Research On Chaotic Cipher Based On The Analysis Of Dynamic Properties

Since the American meteorologist Lorenz discovered Chaos in1965, chaostheory attracted more and more attentions. Chaotic systems have three specialproperties that are sensitive properties dependent on initial values and systemparameters, ergodicity and infinitely long period orbit, which means the chaos haveideal cipher properties. A simple chaotic dynamical system can generate very complexbehaviors with good cryptographic properties, which shows the value of the chaoticcryptography. On the other hand, the traditional cryptographic algorithms based onnumber theory, algebra and algorithm complexity theory are now experiencingchallenges from the progress of the cryptanalysis methods. Some well-knownalgorithms such as DES, MD5, SHA1et al considered secure have been crackedrecent. While, for the development of the quantum computing the asymmetriccryptographic algorithms confront a new significant threat. As the new cipher designtheory is becoming an urgent demand, the chaotic cryptograph from a new perspectiveis a good alternative.However, the existing chaotic cryptographic algorithms still have many problemssuch as poor dynamic properties and structure weakness. None of the existing chaoticcryptographic algorithms have been widely used yet. In response to these problems,this paper was studied from the following aspects:Firstly, the effects on the chaotic cipher from the chaotic dynamic properties arestudied. By the calculation of Lyapunov exponents of the chaotic systems can make itclear that the system in which area is non-chaotic, which can also measure the speedover time of the separation of the adjacent orbit from initial time. The constantpositive Lyapunov exponent spectrum in cryptography is an ideal property. Thelimit-point-set of chaotic system shows a stable long-running nature of the system. In cryptography, the ideal situation of the chaotic system should keep evenly to preventleakage of the characteristics of the system. Research shows that the degree ofcomplexity can be a measure of the degree of the randomness of the random sequencegenerated by the system, which means the higher complexity of the system is thesystem is more difficult to predict.Secondly, based on the analyzing of the dynamic properties of the chaos, a newclass of low-dimensional chaotic system with Markov property of T (x, p, sigma)was propose. It used the interval level transformation as the basis for analysis of thesystem which can make detailed studies by symbolic dynamics. It can be proved thatthe theoretically Lyapunov exponent is positive and the limiting distribution of thechaos is uniform. By adjusting the parameters, the chaos is more complex than theLogistic map and Tent map measuring by approximate entropy, which means thesequence it produced has better randomness.Thirdly, the weaknesses of some existing chaotic image encryption algorithmsare proposed. Mostly, the existing chaotic image encryption algorithms are based onthe Fridrich‘s structure which has two components, a permutation matrix and adiffusion function. When researched in the broken algorithms, it is found that thedesigns of the diffusion functions‘structure are too simple to break. Also, thepermutation matrixes are reversible so that the key stream can be got by chosenplaintext attack, and then the original image can be recovered.Then, a new image encryption algorithm is designed. A new key generationfunction is designed by use of the new chaotic system with ideal dynamical propertyconstructed in this paper, which can generate ideal key stream. What is more, thediffusion function is improved through the mixed operations of the additions indifferent groups, which is hard to break. It can be seen that the algorithm designed inthis article can pass all the tests use in the image encryption algorithms. While, thealgorithm can resist the common attacks either.Finally, the chaotic image encryption algorithm of this paper is realized on theFPGA platform. The realization method of the algorithm is given. The performance ofthe algorithm is optimized in parallel part. The pipeline technology has also beenfurther optimized so that it can have better computing speed while consumed lessresources.The results show that the chaotic system proposed in this article has idealdynamical properties which make the system particularly suitable for the design of cryptographic algorithms. This image encryption algorithm designed in the article issafe and efficiency which has great potential usage.