混沌序列生成新方法及其在数字水印中的应用研究
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摘要
混沌系统具有的确定性系统的内在随机性、对初始条件的高度敏感性、长期不可预测性等特征与密码设计中的混淆与扩散两种基本方法有着本质联系。近年来,混沌密码作为密码学研究的新技术及有力补充,得到了广泛研究。其中大部分研究都集中在混沌序列的产生及分析上,众多文献认为混沌序列具有足够长的周期及良好的随机数统计特性,并将此引入到数字水印等领域。然而,随着研究的深入,人们发现数字化后的混沌系统普遍存在有限精度下的动力学特征退化,从而使得系统进入短周期轨道。这种有限精度下的短周期行为严重破坏了混沌序列的随机数特征,导致其统计学特性变弱,相应的密钥空间缩小。
为改善和消除有限精度下混沌的短周期行为对混沌序列密码的影响,学术界提出了包括改善混沌系统结构、多个混沌系统混和及采用更复杂的混沌系统等三种解决方法。这些方法的共同点是采用实验仿真的形式验证了其有效性,而理论上的分析仅涉及混沌系统的部分特性。本文首次比较完整地分析了混沌系统在有限精度下的动力学特征退化问题及三种解决方法的理论依据,并以切比雪夫序列为例定性分析了这种短周期的存在；以混沌系统的可加性理论研究深入探讨了多个混沌系统在串联混合、交叉混合、级联混合下的规律。为验证规律的有效性,本文设计、实现了一些新的混沌序列产生方法,并将其应用到数字水印领域。本文从三个方面探讨了混沌序列的生成方法及其在数字水印中的应用问题。
(1)混沌动力学特征退化及系统可加性研究
近年来,数字化混沌系统在有限精度下的动力学特征退化现象引起了广泛关注,不少文献指出,Logistic映射、Tent映射、Chebyshev映射等一维混沌映射均存在短周期退化现象,导致生成的混沌序列退化为短周期序列,从而失去其密码学上的应用价值。本文以Chebyshev映射为例,定性分析了这种短周期的存在性,并给出了相应的周期公式。
为解决数字化混沌系统存在的短周期特性,业界普遍采用多个混沌系统混合的方法,并采用实验仿真的形式说明了这些方法的有效性。然而,随着本文作者的进一步研究,发现这些方法的本质特征是利用了混沌系统具备的可加性性质,可惜的是,国内外研究人员很少关心混沌系统的可加性性质。本文先定义了混沌系统的串联混合、交叉混合、级联混合等概念,然后采用定性分析及仿真实验相结合的方式论证了混沌系统在这几种混合方式下存在的特性,研究结果对混沌序列的产生及分析有着良好的指导作用。
(2)混沌序列产生新方法及其性质分析
基于混沌动力学特征退化和系统可加性理论,本文提出了一种改进型切比雪夫混沌序列算法、一类多涡卷蔡氏混沌序列产生方法和一种新型混沌电路随机序列发生器。改进型切比雪夫混沌序列算法充分利用了混沌系统的短周期特性,通过在系统出现短周期时的前—时刻改变方程参数,使得混沌系统避免进入短周期,从而得到周期更长的混沌序列。基于多涡卷蔡氏混沌系统的序列产生新方法说明,采用高维混沌系统既可以从一定程度削弱短周期行为对混沌序列的影响,又能扩大混沌系统的密钥空间。尤为重要的是,高阶多涡卷乃至网格状多涡卷蔡氏混沌系统产生随机序列的原理均一样,因此,在实际应用时,可以根据需要选择不同的多涡卷蔡氏混沌系统。一种新型混沌电路随机序列发生器在作者提出的新的混沌系统的基础上以电子电路形式实现了一种随机数序列发生器,为硬件实现混沌序列发生器提供了新的思路。
(3)混沌序列在数字水印中的应用研究
混沌序列在数字水印领域得到了越来越广泛的研究,这些研究普遍认为低维混沌系统产生的伪随机序列比传统算法生成的伪随机序列性质好、密钥空间大,忽视了混沌的短周期行为对混沌序列密码的影响。本文先讨论了混沌密码系统中的密钥及密钥空间问题,然后给出了混沌密码算法设计过程中有关密钥及密钥空间的若干建议,并在基于Contourlet及SVD的混沌数字水印算法和基于网格状多涡卷蔡氏混沌系统的数字水印两章,探讨了混沌序列在数字水印中的应用问题及应遵循的密码学设计原则和评价体系。
本学位论文的工作得到了禹思敏教授主持的国家自然科学基金(批准号：60871025,61172023)、教育部高等学校博士学科点(博导类)专项科研基金(批准号：20114420110003)、广东省自然科学基金(批准号：8151009001000060,S2011010001018)、广东省科技计划项目(批准号：20098010800037)的资助。
Chaos system has the following features, such as intrinsic randomness of deterministic system, highly sensitive to initial conditions, long-term unpredictability. They have essentially relationship with confusion and diffusion in cipher design.As a new technology and a strong complement to Cryptography, chaotic cryptography has got a lot of research in recent years. Most studies have focused on the generation and analysis of chaotic sequence, numerous literatures suggests that the chaotic sequence has a sufficiently long cycle and good statistical properties. However, with the in-depth study, people found that the dynamic characteristics of chaotic system will be degraded in finite precision, which make the system into short-period. The short-cycle behavior serious damage to the randomness of the chaotic sequence, lead to its statistical characteristics weakened and the corresponding key space narrowed.
In order to improve and eliminate the impacts on chaotic sequence, three methods were put forward by researchers, including improving chaotic system structure, mixing multi-chaotic system and using more complex chaotic system. The common characteristics of these methods are the use of experimental simulation verifies its effectiveness, and almost with no theoretical analysis. This paper was first qualitative analysis these characteristics, and used chebyshev sequence as an example in order to qualitative analysis the feature of short-period, then discusses the law of multiple chaotic systems which were series connected, cross-mixed, cascade mixed. In order to verify the validity of the law, we design some new chaotic sequence generating method and apply them in digital watermarking.
This article discussed the three aspects about chaotic sequence generation method and its application in digital watermarking.
(1) dynamics characteristics degradation and addition of chaos
In recent years, dynamics characteristics degradation of digital chaotic in finite precision has aroused widespread interest. A lot of literatures show that it appears in Logistic map, Tent map and Chebyshev Map, the chaotic sequences which were generated by these map have always lost their value on cryptography. In order to analysis the short-period feature of chaos, we used chebyshev sequence as an example to qualitative analysis of the existence and give the cycle formula of chebyshev sequence finally.
To solve the short-period characteristics of digital chaotic system, multi-chaotic system mixed method was commonly used in the industry and experimental simulation always used to illustrates the effectiveness of these methods. It is found that the essential characteristic of these methods is the addition ability of chaos.Unfortunately, few researchers at home and abroad concerned about the addition ability of chaos. This article first defined a series of concept about chaotic system, such as series connected, cross-mixed, cascade mixed. Then using qualitative analysis and simulation experiments to demonstrate the characteristics of chaotic systems exist in several mixed mode. This is a very good guide to design and analysis the chaotic sequences.
(2) The new generation method and property analysis of chaotic sequences
Based on the characteristics of degradation and addition ability of chaos, we present some methods, such as an improved chaotic sequence algorithm based on Chebyshev map, a class of multi-scroll Chua chaotic sequence generating method and a new kind of chaotic circuit random sequence generator. The improved chaotic sequence algorithm with Chebyshev by changes the equation parameters before the system enter in short-period, which makes the system not enter the short cycle and got longer sequence. The new method to generate chaotic sequence by multi-scroll Chua chaotic system shows that the short-period problem can be weakened by high-dimensional chaotic system as well as enlarging the key space. Particularly important, the principle of generating random sequence by high-dimensional is the same as by low-dimensional chaotic system. Therefore, we can choose a different multi-scroll Chua chaotic system to generate a random sequence according to the actual situation. In this paper, we first propose a new fourth-order chaotic system, and then implement a random sequence, which provides a new way for the hardware implementation of chaotic sequence.
(3) The research of chaos sequence in digital watermarking
In the field of digital watermarking, chaotic sequence has got more and more researched. These studies generally agreed that the nature of the pseudo-random sequences generated by chaotic systems is better than by traditional algorithms, and ignore the impact of the short-period behavior of the chaotic sequence. In this article, we firstly discuss the key and key space of chaotic cryptosystems and then given some suggestions to design a chaotic cryptographic algorithm, we give two algorithms, one is a digital watermarking algorithm with dual transform domain based on Contourlet and SVD, another one is a digital watermarking algorithm based on grid multi-scroll Chua's chaotic attractors. Furthermore, we point out that cryptography design principles and evaluation method should be followed in the chaotic digital watermark design.
This work was supported by the National Natural Science Foundation of China under Grant Nos.60871025and61172023, the Specialized Research Foundation of Doctoral Subject Point of Education Ministry under Grant No.20114420110003, the Natural Science Foundation of Guangdong Province under Grant Nos.8151009001000060and S2011010001018, and the Science and Technology Program of Guangdong Province under Grant No.2009B010800037.
为改善和消除有限精度下混沌的短周期行为对混沌序列密码的影响,学术界提出了包括改善混沌系统结构、多个混沌系统混和及采用更复杂的混沌系统等三种解决方法。这些方法的共同点是采用实验仿真的形式验证了其有效性,而理论上的分析仅涉及混沌系统的部分特性。本文首次比较完整地分析了混沌系统在有限精度下的动力学特征退化问题及三种解决方法的理论依据,并以切比雪夫序列为例定性分析了这种短周期的存在；以混沌系统的可加性理论研究深入探讨了多个混沌系统在串联混合、交叉混合、级联混合下的规律。为验证规律的有效性,本文设计、实现了一些新的混沌序列产生方法,并将其应用到数字水印领域。本文从三个方面探讨了混沌序列的生成方法及其在数字水印中的应用问题。
(1)混沌动力学特征退化及系统可加性研究
近年来,数字化混沌系统在有限精度下的动力学特征退化现象引起了广泛关注,不少文献指出,Logistic映射、Tent映射、Chebyshev映射等一维混沌映射均存在短周期退化现象,导致生成的混沌序列退化为短周期序列,从而失去其密码学上的应用价值。本文以Chebyshev映射为例,定性分析了这种短周期的存在性,并给出了相应的周期公式。
为解决数字化混沌系统存在的短周期特性,业界普遍采用多个混沌系统混合的方法,并采用实验仿真的形式说明了这些方法的有效性。然而,随着本文作者的进一步研究,发现这些方法的本质特征是利用了混沌系统具备的可加性性质,可惜的是,国内外研究人员很少关心混沌系统的可加性性质。本文先定义了混沌系统的串联混合、交叉混合、级联混合等概念,然后采用定性分析及仿真实验相结合的方式论证了混沌系统在这几种混合方式下存在的特性,研究结果对混沌序列的产生及分析有着良好的指导作用。
(2)混沌序列产生新方法及其性质分析
基于混沌动力学特征退化和系统可加性理论,本文提出了一种改进型切比雪夫混沌序列算法、一类多涡卷蔡氏混沌序列产生方法和一种新型混沌电路随机序列发生器。改进型切比雪夫混沌序列算法充分利用了混沌系统的短周期特性,通过在系统出现短周期时的前—时刻改变方程参数,使得混沌系统避免进入短周期,从而得到周期更长的混沌序列。基于多涡卷蔡氏混沌系统的序列产生新方法说明,采用高维混沌系统既可以从一定程度削弱短周期行为对混沌序列的影响,又能扩大混沌系统的密钥空间。尤为重要的是,高阶多涡卷乃至网格状多涡卷蔡氏混沌系统产生随机序列的原理均一样,因此,在实际应用时,可以根据需要选择不同的多涡卷蔡氏混沌系统。一种新型混沌电路随机序列发生器在作者提出的新的混沌系统的基础上以电子电路形式实现了一种随机数序列发生器,为硬件实现混沌序列发生器提供了新的思路。
(3)混沌序列在数字水印中的应用研究
混沌序列在数字水印领域得到了越来越广泛的研究,这些研究普遍认为低维混沌系统产生的伪随机序列比传统算法生成的伪随机序列性质好、密钥空间大,忽视了混沌的短周期行为对混沌序列密码的影响。本文先讨论了混沌密码系统中的密钥及密钥空间问题,然后给出了混沌密码算法设计过程中有关密钥及密钥空间的若干建议,并在基于Contourlet及SVD的混沌数字水印算法和基于网格状多涡卷蔡氏混沌系统的数字水印两章,探讨了混沌序列在数字水印中的应用问题及应遵循的密码学设计原则和评价体系。
本学位论文的工作得到了禹思敏教授主持的国家自然科学基金(批准号：60871025,61172023)、教育部高等学校博士学科点(博导类)专项科研基金(批准号：20114420110003)、广东省自然科学基金(批准号：8151009001000060,S2011010001018)、广东省科技计划项目(批准号：20098010800037)的资助。
Chaos system has the following features, such as intrinsic randomness of deterministic system, highly sensitive to initial conditions, long-term unpredictability. They have essentially relationship with confusion and diffusion in cipher design.As a new technology and a strong complement to Cryptography, chaotic cryptography has got a lot of research in recent years. Most studies have focused on the generation and analysis of chaotic sequence, numerous literatures suggests that the chaotic sequence has a sufficiently long cycle and good statistical properties. However, with the in-depth study, people found that the dynamic characteristics of chaotic system will be degraded in finite precision, which make the system into short-period. The short-cycle behavior serious damage to the randomness of the chaotic sequence, lead to its statistical characteristics weakened and the corresponding key space narrowed.
In order to improve and eliminate the impacts on chaotic sequence, three methods were put forward by researchers, including improving chaotic system structure, mixing multi-chaotic system and using more complex chaotic system. The common characteristics of these methods are the use of experimental simulation verifies its effectiveness, and almost with no theoretical analysis. This paper was first qualitative analysis these characteristics, and used chebyshev sequence as an example in order to qualitative analysis the feature of short-period, then discusses the law of multiple chaotic systems which were series connected, cross-mixed, cascade mixed. In order to verify the validity of the law, we design some new chaotic sequence generating method and apply them in digital watermarking.
This article discussed the three aspects about chaotic sequence generation method and its application in digital watermarking.
(1) dynamics characteristics degradation and addition of chaos
In recent years, dynamics characteristics degradation of digital chaotic in finite precision has aroused widespread interest. A lot of literatures show that it appears in Logistic map, Tent map and Chebyshev Map, the chaotic sequences which were generated by these map have always lost their value on cryptography. In order to analysis the short-period feature of chaos, we used chebyshev sequence as an example to qualitative analysis of the existence and give the cycle formula of chebyshev sequence finally.
To solve the short-period characteristics of digital chaotic system, multi-chaotic system mixed method was commonly used in the industry and experimental simulation always used to illustrates the effectiveness of these methods. It is found that the essential characteristic of these methods is the addition ability of chaos.Unfortunately, few researchers at home and abroad concerned about the addition ability of chaos. This article first defined a series of concept about chaotic system, such as series connected, cross-mixed, cascade mixed. Then using qualitative analysis and simulation experiments to demonstrate the characteristics of chaotic systems exist in several mixed mode. This is a very good guide to design and analysis the chaotic sequences.
(2) The new generation method and property analysis of chaotic sequences
Based on the characteristics of degradation and addition ability of chaos, we present some methods, such as an improved chaotic sequence algorithm based on Chebyshev map, a class of multi-scroll Chua chaotic sequence generating method and a new kind of chaotic circuit random sequence generator. The improved chaotic sequence algorithm with Chebyshev by changes the equation parameters before the system enter in short-period, which makes the system not enter the short cycle and got longer sequence. The new method to generate chaotic sequence by multi-scroll Chua chaotic system shows that the short-period problem can be weakened by high-dimensional chaotic system as well as enlarging the key space. Particularly important, the principle of generating random sequence by high-dimensional is the same as by low-dimensional chaotic system. Therefore, we can choose a different multi-scroll Chua chaotic system to generate a random sequence according to the actual situation. In this paper, we first propose a new fourth-order chaotic system, and then implement a random sequence, which provides a new way for the hardware implementation of chaotic sequence.
(3) The research of chaos sequence in digital watermarking
In the field of digital watermarking, chaotic sequence has got more and more researched. These studies generally agreed that the nature of the pseudo-random sequences generated by chaotic systems is better than by traditional algorithms, and ignore the impact of the short-period behavior of the chaotic sequence. In this article, we firstly discuss the key and key space of chaotic cryptosystems and then given some suggestions to design a chaotic cryptographic algorithm, we give two algorithms, one is a digital watermarking algorithm with dual transform domain based on Contourlet and SVD, another one is a digital watermarking algorithm based on grid multi-scroll Chua's chaotic attractors. Furthermore, we point out that cryptography design principles and evaluation method should be followed in the chaotic digital watermark design.
This work was supported by the National Natural Science Foundation of China under Grant Nos.60871025and61172023, the Specialized Research Foundation of Doctoral Subject Point of Education Ministry under Grant No.20114420110003, the Natural Science Foundation of Guangdong Province under Grant Nos.8151009001000060and S2011010001018, and the Science and Technology Program of Guangdong Province under Grant No.2009B010800037.
引文
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