混沌与密码学之间存在紧密联系,诸如初值敏感性、遍历性等混沌系统的基本特性与密码学中的混淆与扩散概念具有本质联系。目前国内外均将混沌密码学作为信息安全新技术加以支持,力图在信息安全科学理论和技术上的实现源头创新。但无论数字混沌(伪混沌)密码还是模拟混沌保密通信技术研究现状表明,仅依赖于这些属性并不能保证密码系统的安全性,研究表明确定性随机在混沌与随机过程之间的桥梁作用将是提高混沌密码安全性的关键因素。本文主要研究确定性随机理论及在混沌密码学中的应用可行性。主要工作包括: 第一章在简要介绍混沌、确定性随机理论基础上,讨论了数字混沌密码的研究现状。 第二章研究了基于搜索机制的数字混沌序列密码算法,并详细分析了包括一次一密攻击,分叉图攻击,熵攻击,差分攻击以及基于符号动力学的攻击方法。研究表明此类序列密码方案混淆了混沌与伪混沌概念,即在方案设计与安全性分析上,错误地利用了诸如长期不可预测等混沌的动力学特性,在安全性方面没有超出传统密码学领域中关于线性同余发生器研究的范畴。 第三章研究了对统计与差分攻击具有高免疫力的三维CAT变换混沌图像加密算法,并针对此算法提出一种选择明文攻击方法,通过符号动力学理论及设计特定的明文图像,可构造出等价的三维Cat变换矩阵及扩散操作的初始密钥值。分析与实验结果表明基于三维Cat变换的混沌图像加密算法无法抵抗选择明文攻击。另外本文提出的分析方法可以有效攻击多种混沌图像加密算法。 第四章深入研究并发展了确定性随机理论。首先,证明了不存在确定性随机的可实现模型,并给出了基于斜抛物线映射与不可逆非线性变换所构造李沙育映射可以完整描述渐进确定性随机的条件。在此基础上不但更正了确定性随机理论创始人提出的自治系统可以产生确定性随机的错误观点,而且证明其仅为渐近确定性随机近似描述模型。其次,不但更正了国际上关于显式函数xn = sin 2(πθzn)无法产生确定性随机的错误观点,而且证明z为互质假分数时便可以产生确定性随机现象。最后,在详细分析李沙育映射不动点、倍周期分叉、Lyapunov指数谱、符号动力学等相关动力学特性等基础上,不但证明确定性随机序列的Lyapunov指数为无穷大而不是目前认为的lnz ,而且发现渐近确定性随机可实现模型将退化为一类特殊高维混沌系统。进一步,基于渐近确定性随机可实现模型构造了一种混沌序列密码,分析与实验结果表明可抵御针对伪混沌密码分析方法的攻击。
The use of chaos in cryptography depends on the natural relation between the two research aeras: Shannon’s classic paper on the theory of secreacy systems has indicated that rolled-out and folded-over operations can be used to design strong encryption scheme, intuitively, such operations consist with those elemential properties of chaos as ergodicity and initial value sensitivity. Chaotic cryptography has been well studied uptodate and makes it possible to design new type of cryptographic functions beyond number theory. Unfortunately, a lot of negtive cryptanalysises show that the security of chaos synchronization based security communication and pseudo-chaotic ciphers is very weak. A new nonlinear dynamic phenomenon named deterministic randomness, which bridges chaos and randomness, may play the key role in increasing the security of chaotic ciphers. This dissertation mainly focuses on the investigation of deterministic randomness theory and its application to chaotic ciphers. Firstly, we give a brief introduction to chaos and deterministic randomness theory, and furthermore discuss the research situations of pseudo-chaotic stream ciphers. Secondly, the pseudo-chaotic stream cipher based on searching plaintext in a pseudo-random symbolic sequence is analyzed, by several successful cryptanalytic attacks such as one-time pad attacks, bifurcation diagram attacks, entropy attacks, differential attacks and key recovery attacks etc.. Since chaos and psedo-chaos are confused conceptually in the research of pseudo-chaotic stream cipher, in other words, such property of chaos as long-term unpredictability is used to design and analyze psedo-chaoic stream cipher, in fact, psedo-chaos should be characterized by predictability. Cryptanalytical results show that pseudo-chaotic stream cipher even dosen’t go beyond the scope of linear congruential algorithm in fields of conventional cryptography. Then we investigate a recent proposed three dimensional Cat map based symmetric image encryption algorithm which employs the Cat map to shuffle the positions of image pixels and uses Logistic map to diffuse the relationship between the cipher-image and the plain-image, and significantly resist the statistical and differential attacks. However, with the knowledge of symbolic dynamics, fundamental weaknesses of the cryptosystem are pointed out under specially designed plain-images, and then a successful chosen-plain-text cryptanalytic attack is described. Both theoretical and experimental results show that the intruder can access to the equivalent keys of such a scheme and other variants with little difficulty. The lack of security discourages the use of these algorithms for practical applications Finally we focus our investigation on the deterministic randomness theory and its application to cryptography. The conditions for generating asymptotic deterministic randomness based on Lissajous map, which constructed by the skewed parabola map and the non-invertible non-linearities transform, are presented. It should be noted that any realizable model for deterministic randomness will degenerate to some special high dimensional chaotic system. With help of asymptotic deterministic randomness theory, we find that the well-known autonomous system that has been stated to generate deterministic randomness can only act as an approximative realizable model of which, and then several popular statements such as function xn = sin 2(πθzn) can not generate deterministic randomness, and corresponding Lyapunov exponent is ln z etc. have been rectified. In other words, we prove that such function can generate deterministic randomness only when the value of parameter z belongs to some relative prime fraction number larger than one. Furthermore, we also analyze the underlying dynamics such as fixed point, bifurcation process, Lyapunov exponent spectrum, and symbolic dynamics etc. in details. Fortunately and most important, it seems that the asymptotic deterministic randomness based ciphers can resist various attacks which can break almost all pseudo-chaotic stream ciphers sucessfully, and a simplified scheme is provided.
