Entropy in Dynamic Systems

In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed

Fast, parallel and secure cryptography algorithm using Lorenz's attractor

A novel cryptography method based on the Lorenz's attractor chaotic system is presented. The proposed algorithm is secure and fast, making it practical for general use. We introduce the chaotic operation mode, which provides an interaction among the password, message and a chaotic system. It ensures that the algorithm yields a secure codification, even if the nature of the chaotic system is known. The algorithm has been implemented in two versions: one sequential and slow and the other, parallel and fast. Our algorithm assures the integrity of the ciphertext (we know if it has been altered, which is not assured by traditional algorithms) and consequently its authenticity. Numerical experiments are presented, discussed and show the behavior of the method in terms of security and performance. The fast version of the algorithm has a performance comparable to AES, a popular cryptography program used commercially nowadays, but it is more secure, which makes it immediately suitable for general purpose cryptography applications. An internet page has been set up, which enables the readers to test the algorithm and also to try to break into the cipher in

Using a Micro-Integrator to Eliminate the Numerical Butterfly Effect in Non-Linear Chaotic Partial Differential Equations

Chaos theory is a relatively new scientific paradigm for the analysis, simulation and prediction of non-linear phenomena whose initial conditions determine the behavior of their entire time series representation. It finds many applications in mathematics, science, and engineering. These include, but are not limited, to data encryption and decryption, designing secure communication systems, predicting weather patterns, noise fluctuations on data lines, understanding turbulence in fluid flow, and analyzing quantum wells. Systems that exhibit chaos are called chaotic systems. In computing solutions to nonlinear chaotic partial differential equation sets, slight deviations in step size could lead to completely diverging trajectories as the systems time series progresses. This is called the numerical butterfly effect. Smaller step sizes produce arrays closer to the desired continuous time solution, but they require more sampling points and as a result more memory. The Micro-Integrator produces results with a high level of accuracy while using only a fraction of the amount of memory required by conventional numerical integration methods. The reduction in memory requirements by the Micro-Integrator was quantified by introducing a performance factor \u27η\u27 that was mathematically equal to the ratio of the amount of memory required for computing without the Micro-Integrator to that required for computing with it. Recorded values of the performance factor from the tests ranged from 5 to 4 10 , out of which 75% were above 3 10 . The performance factor was also found to depend on the type of chaotic system, the numerical method, and the time window for computation. Less computationally efficient numerical methods resulted in higher performance factors than the more efficient ones

From continuous-time chaotic systems to pseudo random number generators: Analysis and generalized methodology

The use of chaotic systems in electronics, such as Pseudo-Random Number Generators (PRNGs), is very appealing. Among them, continuous-time ones are used less because, in addition to having strong temporal correlations, they require further computations to obtain the discrete solutions. Here, the time step and discretization method selection are first studied by conducting a detailed analysis of their effect on the systems’ statistical and chaotic behavior. We employ an approach based on interpreting the time step as a parameter of the new “maps”. From our analysis, it follows that to use them as PRNGs, two actions should be achieved (i) to keep the chaotic oscillation and (ii) to destroy the inner and temporal correlations. We then propose a simple methodology to achieve chaos-based PRNGs with good statistical characteristics and high throughput, which can be applied to any continuous-time chaotic system. We analyze the generated sequences by means of quantifiers based on information theory (permutation entropy, permutation complexity, and causal entropy × complexity plane). We show that the proposed PRNG generates sequences that successfully pass Marsaglia Diehard and NIST (National Institute of Standards and Technology) tests. Finally, we show that its hardware implementation requires very few resources.Fil: de Micco, Luciana. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones Científicas y Tecnológicas en Electrónica. Universidad Nacional de Mar del Plata. Facultad de Ingeniería. Instituto de Investigaciones Científicas y Tecnológicas en Electrónica; ArgentinaFil: Antonelli, Maximiliano. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones Científicas y Tecnológicas en Electrónica. Universidad Nacional de Mar del Plata. Facultad de Ingeniería. Instituto de Investigaciones Científicas y Tecnológicas en Electrónica; ArgentinaFil: Rosso, Osvaldo Anibal. Universidade Federal de Alagoas; Brasi

A universal variable extension method for designing multi-scroll/wing chaotic systems

© 2023 IEEE. This is the accepted manuscript version of an article which has been published in final form at https://doi.org/10.1109/TIE.2023.3299020Developing a universal design method to construct different multiscroll/wing chaotic systems (MS/WCSs) has been challenging. This article proposes a general design method for MS $/$ WCSs called the universal variable extension method (UVEM). It is a simple but effective approach that generates one-direction (1-D) and 2-D multiscroll/wing chaotic attractors. Using any double-scroll/wing chaotic system as the basic system, the UVEM is able to construct different MS/WCSs. Employing Chua's chaotic system and Lorenz chaotic system as two examples, we construct two MSCSs (including 1-D and 2-D) and two MWCSs (including 1-D and 2-D), respectively. Theoretical analysis and numerical simulation show that the constructed MS/WCSs not only can generate 1-D and 2-D multiscroll/wing chaotic attractors but also have 1-D and 2-D initial boosting behaviors. This means that the MS/WCSs designed by the UVEM are very sensitive to their initial states, and have better unpredictability and more complex chaotic behaviors. To show the simplicity of UVEM in hardware implementation, we develop a field-programmable gate array-based digital hardware platform to implement the designed MS $/$ WCSs. Finally, a new pseudorandom number generator is proposed to investigate the application of the MS/WCSs. All P-values obtained by the NIST SP800-22 test are larger than 0.01, which indicates that the MS/WCSs designed by UVEM have high randomness.Peer reviewe

Discrete one-dimensional piecewise chaotic systems without fixed points

Acknowledgements The authors are thankful to the anonymous reviewers for their constructive feedback.Peer reviewe

Kriptolojik uygulamalar için FPGA tabanlı yeni kaotik osilatörlerin ve gerçek rasgele sayı üreteçlerinin tasarımı ve gerçeklenmesi

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.Bu tez çalışmasında, gerçek zamanlı, yüksek çalışma frekansı ve bit üretim hızına sahip Gerçek Rasgele Sayı Üreteçleri (GRSÜ), FPGA tabanlı kaotik osilatörler kullanılarak tasarlanmış ve gerçekleştirilmiştir. Bu amaçla tezin ilk aşamasında, çeşitli sistem parametrelerinin karşılaştırılması ve değerlendirilmesi amacıyla iki farklı kaotik sistem dört farklı nümerik diferansiyel denklem çözüm metodu ile modellenerek sistemlerin dinamik davranışları incelenmiş ve kaos analizleri yapılmıştır. İkinci aşamada, seçilen kaotik sistemler bir ECAD programında şematik giriş yapılarak analog devre elemanları ile modellenmiştir. Nümerik benzetim sonuçları ile ECAD benzetim sonuçları karşılaştırılmıştır. Elde edilen sonuçlara göre analog elemanlar kullanılarak yapılan ECAD benzetimi ile Matlab destekli nümerik model sonuçları birbiri ile uyumlu değerler üretmiştir. Sonraki aşamada, kaotik sistemler dört farklı diferansiyel denklem çözüm metotlarından yararlanılarak, 32-bit IEEE 754-1985 kayan noktalı sayı standardında VHDL programlama dili ile FPGA üzerinde modellenmiştir. Tasarımlar Virtex–6 ailesi XC6VLX550T-2FF1759 çipi için Xilinx ISE Design Tools 14.2 benzetim programı kullanılarak sentezlenmiştir. Elde edilen sonuçlara göre FPGA-tabanlı kaotik osilatörlerin maksimum çalışma frekansları yaklaşık olarak 390-464 MHz arasında değişmektedir. Buna göre kaotik osilatör ünitesi 1 milyon veriyi 46 ms gibi çok kısa bir sürede hesaplayabilmektedir. Bu aşamada, FPGA tabanlı ünitelerin ürettiği sonuçların doğruluğunu test etmek amacıyla RMSE yöntemi kullanılarak hassasiyet analizleri de yapılmıştır. Dördüncü aşamada, FPGA-tabanlı örnek kaotik sistemler kullanılarak GRSÜ tasarımı gerçekleştirilmiştir. Genel olarak iki farklı kaotik sistem, kaotik osilatör tasarımında dört ayrı algoritma ve kuantalama için üç değişik yöntem sunularak toplamda 24 farklı gerçek rasgele sayı üreteci ünitesi tasarlanmıştır. Tasarımlardan elde edilen sonuçlara göre, ünitelerin maksimum çalışma frekansları 339-401 MHz ve bit üretim hızları 53-132 Mbit/s arasında değişmektedir. Son aşamada, FPGA tabanlı GRSÜ'den elde edilen sayı dizileri FIPS-140-1 ve NIST-800-22 istatistiksel rasgelelik testleri kullanılarak test edilmiş ve tüm testlerden başarılı olmuştur.In this thesis, real-time True Random Number Generators (TRNGs) with high operating frequency and bit generation rate have been designed and implemented using FPGA-based chaotic oscillators. In the first stage, two separate chaotic systems have been determined and their dynamical behavioral and chaotic analysis have been investigated to compare various system parameters using by four diverse numerical differential equation solution methods. In the second stage, the chaotic systems have been modelled using analog components in an ECAD program. After that numerical and ECAD simulation results have been compared and the results obtained from each simulation proves that both approaches have produced compatible outcomes. In the next stage, the chaotic systems have been modelled in VHDL in 32-bit IEEE 754-1985 floating point number standard using by four diverse numerical differential equation solution methods. The designs have been synthesized for Virtex–6 using Xilinx ISE Design Tools 14.2. According to the syntheses results, the maximum operating frequency of the FPGA-based chaotic oscillators varies between 390 MHz and 464 MHz. Accordingly, the chaotic oscillator unit has been able to calculate 1 million data sets in 46 ms. In this stage, in order to test accuracy of results produced by FPGA-based units, the sensitivity analysis have been also performed by employing RMSE method. In the fourth stage, TRNG designs have been implemented using FPGA-based chaotic systems. 24 different TRNG units have been designed and implemented by employing two distinct chaotic systems, four different algorithms in the design of the chaotic oscillators and three diverse quantification methods. According to the results, operating frequency of the units varies between 339 MHz and 401 MHz and the bit-rates varies between 53 Mbit/s and 132 Mbit/s

Yeni kaotik sistemler ile rasgele sayı üreteci tasarımı ve çoklu-ortam verilerinin yüksek güvenlikli şifrelenmesi

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.Bu tez çalışmasında, literatürdeki şifreleme algoritmalarından daha hızlı ve güvenli olan kaos tabanlı bir şifreleme algortimasının tasarımı ile çoklu ortam verilerinin şifrelenmesi amaçlanmıştır. Tezin ilk aşamasında, literatürde olmayan yeni kaotik sistemlerin tasarım ve analizleri için; öngörülen sistemlerin denge noktaları bulunmuş, zaman serileri ve faz portreleri elde edilmiş, lyapunov üstelleri hesaplatılmış, parametre değişimine göre lyapunov üstellleri spektrumları ve çatallaşma diyagramları çizdirilmiştir. Ardından tasarlanan yeni kaotik sistemlere ait elektronik devreler modellenerek, ORCAD-PSpice simülasyonları ve deneysel olarak devre uygulamaları gerçekleştirilmiştir. Yapılan tüm dinamik analizler, devre simülasyonları, devre gerçeklemelerine ait çıkışlar ve karşılaştırmalar ile sistemlerin kaotik yapıları ispatlanmış, daha iyi anlaşılmış ve denklemlere son halleri verilmiştir. İkinci aşamada, öncelikle sürekli zamanlı kaotik sistemler Runge Kutta-4 yöntemi ile ayrıklaştırılmış ve elde edilen sayılar ikili sayı formatına çevrilerek yeni ve özgün RSÜ tasarımları yapılmıştır. Tasarlanan RSÜ'ler uluslararası en üst standart olan NIST-800-22 ve FIPS-140-1 istatistiksel testlerinden başarıyla geçirilerek, yeni bir kaos tabanlı şifreleme algoritmasının geliştirilmesinde temel alınmıştır. Üçüncü aşamada; yeni kaotik RSÜ tabanlı özgün bir şifreleme algoritması geliştirilerek sinyal, metin, ses, resim ve video gibi farklı çoklu-ortam verileri ayrı ayrı şifrelenmiştir. Şifrelenen multimedya verilerinin korelasyon, histogram gibi güvenlik analizleri yapılarak başarımları ölçülmüştür. Son aşamada ise; AVR Studio 5.1 programı ile yeni kaotik RSÜ tabanlı özgün şifreleme yöntemi ile güncel literatürdeki bazı şifreleme yöntemleri, bellek ve hız bakımından karşılaştırılarak, gerçek ortam uygulamaları için performans değerlendirmeleri sunulmuştur. Sonuç olarak; geliştirilen kaos tabanlı şifreleme yönteminin yeni ve özgün özellikleri şunlardır: dinamik yapısı karmaşık ve rasgeleliği yüksek yeni kaotik sistemler içermektedir, NIST-800-22 ve FIPS-140-1 rasgelelik testleri ile daha üstün rasgeleliğe sahip olan özgün RSÜ tabanlı bir yapıdadır, daha kolay ve hızlı işlenebilen bir algoritma yapısına sahip olduğundan diğer şifreleme algoritmalarına (AES, Skipjack, RC5, vb.) göre genel olarak daha hızlı, bellek olarak çok daha az yer kaplamakta ve en önemlisi de tüm çoklu ortam verilerini (ses, görüntü, video, metin, vb.) daha güvenli olarak şifrelemektedir.In this thesis, the encryption of multimedia data with the design of a new chaos based encryption algorithm, which is much more secure and faster than the encryption algorithm in the literature, is aimed. At the first stage of the thesis, to design and analyze the novel chaotic system, the equilibrium points are found, time series and phase portraits are acquired, lypapunov exponents of the systems are calculated, the spectrums of lypanov exponents with respect to the system parameters and bifurcation diagram of the systems are plotted. Then, the electronic circuit model of the designed chaotic systems are simulated in ORCAD-PSpice and the circuits are realized at the laboratory. At the second stage, the continuous time chaotic systems are discretized with Runge Kutta-4 numerical algorithm. The new novel RNGs are designed by converting the numbers obtained after the discretization process into binary. The RNGs pass the NIST-800-22 and FIPS-140-1 statistical tests, which are the highest international standards, successfully and then these RNGs are used as a base for developing a new chaos based encryption algorithm. At the third stage; by developing a new novel chaotic RNG based encryption algorithm, multimedia data like signal, text, audio, image and video is encrypted. Security analyses like correlation and histogram analysis, of the encrypted data are made to evaluate the performance of the encryption. At the last stage, the performance analyses of the new chaotic RNG based encryption algorithm and some encryption algorithms in the recent literature with are made AVR Studio 5.1 program and comparison with respect to speed and memory is done to present the evaluation of the performance of the new encryption algorithm for real time application. In conclusion; the novelty of the developed chaos based encryption method is: it consist chaotic systems with complex dynamic structure and high randomness, The designed original RNGs are more random structure with successfull NIST-800-22 and FIPS-140-1 statistical tests, due to the algorithm has a structure that is easier and faster to process, the new chaos based algorithm is faster and uses less memory than other encryption algorithm (AES, Skipjack, RC5) and more importantly for all type of the multimedia data(audio, image, video, text) the new chaos based encryption algorithm provides more reliable encryption than other algorithms

Security and robustness of a modified parameter modulation communication scheme

Please read the abstract in the section front of this documentThesis (PhD)--University of Pretoria, 2009.Electrical, Electronic and Computer EngineeringUnrestricte

Entropy in Image Analysis III

Image analysis can be applied to rich and assorted scenarios; therefore, the aim of this recent research field is not only to mimic the human vision system. Image analysis is the main methods that computers are using today, and there is body of knowledge that they will be able to manage in a totally unsupervised manner in future, thanks to their artificial intelligence. The articles published in the book clearly show such a future