GENEROWANIE SEKWENCJI LOSOWYCH O ZWIĘKSZONEJ SILE KRYPTOGRAFICZNEJ

Random sequences are used in various applications in construction of cryptographic systems or formations of noise-type signals. For these tasks there is used the program generator of random sequences which is the determined device. Such a generator, as a rule, has special requirements concerning the quality of the numbers formation sequence. In cryptographic systems, the most often used are linearly – congruent generators, the main disadvantage of which is the short period of formation of pseudo-random number sequences. For this reason, in the article there is proposed the use of chaos generators as the period of the formed selection in this case depends on the size of digit net of the used computing system. It is obvious that the quality of the chaos generator has to be estimated through a system of the NIST tests. Therefore, detailed assessment of their statistical characteristics is necessary for practical application of chaos generators in cryptographic systems. In the article there are considered various generators and there is also given the qualitative assessment of the formation based on the binary random sequence. Considered are also the features of testing random number generators using the system. It is determined that not all chaos generators meet the requirements of the NIST tests. The article proposed the methods for improving statistical properties of chaos generators. The method of comparative analysis of random number generators based on NIST statistical tests is proposed, which allows to select generators with the best statistical properties. Proposed are also methods for improving the statistical characteristics of binary sequences, which are formed on the basis of various chaos generators.Sekwencje losowe wykorzystywane są do tworzenia systemów kryptograficznych lub do formowania sygnałów zakłócających. Do tych zadań wykorzystywany jest generator sekwencji losowych, który jest urządzeniem deterministycznym. Taki generator z reguły ma specjalne wymagania dotyczące jakości tworzenia sekwencji liczbowej. W systemach kryptograficznych najczęściej stosuje się generatory liniowo-przystające, których główną wadą jest krótki okres formowania pseudolosowych sekwencji liczbowych. Z tego powodu w artykule zaproponowano użycie generatora chaotycznego, jako że okres próbkowania w tym przypadku zależy od rozmiaru siatki bitowej w używanym systemie obliczeniowym. Oczywistym jest, że należy oszacować jakość generatora chaotycznego za pomocą systemu testów NIST, dlatego też do praktycznego zastosowania generatorów chaotycznych w systemach kryptograficznych wymagana jest szczegółowa ocena ich cech statystycznych. W artykule rozważono różne generatory, a także podano ocenę jakościową procesu formacji na podstawie losowej sekwencji binarnej. Rozważano również funkcje testowania generatorów liczbowych przy użyciu systemu. Stwierdzono, że nie wszystkie generatory chaotyczne spełniają wymagania testów NIST. W artykule zaproponowano metody poprawy właściwości statystycznych generatorów chaotycznych, tak jak również metodę analizy porównawczej generatorów liczb losowych, która oparta jest na testach statystycznych NIST, i która pozwala wybrać generatory o najlepszych cechach statystycznych. Przedstawiono także metody poprawy właściwości statystycznych sekwencji binarnych, które powstają na podstawie różnych generatorów chaotycznych

Divergent Predictive States: The Statistical Complexity Dimension of Stationary, Ergodic Hidden Markov Processes

Even simply-defined, finite-state generators produce stochastic processes that require tracking an uncountable infinity of probabilistic features for optimal prediction. For processes generated by hidden Markov chains the consequences are dramatic. Their predictive models are generically infinite-state. And, until recently, one could determine neither their intrinsic randomness nor structural complexity. The prequel, though, introduced methods to accurately calculate the Shannon entropy rate (randomness) and to constructively determine their minimal (though, infinite) set of predictive features. Leveraging this, we address the complementary challenge of determining how structured hidden Markov processes are by calculating their statistical complexity dimension -- the information dimension of the minimal set of predictive features. This tracks the divergence rate of the minimal memory resources required to optimally predict a broad class of truly complex processes.Comment: 16 pages, 6 figures; Supplementary Material, 6 pages, 2 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/icfshmp.ht

From Chaos to Pseudorandomness: A Case Study on the 2-D Coupled Map Lattice

Applying the chaos theory for secure digital communications is promising and it is well acknowledged that in such applications the underlying chaotic systems should be carefully chosen. However, the requirements imposed on the chaotic systems are usually heuristic, without theoretic guarantee for the resultant communication scheme. Among all the primitives for secure communications, it is well accepted that (pseudo) random numbers are most essential. Taking the well-studied 2-D coupled map lattice (2D CML) as an example, this article performs a theoretical study toward pseudorandom number generation with the 2D CML. In so doing, an analytical expression of the Lyapunov exponent (LE) spectrum of the 2D CML is first derived. Using the LEs, one can configure system parameters to ensure the 2D CML only exhibits complex dynamic behavior, and then collect pseudorandom numbers from the system orbits. Moreover, based on the observation that least significant bit distributes more evenly in the (pseudo) random distribution, an extraction algorithm E is developed with the property that when applied to the orbits of the 2D CML, it can squeeze uniform bits. In implementation, if fixed-point arithmetic is used in binary format with a precision of z bits after the radix point, E can ensure that the deviation of the squeezed bits is bounded by 2(-z) . Further simulation results demonstrate that the new method not only guides the 2D CML model to exhibit complex dynamic behavior but also generates uniformly distributed independent bits with good efficiency. In particular, the squeezed pseudorandom bits can pass both NIST 800-22 and TestU01 test suites in various settings. This study thereby provides a theoretical basis for effectively applying the 2D CML to secure communications

Islands in the Gap: Intertwined Transport and Localization in Structurally Complex Materials

Localized waves in disordered one-dimensional materials have been studied for decades, including white-noise and correlated disorder, as well as quasi-periodic disorder. How these wave phenomena relate to those in crystalline (periodic ordered) materials---arguably the better understood setting---has been a mystery ever since Anderson discovered disorder-induced localization. Nonetheless, together these revolutionized materials science and technology and led to new physics far beyond the solid state. We introduce a broad family of structurally complex materials---chaotic crystals---that interpolate between these organizational extremes---systematically spanning periodic structures and random disorder. Within the family one can tune the degree of disorder to sweep through an intermediate structurally disordered region between two periodic lattices. This reveals new transport and localization phenomena reflected in a rich array of energy-dependent localization degree and density of states. In particular, strong localization is observed even with a very low degree of disorder. Moreover, markedly enhanced localization and delocalization coexist in a very narrow range of energies. Most notably, beyond the simply smoothed bands found in previous disorder studies, islands of transport emerge in band gaps and sharp band boundaries persist in the presence of substantial disorder. Finally, the family of materials comes with rather direct specifications of how to assemble the requisite material organizations.Comment: 7 pages, 3 figures, supplementary material; http://csc.ucdavis.edu/~cmg/compmech/pubs/talisdm.ht