基于浮点格式的数字混沌系统周期研究
摘要
数字混沌系统的周期及其分布是混沌退化与混沌抗退化机制研究的必要内容,是衡量混沌系统抗退化能力以及能否用于混沌加密算法的重要指标。一维混沌系统的平均周期与计算精度的经验关系,以及周期分布范围的经验式先后被提出,但这些成果均是基于定点格式的,虽沿用至今,却只有定性意义,实际意义有限,特别是要作为混沌抗退化机制研究的定量评估标准,理论依据不充分。
为获得一个具有理论和实际应用价值的关于数字混沌系统的周期及周期分布的经验关系,本文研究了计算机迭代下基于浮点格式的数字混沌系统的周期及其分布规律。鉴于标准浮点数格式种类有限,通过定义一套与标准浮点数格式相匹配的非标准浮点数格式,将精度变量引入浮点数格式中,同时在计算上用标准浮点数格式代替非标准浮点数格式,解决了非标准浮点数格式下的计算算法难题；提出了一种搜索数字混沌系统周期的快速算法,使得在有限时间内采用多种不同精度的浮点数格式测算各种混沌系统的周期成为可能；使用VisualC++6.0编程,统计测算了11种一维混沌系统、4种二维混沌系统以及基于切延迟椭圆反射腔映射混沌系统(Tangent-Delay Ellipse Reflecting Cavity map System, TD-ERCS)在不同迭代初始值、不同系统参数和不同浮点计算精度下的轨道周期；通过线性拟合方法获得了浮点格式下数字混沌系统周期随计算精度变化的经验关系,研究发现了影响数字混沌系统周期分布的主要原因,纠正了多年来沿用的基于定点格式的相应经验关系,为混沌抗退化机制研究提供了一个合理的可用于实验测试的参考标准。本文的研究结果表明,对于混沌系统而言,基于定点格式的结论不能简单随意推广到浮点格式。
The periods and their distribution of a digital chaotic system, essential content of chaotic degradation and anti-degradation mechanism research, is an important indicator of assessing the capacity of anti-dagradation and the safety of a chaotic encryption algorithm. The existing conclusions about empirical relation of 1-D chaotic system's average period and the distribution range of periods are all based on fix-point formats. If using them as the quantitative assessment criteria of chaotic anti-dagration, theoretical basis is not sufficient.
To get an empirical relation with more theoretical and applicative value, the periods and their distribution of digital chaotic systems based on floating-point formats are studied in this paper. Because standard floating-point formats are not sufficient, non-standard floating-point formats matching with the standard ones are constructed, by putting precision variables into floating-point formats and using standard floating-point formats for storing non-standard ones. A fast searching algorithm is designed, making it possible to find out the chaotic periods under different floating-point precisions. Using Visual C++ 6.0, the periods of eleven 1-D chaotic systems, four 2-D chaotic systems and TD-ERCS (Tangent-Delay Ellipse Reflecting Cavity map System) are measured, by condition of different initial iterating values, different system parameters, or different floating-point computing precisions. Factors impacting periods are figured out. Using a linear fitting method, the distribution relationship between periods and numerical precisions is obtained, which corrects the one based on fixed-point formats. Moreover, it gives a rational criterion for the study of chaotic anti-degradation mechanism. This paper also shows that for chaotic systems, conclusions based on fixed-point formats can not be simply extended to ones based on floating-point formats.
为获得一个具有理论和实际应用价值的关于数字混沌系统的周期及周期分布的经验关系,本文研究了计算机迭代下基于浮点格式的数字混沌系统的周期及其分布规律。鉴于标准浮点数格式种类有限,通过定义一套与标准浮点数格式相匹配的非标准浮点数格式,将精度变量引入浮点数格式中,同时在计算上用标准浮点数格式代替非标准浮点数格式,解决了非标准浮点数格式下的计算算法难题；提出了一种搜索数字混沌系统周期的快速算法,使得在有限时间内采用多种不同精度的浮点数格式测算各种混沌系统的周期成为可能；使用VisualC++6.0编程,统计测算了11种一维混沌系统、4种二维混沌系统以及基于切延迟椭圆反射腔映射混沌系统(Tangent-Delay Ellipse Reflecting Cavity map System, TD-ERCS)在不同迭代初始值、不同系统参数和不同浮点计算精度下的轨道周期；通过线性拟合方法获得了浮点格式下数字混沌系统周期随计算精度变化的经验关系,研究发现了影响数字混沌系统周期分布的主要原因,纠正了多年来沿用的基于定点格式的相应经验关系,为混沌抗退化机制研究提供了一个合理的可用于实验测试的参考标准。本文的研究结果表明,对于混沌系统而言,基于定点格式的结论不能简单随意推广到浮点格式。
The periods and their distribution of a digital chaotic system, essential content of chaotic degradation and anti-degradation mechanism research, is an important indicator of assessing the capacity of anti-dagradation and the safety of a chaotic encryption algorithm. The existing conclusions about empirical relation of 1-D chaotic system's average period and the distribution range of periods are all based on fix-point formats. If using them as the quantitative assessment criteria of chaotic anti-dagration, theoretical basis is not sufficient.
To get an empirical relation with more theoretical and applicative value, the periods and their distribution of digital chaotic systems based on floating-point formats are studied in this paper. Because standard floating-point formats are not sufficient, non-standard floating-point formats matching with the standard ones are constructed, by putting precision variables into floating-point formats and using standard floating-point formats for storing non-standard ones. A fast searching algorithm is designed, making it possible to find out the chaotic periods under different floating-point precisions. Using Visual C++ 6.0, the periods of eleven 1-D chaotic systems, four 2-D chaotic systems and TD-ERCS (Tangent-Delay Ellipse Reflecting Cavity map System) are measured, by condition of different initial iterating values, different system parameters, or different floating-point computing precisions. Factors impacting periods are figured out. Using a linear fitting method, the distribution relationship between periods and numerical precisions is obtained, which corrects the one based on fixed-point formats. Moreover, it gives a rational criterion for the study of chaotic anti-degradation mechanism. This paper also shows that for chaotic systems, conclusions based on fixed-point formats can not be simply extended to ones based on floating-point formats.
引文
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