The Investigation Of Deterministic Randomness Theory And Its Application To Chaotic Ciphers

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 applicationsFinally 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.