaos Soliton & Fract. 14 633
[21] Robinson R C 2004 An Introduction to Dynamical Systems: Continuous and Discrete (Prentice Hall Press) p459-499
[22] Chen X J, Li Z, Bai B M, Cai J P 2011 ACcta Phys. Sin. 60 064215 (in Chinese) [陈小军, 李赞, 白宝明, 蔡觉平 2011 60 064215]
[23] Lasota A, Yorke J 1973 Transactions Amer Math Soc. 186 481
[24] Li T Y, Yorke J 1978 Transactions Amer Math Soc. 235 183
[25] Eckmann J P, Ruelle D 1985 Rev. Mod. Phys. 57 617
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[1] Matthews R 1989 Cryptologia 13 29
[2] Baptista M S 1998 Phys. Lett. A 240 50
[3] Xiao D, Liao X F, Deng S J 2008 Phys. Lett. A 372 4682
[4] Li P Y, Gu L, Sui Y X, Yang H J 2010 Optics and Precision Engineering 18 2102 (in Chinese) [李佩玥, 古力, 隋永新, 杨怀江 2010 光学 精密工程 18 2102]
[5] Kanso A, Ghebleh M 2012 Commun Nonlinear Sci Numer Simulat 17 2943
[6] Hu H P, Liu S H, Wang Z X, Wu X G 2004 Chinese Journal of Computers 27 408 (in Chinese) [胡汉平, 刘双红, 王祖喜, 吴晓刚 2004 计算机学报 27 408]
[7] Ye R S 2011 Optics Communications 284 5290
[8] Tong X J 2013 Commun Nonlinear Sci. Numer Simulat 18 1725
[9] Wang X M, Zhang J S, Zhang W F 2003 Acta Phys. Sin. 52 2737 (in Chinese) [王小敏, 张家树, 张文芳 2003 52 2737]
[10] Li Z, Cai J P, Chen X J, Lu X F 2009 Wireless Communications and Networking Conference 2009 IEEE 4 p1
[11] Li Z, Cai J P, Lu X F, Si J B 2009 Communications 2009 ICC 09. IEEE International Conference on 6 p1
[12] Li Z, Cai J P, Chang Y L 2009 IEEE Trans Commun 57 812
[13] Xiao G Z, Wei S M, Lam K Y, Imamura K 2000 IEEE Trans Inform Theory 46 2203
[14] Kolokotronis N, Kalouptsidis N 2003 IEEE Trans Inform Theory 49 3047
[15] Kurosawa K, Sato F, Sakata T, Kishimoto W 2000 IEEE Trans Inform Theory 46 694
[16] Lauder A G B, Paterson K G 2003 IEEE Trans Inform Theory 49 273
[17] Cai J P, Li Z, Song W T 2003 Acta Phys. Sin. 52 1871 (in Chinese) [蔡觉平, 李赞, 宋文涛 2003 52 1871]
[18] Pincus S M 1995 Chaos 5 110
[19] Xiao F H, Yan G R, Han Y H 2004 Acta Phys. Sin. 53 2877 (in Chinese) [肖方红, 阎桂荣, 韩宇航 2004 53 2877]
[20] Azad R K, Rao J S, Ramaswamy R 2002 Chaos Soliton & Fract. 14 633
[21] Robinson R C 2004 An Introduction to Dynamical Systems: Continuous and Discrete (Prentice Hall Press) p459-499
[22] Chen X J, Li Z, Bai B M, Cai J P 2011 ACcta Phys. Sin. 60 064215 (in Chinese) [陈小军, 李赞, 白宝明, 蔡觉平 2011 60 064215]
[23] Lasota A, Yorke J 1973 Transactions Amer Math Soc. 186 481
[24] Li T Y, Yorke J 1978 Transactions Amer Math Soc. 235 183
[25] Eckmann J P, Ruelle D 1985 Rev. Mod. Phys. 57 617
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