Periodic Detection And Resistance In Chaotic Sequence Ciphers

Due to determinable and similar to traditional cipher design requirements, chaotic sequence ciphers which generated by chaotic map or chaotic dynamic characteristics have been widely concerned, studied and used for encryption methods in communication, information security, or other computer applications fields. On one hand, the main benefit of chaotic sequence ciphers is that dynamic determinable characteristics of chaotic maps will make calculation costs far less than other traditional ciphers. On the other hand, chaotic systems are intrinsic randomness, ergodicity and sensitive to initial conditions and parameters. Trajectories from adjacent points will be separated far away after several iterating steps. Thus little changes making huge different chaotic ciphers from a same chaotic map is well satisfied with diffusion and confusion in conventional encryption systems. Such as chaotic random number generators based Logistic map or 1-D chaotic maps can easily get one pseudo-random bit by at most 3-5 floating operations. Even initial states are known, it is extremely difficult to predict the pseudo-random bits from a chaotic sequence cipher after a while.Compared with other sequence ciphers, there are two flaws in chaotic sequence ciphers. One is lack of the analysis theories on the security and characteristics of chaotic encryption systems. Until now, it is hard to get the relationship between chaotic sequence ciphers and random sequences by using sophisticated mathematics. Evaluating the security of chaotic sequence ciphers is an unresolved question. The other is that most chaotic sequence ciphers have weakened randomness behaviors. For example, the cycle length or periodicity of a chaotic sequence cipher will be far less than the elements number of chaotic system`s range. That is also called the short period phenomenon in chaotic systems.Thus, this dissertation will focus on studying the method of determining and locating weakened randomness. Comparing with the characteristics of Bernoulli sequences, this dissertation emphasizes on getting periodicities which affect the randomness of chaotic sequence ciphers and finding their statistical regularities. In this dissertation, other purpose is to eliminate or reduce the influence of such factors withlittle increasing calculating costs.The main research contents and achievements are shown as follow:Firstly, according to the logic relation of Bernoulli sequences, a novel extend definition of period phenomena is proposed. Obeyed these definitions, the relations between reconstruction sequences and local-period phenomena of chaotic binary sequences are also proposed. Furthermore, a binary sequence`s periodic detection method named BSPD(Binary Sequence`s Periodic Detection) is presented and proven,while the correctness of BSPD is also proven for detecting local-period phenomena of longer period pattern.Secondly, in order to reduce the BSPD`s limitation which is only detecting longer period pattern and not all notable local-period phenomena, a new set of definitions and statistical characteristics of period phenomena based on frequence are given as well as Accuracy-Period phenomenon, Period Symbol, Approximate-Period phenomenon,Notable Local-Period phenomenon. Based on these definitions and statistical characteristics, the relations between the weakened randomness phenomena and statistical characteristics of chaotic binary sequences are proposed. Furthermore, based on Pearson`s theorem, a new detecting method named PCDA(Periodicity Component Detecting Algorithm) is presented to reduce limitations of BSPD by finitely increasing calculation time of algorithm.Thirdly, the dissertation discusses that the impact on the randomness of chaotic sequences cipher by different quantification methods, through local-period phenomena detection and analysis of chaotic system with traditional quantification methods. Here we proved that two results by BSPD detecting chaotic sequence ciphers, one is that those phenomena certainly ubiquitous, such as the chaotic systems have the short-term forecasting character and quantification methods for the same real-value chaotic system can make the randomness of chaotic sequences cipher different based on Logistic-map.Moreover, threshold methods have less impact than other quantification methods through BSPD detecting test and statistical analysis.At last, based on the bifurcation theory, and the new idea that random sequences cannot only be generated by chaotic iterating steps, but can be separated from theelement range of digital chaotic maps, a new chaotic key stream generator which named EP-PRNG is designed and proven by using accuracy n-bits Logistic map. When chaotic sequence ciphers are constructed by EP-PRNG, it is proven that the period lengths of ciphers are larger than22n?. Furthermore, simulation results show that when EP-PRNG use 24-bits Logistic map, the chaotic sequence ciphers are not only have long-period,but also have few weakened randomness phenomena which can be located by BSPD.