基于分区初等元胞自动机的二维伪随机耦合映像格系统及其动态特性

doi:10.11959/j.issn.1000-436x.2022001

摘要/Abstract

摘要：

针对基于耦合映像格的时空混沌系统中，某些控制参数会导致弱混沌的现象，以及系统生成序列频率分布不均的情况，提出了一种基于分区初等元胞自动机的二维伪随机耦合映像格系统。首先，基于初等元胞自动机，设计了高维的分区初等元胞自动机。然后，根据该高维自动机的输出，实现了伪随机的耦合方案，同时将自动机的输出作为扰动添加至时空混沌系统中。利用K熵、分岔图等对二维伪随机耦合映像格系统的动态特性进行了对比分析，同时对系统生成序列的分布特性、相关性以及随机性进行了研究。结果表明，该系统建立了更强更广泛的混沌特性，拥有良好的复杂性、遍历性和非周期性。此外，该系统生成的序列具有更均匀的分布和序列之间更低的相关性，并拥有良好的伪随机性。因此，二维伪随机耦合映像格系统在密码系统和混沌保密通信中具有广阔的应用前景。

关键词: 时空混沌系统, 耦合映像格, 初等元胞自动机, 分岔图, 均匀性

Abstract:

To solve the weak chaos in the spatiotemporal chaotic system based on coupled map lattices under some control parameters and the un-uniformity of sequences generated by the coupled map lattices, a two-dimensional pseudo-random coupled map lattices (2D-PRCML) system was proposed.Firstly, the two-dimensional partitioned elementary cellular automata (2D-PECA) was designed to establish pseudo-random coupling.Secondly, iterative results of 2D-PECA were utilized to perturb the 2D-PRCML system.The chaotic behaviors of the proposed system, such as the bifurcation diagram, Kolmogorov-Sinai entropy, were investigated.Moreover, the uniformity of sequences generated by the 2D-PRCML system was discussed, and the correlation coefficients between any two sequences generated by different lattices were acquired.The analyses and tests indicate that the 2D-PRCML system exhibits stronger chaotic behavior.Furthermore, the sequence generated by the proposed system possesses better uniformity, randomness, and unpredictability.The outstanding properties of the 2D-PRCML system prove that it is more suitable for applying in cryptography and chaotic secure communication.

Key words: spatiotemporal chaotic system, coupled map lattices, partitioned elementary cellular automata, bifurcation diagram, uniformity

中图分类号:

TN918

董有恒, 赵耿, 马英杰. 基于分区初等元胞自动机的二维伪随机耦合映像格系统及其动态特性[J]. 通信学报, 2022, 43(1): 71-82.

Youheng DONG, Geng ZHAO, Yingjie MA. Two-dimensional pseudo-random coupled map lattices system based on partitioned elementary cellular automata and its dynamic properties[J]. Journal on Communications, 2022, 43(1): 71-82.

图/表 12

表1

表2

图1

图2

图3

图4

图5

图6

图7

图8

图9

表3

参考文献 51

[1]	ALAWIDA M , SAMSUDIN A , TEH J S . Enhanced digital chaotic maps based on bit reversal with applications in random bit generators[J]. Information Sciences, 2020,512: 1155-1169.
[2]	黄春光, 程海, 丁群 . 基于PUF的Logistic混沌序列发生器[J]. 通信学报, 2019,40(3): 182-189.
	HUANG C G , CHENG H , DING Q . Logistic chaotic sequence generator based on physical unclonable function[J]. Journal on Communications, 2019,40(3): 182-189.
[3]	LORENZ E N . Deterministic nonperiodic flow[J]. Journal of the Atmospheric Sciences, 1963,20(2): 130-141.
[4]	LIU C Y , DING Q . A modified algorithm for the logistic sequence based on PCA[J]. IEEE Access, 2020,8: 45254-45262.
[5]	LIU Y , QIN Z , LIAO X F ,et al. A chaotic image encryption scheme based on Hénon-Chebyshev modulation map and genetic operations[J]. International Journal of Bifurcation and Chaos, 2020,30(6): 2050090.
[6]	NASKAR P K , BHATTACHARYYA S , NANDY D ,et al. A robust image encryption scheme using chaotic tent map and cellular automata[J]. Nonlinear Dynamics, 2020,100(3): 2877-2898.
[7]	HAMDI M , MIRI J , MOALLA B . Hybrid encryption algorithm (HEA) based on chaotic system[J]. Soft Computing, 2021,25(3): 1847-1858.
[8]	SAHASRABUDDHE A , LAIPHRAKPAM D S . Multiple images encryption based on 3D scrambling and hyper-chaotic system[J]. Information Sciences, 2021,550: 252-267.
[9]	WANG M X , WANG X Y , ZHAO T T ,et al. Spatiotemporal chaos in improved cross coupled map lattice and its application in a bit-level image encryption scheme[J]. Information Sciences, 2021,544: 1-24.
[10]	WANG X Y , YANG J J , GUAN N N . High-sensitivity image encryption algorithm with random cross diffusion based on dynamically random coupled map lattice model[J]. Chaos,Solitons ＆ Fractals, 2021,143:110582.
[11]	ZHANG X , YE R S . A novel RGB image encryption algorithm based on DNA sequences and chaos[J]. Multimedia Tools and Applications, 2021,80(6): 8809-8833.
[12]	邓晓衡, 廖春龙, 朱从旭 ,等. 像素位置与比特双重置乱的图像混沌加密算法[J]. 通信学报, 2014,35(3): 216-223.
	DENG X H , LIAO C L , ZHU C X ,et al. Image encryption algorithms based on chaos through dual scrambling of pixel position and bit[J]. Journal on Communications, 2014,35(3): 216-223.
[13]	石航, 王丽丹 . 一种基于压缩感知和多维混沌系统的多过程图像加密方案[J]. 物理学报, 2019,68(20): 39-52.
	SHI H , WANG L D . Multi-process image encryption scheme based on compressed sensing and multi-dimensional chaotic system[J]. Acta Physica Sinica, 2019,68(20): 39-52.
[14]	ZHOU P Z , DU J X , ZHOU K ,et al. 2D mixed pseudo-random coupling PS map lattice and its application in S-box generation[J]. Nonlinear Dynamics, 2021,103(1): 1151-1166.
[15]	韩丹丹, 闵乐泉, 赵耿 ,等. 一维鲁棒混沌映射及 S 盒的设计[J]. 电子学报, 2015,43(9): 1770-1775.
	HAN D D , MIN L Q , ZHAO G ,et al. One-dimensional robust chaotic map and the construction of S-box[J]. Acta Electronica Sinica, 2015,43(9): 1770-1775.
[16]	DRIDI F , EL-ASSAD S , EL-HADJ YOUSSEF W ,et al. The design and FPGA-based implementation of a stream cipher based on a secure chaotic generator[J]. Applied Sciences, 2021,11(2): 625.
[17]	LIU Z , WANG Y , ZHAO Y ,et al. A stream cipher algorithm based on 2D coupled map lattice and partitioned cellular automata[J]. Nonlinear Dynamics, 2020,101(2): 1383-1396.
[18]	陈铁明, 蒋融融 . 混沌映射和神经网络互扰的新型复合流密码[J]. 物理学报, 2013,62(4): 040301.
	CHEN T M , JIANG R R . New hybrid stream cipher based on chaos and neural networks[J]. Acta Physica Sinica, 2013,62(4): 040301.
[19]	LI S J , CHEN G R , MOU X Q . On the dynamical degradation of digital piecewise linear chaotic maps[J]. International Journal of Bifurcation and Chaos, 2005,15(10): 3119-3151.
[20]	BECK C , ROEPSTORFF G . Effects of phase space discretization on the long-time behavior of dynamical systems[J]. Physica D:Nonlinear Phenomena, 1987,25(1-3): 173-180.
[21]	BINDER P M , JENSEN R V . Simulating chaotic behavior with finite-state machines[J]. Physical Review A:General Physics, 1986,34(5): 4460-4463.
[22]	FLORES-VERGARA A , GARCíA-GUERRERO E E , INZUNZA-GONZáLEZ E ,et al. Implementing a chaotic cryptosystem in a 64-bit embedded system by using multiple-precision arithmetic[J]. Nonlinear Dynamics, 2019,96(1): 497-516.
[23]	CHEN C , SUN K H , HE S B . An improved image encryption algorithm with finite computing precision[J]. Signal Processing, 2020,168:107340.
[24]	LUO Y L , LIU Y Q , LIU J X ,et al. Counteracting dynamical degradation of a class of digital chaotic systems via Unscented Kalman Filter and perturbation[J]. Information Sciences, 2021,556: 49-66.
[25]	ZHANG Y Q , WANG X Y , LIU L Y ,et al. Spatiotemporal chaos of fractional order logistic equation in nonlinear coupled lattices[J]. Communications in Nonlinear Science and Numerical Simulation, 2017,52: 52-61.
[26]	ZHANG Y Q , HE Y , WANG X Y . Spatiotemporal chaos in mixed linear-nonlinear two-dimensional coupled logistic map lattice[J]. Physica A:Statistical Mechanics and Its Applications, 2018,490: 148-160.
[27]	王永, 马键滨, 陈燕 ,等. 一种新的基于时空混沌的伪随机数发生器[J]. 计算机工程与应用, 2018,54(11): 97-102.
	WANG Y , MA J B , CHEN Y ,et al. New pseudorandom number generator based on spatiotemporal chaos[J]. Computer Engineering and Applications, 2018,54(11): 97-102.
[28]	王永, 赵毅, . 基于分段logistic映射的二维耦合映像格子模型的密码学相关特性分析[J]. 电子学报, 2019,47(3): 657-663.
	WANG Y , ZHAO Y , GAO J ,et al. Cryptographic feature analysis on 2D coupled map lattices based on piecewise logistic map[J]. Acta Electronica Sinica, 2019,47(3): 657-663.
[29]	CHATé H , MANNEVILLE P . Spatio-temporal intermittency in coupled map lattices[J]. Physica D:Nonlinear Phenomena, 1988,32(3): 409-422.
[30]	KANEKO K . Pattern dynamics in spatiotemporal chaos[J]. Physica D:Nonlinear Phenomena, 1989,34(1/2): 1-41.
[31]	WANG X Y , GUAN N N , ZHAO H Y ,et al. A new image encryption scheme based on coupling map lattices with mixed multi-chaos[J]. Scientific Reports, 2020,10:9784.
[32]	WANG M X , WANG X Y , WANG C P ,et al. Spatiotemporal chaos in cross coupled map lattice with dynamic coupling coefficient and its application in bit-level color image encryption[J]. Chaos,Solitons ＆Fractals, 2020,139:110028.
[33]	ZHANG Y Q , HE Y , LI P ,et al. A new color image encryption scheme based on 2DNLCML system and genetic operations[J]. Optics and Lasers in Engineering, 2020,128:106040.
[34]	HE Y , ZHANG Y Q , WANG X Y . A new image encryption algorithm based on two-dimensional spatiotemporal chaotic system[J]. Neural Computing and Applications, 2020,32(1): 247-260.
[35]	PENG Y X , SUN K H , HE S B . An improved return maps method for parameter estimation of chaotic systems[J]. International Journal of Bifurcation and Chaos, 2020,30(4): 2050058.
[36]	DONG Y H , ZHAO G . A spatiotemporal chaotic system based on pseudo-random coupled map lattices and elementary cellular automata[J]. Chaos,Solitons ＆ Fractals, 2021,151:111217.
[37]	LI W , PACKARD N . The structure of the elementary cellular automata rule space[J]. Complex Systems. 2000,4(3): 281-297.
[38]	KANEKO K . Spatiotemporal chaos in one- and two-dimensional coupled map lattices[J]. Physica D:Nonlinear Phenomena, 1989,37(1-3): 60-82.
[39]	NEUMANN J , BURKS A W . Theory of self-reproducing automata[M]. Urbana: University of Illinois Press, 1966.
[40]	LANGTON C G . Self-reproduction in cellular automata[J]. Physica D:Nonlinear Phenomena, 1984,10(1-2): 135-144.
[41]	WOLFRAM S . Cellular automata as models of complexity[J]. Nature, 1984,311(5985): 419-424.
[42]	WOLFRAM S , MALLINCKRODT A J . Cellular automata and complexity[J]. Computers in Physics, 1995,9(1): 55.
[43]	WOLF A , SWIFT J B , SWINNEY H L ,et al. Determining Lyapunov exponents from a time series[J]. Physica D:Nonlinear Phenomena, 1985,16(3): 285-317.
[44]	万求真, 周昭腾 . 具有多参数恒 Lyapunov 指数谱的新型统一混沌系统[J]. 通信学报, 2020,41(6): 202-213.
	WAN Q Z , ZHOU Z T . Novel unified chaotic system with constant Lyapunov expective spectrum with multiple parameters[J]. Journal on Communications, 2020,41(6): 202-213.
[45]	WANG X Y , FENG L , WANG S B ,et al. Spatiotemporal chaos in coupled logistic map lattice with dynamic coupling coefficient and its application in image encryption[J]. IEEE Access, 2018,6: 39705-39724.
[46]	WANG X Y , ZHAO H Y , FENG L ,et al. High-sensitivity image encryption algorithm with random diffusion based on dynamic-coupled map lattices[J]. Optics and Lasers in Engineering, 2019,122: 225-238.
[47]	ZHANG Y Q , WANG X Y . Spatiotemporal chaos in Arnold coupled logistic map lattice[J]. Nonlinear Analysis:Modelling and Control, 2013,18(4): 526-541.
[48]	YAN J , BECK C . Distinguished correlation properties of Chebyshev dynamical systems and their generalisations[J]. Chaos,Solitons ＆Fractals:X, 2020,5:100035.
[49]	BASSHAM L E I , RUKHIN A L , SOTO J ,et al. A statistical test suite for random and pseudorandom number generators for cryptographic applications[R]. National Institute of Standards and Technology, 2010.
[50]	HU G Z , LI B B . Coupling chaotic system based on unit transform and its applications in image encryption[J]. Signal Processing, 2021,178:107790.
[51]	王兴元 . 混沌系统的同步及在保密通信中的应用[M]. 北京: 科学出版社, 2012.
	WANG X Y . Synchronization of chaotic systems and its application in secure communication[M]. Beijing: Science Press, 2012.

迭代结果	二进制数
S_t(i-1)	1	1	1	1	0	0	0	0
S_t(i)	1	1	0	0	1	1	0	0
S_t(i+1)	1	0	1	0	1	0	1	0
S_t+1(i)	1	0	0	1	0	1	1	0

种类	规则编号
	18(183), 22(151), 30(86, 135, 149), 45(75, 89, 101),
全局混沌	60(102, 153, 195), 90(165), 105, 106(120, 169, 225),
	129(126), 137(110, 124, 193), 146(182), 150, 161(122)