Generation and statistical analysis of chaos-based pseudorandom sequences

True random numbers have gained wide applications in many areas like: com- puter simulation, Monte Carlo integration, cryptography, randomized compu- tation, radar ranging, and other areas. The generation of random numbers is impractical in real life because of difficulty in reproduction, even under the most legitimate requirements. Unfortunately, the output of physical random sources cannot be reproduced, and therefore cannot be used directly for cryptographic purposes. A deterministically generated pseudorandom (appear to be random) numbers are therefore relied upon. Two constructions for generating pseudo- random sequences were considered, viz: Linear feedback shift registers (LFSR) and chaos theory (discrete chaotic maps). A class of one dimensional (1D) chaotic maps has been considered for the generation of binary sequences. From within the class of these 1D maps, we dwell on those that satisfies the equidis- tributivity property (EDP) and constant summation property (CSP). Statistical analysis shows that there exist reasonable cross(auto)correlations within the gen- erated sequences. These correlations are catastrophic in cryptography. Despite these short comings, the process of sequence generation using chaos theory is in- deed rich in nonlinearity, which is a fundamental requirement for cryptography. A newly proposed nonlinear controlled chaotic generator (NCCG) is designed based on the combination of a chaotic map and a LFSR is presented. The gen- erator exhibits all the good qualities of a nonlinear combiner generator which addresses one of the major shortcoming of chaos based sequences- short period. Due to the influence the nonlinear combiner generator may have on the gener- ated sequences, it was tested against fast correlation attack, one of the major attacks known to weaken nonlinear combiner based sequences. The sequence is passed through the National Institute of Standards and Technology (NIST) test suites, which looked for characteristics of a truly random sequence. The gener- ated sequences were found to have passed all the prescribed tests in the suite (exhibits behavior that is expected from a truly random sequence.), thereby, suggesting its ability to be implemented in a cryptographic algorithm. The pro- posed generator has been analyzed in two phases with the first phase subjected to correlation (fast) attack and the second phase by convolutional encoder based correlation attack. It was reported that the initial state of the LFSRs in the combiner generator cannot be recovered through this attacks within available resources. Thus, we conclude that from the results of the statistical analysis, the number of observed keystream symbols cannot be recovered. This recovery is necessary for a successful attack, aimed at determining the initial state of the LFSR. If one is not able to predict the sequence generated by the combiner generator, then the clocking nature of the two chaos based binary generators cannot be understood. Therefore the final binary sequence realized from the generator (NCCG) will be appreciably resistant to the cryptanalytic algorithms considered.

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