Symmetry in Chaotic Systems and Circuits

Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled “Symmetry in Chaotic Systems and Circuits”, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue

OPTOELECTRONIC EXPERIMENTS ON RANDOM BIT GENERATORS AND COUPLED DYNAMICAL SYSTEMS

Optoelectronic systems have many important applications, and they have become ubiquitous in the contexts of communications and sensing. In recent years, optical and optoelectronic systems have been of interest for two newer purposes: generators of random bits and experimental dynamical systems used to understand chaos theory and synchronization. Random bit generators are needed for secure communication, encryption, and Monte Carlo simulations. Algorithm-based pseudorandom number generators are susceptible to being hacked or producing incorrect numerical results in simulations, so physical noise-based sources of random numbers are needed. We have constructed a random bit generator based on amplied spontaneous emission (ASE), with generation rates of 12.5 Gbit/sec [1]. We develop an understanding of the mechanism behind generating random bits from ASE, and we demonstrate its suitability as a random number generator by standard statistical testing used to evaluate the random bits. This is the first use of ASE as a physical random number generator (RNG). Coupled dynamical systems are present in numerous contexts in the natural and man-made world. From neurons in the brain to coupled lasers to pedestrians on a bridge, it is important to understand how coupled dynamical systems or oscillators can synchronize in dierent ways. While many studies of coupled dynamical systems are conducted analytically and numerically, experimental studies are crucial for understanding how systems with real noise and features, which may not be accounted for in the models, actually synchronize. Experimental dynamical systems can display phenomena not previously studied or expected, guiding the development of more sophisticated models and the direction of analytical and numerical work, and experiments offer means for quickly exploring parameter space. Sorrentino and Ott first proposed a theoretical formulation that described a counterintuitive phenomenon they referred to as group synchrony [2]. We show an experimental realization of group synchrony, in which the oscillators are grouped based on different parameters for each group [3]. Despite being coupled only to the oscillators in the dissimilar group, oscillators in the same group identically synchronize, through the mediation provided by the other group. Unidirectional rings of oscillators have been studied in order to understand synchronization between coupled neurons, which can contribute to functions such as locomotion [4, 5]. We show an experimental realization of a uni-directional ring coupling conguration, with tunable coupling delays [6]. By changing the coupling delays, we show that it is possible to obtain dierent synchronization states. We compare experimental results to numerical simulations and calculations of the stability of the synchronous states. We present an experiment of four delay-coupled optoelectronic oscillators as the first experimental observations of both of these novel synchronization phenomena in simple networks of coupled oscillators

Multiphysics modelling and experimental validation of microelectromechanical resonator dynamics

The modelling of microelectromechanical systems provides a very challenging task in microsystems engineering. This field of research is inherently multiphysics of nature, since different physical phenomena are tightly intertwined at microscale. Typically, up to four different physical domains are usually considered in the analysis of microsystems: mechanical, electrical, thermal and fluidic. For each of these separate domains, well-established modelling and analysis techniques are available. However, one of the main challenges in the field of microsystems engineering is to connect models for the behavior of the device in each of these domains to equivalent lumped or reduced-order models without making unacceptably inaccurate assumptions and simplifications and to couple these domains correctly and efficiently. Such a so-called multiphysics modelling framework is very important for simulation of microdevices, since fast and accurate computational prototyping may greatly shorten the design cycle and thus the time-to-market of new products. This research will focus on a specific class of microsystems: microelectromechanical resonators. MEMS resonators provide a promising alternative for quartz crystals in time reference oscillators, due to their small size and on-chip integrability. However, because of their small size, they have to be driven into nonlinear regimes in order to store enough energy for obtaining an acceptable signal-to-noise ratio in the oscillator. Since these resonators are to be used as a frequency reference in the oscillator circuits, their steady-state (nonlinear) dynamic vibration behaviour is of special interest. A heuristic modelling approach is investigated for two different MEMS resonators, a clamped-clamped beam resonator and a dog-bone resonator. For the clamped-clamped beam resonator, the simulations with the proposed model shows a good agreement with experimental results, but the model is limited in its predictive capabilities. For the dogbone resonator, the proposed heuristic modelling approach does not lead to a match between simulations and experiments. Shortcomings of the heuristic modelling approach serve as a motivation for a first-principles based approach. The main objective of this research is to derive a multiphysics modelling framework for MEMS resonators that is based on first-principles formulations. The framework is intended for fast and accurate simulation of the steady-state nonlinear dynamic behaviour of MEMS resonators. Moreover, the proposed approach is validated by means of experiments. Although the multiphysics modelling framework is proposed for MEMS resonators, it is not restricted to this application field within microsystems engineering. Other fields, such as (resonant) sensors, switches and variable capacitors, allow for a similar modelling approach. In the proposed framework, themechanical, electrical and thermal domains are included. Since the resonators considered are operated in vacuum, the fluidic domain (squeeze film damping) is not included. Starting from a first-principles description, founded on partial differential equations (PDEs), characteristic nonlinear effects from each of the included domains are incorporated. Both flexural and bulk resonators can be considered. Next, Galerkin discretization of the coupled PDEs takes place, to construct reduced-order models while retaining the nonlinear effects. The multiphysics model consists of the combined reduced-order models from the different domains. Designated numerical tools are used to solve for the steady-state nonlinear dynamic behaviour of the combined model. The proposed semi-analytical (i.e. analytical-numerical) multiphysics modeling framework is illustrated for a full case study of an electrostatically actuated single-crystal silicon clamped-clamped beam MEMS resonator. By means of the modelling framework, multiphysics models of varying complexity have been derived for this resonator, including effects like electrostatic actuation, fringing fields, shear deformation, rotary inertia, thermoelastic damping and nonlinear material behaviour. The first-principles based approach allows for addressing the relevance of individual effects in a straightforward way, such that the models can be used as a (pre-)design tool for dynamic response analysis. The method can be considered complementary to conventional finite element simulations. The multiphysics model for the clamped-clamped beam resonator is validated by means of experiments. A good match between the simulations and experiments is obtained, thereby giving confidence in the proposed modelling framework. Finally, next to themodelling approach for MEMS resonators, a technique for using these nonlinear resonators in an oscillator circuit setting is presented. This approach, called phase feedback, allows for operation of the resonator in its nonlinear regime. The closedloop technique enables control of both the frequency of oscillation and the output power of the signal. Additionally, optimal operation points for oscillator circuits incorporating a nonlinear resonator can be defined

Digital Design of New Chaotic Ciphers for Ethernet Traffic

Durante los últimos años, ha habido un gran desarrollo en el campo de la criptografía, y muchos algoritmos de encriptado así como otras funciones criptográficas han sido propuestos.Sin embargo, a pesar de este desarrollo, hoy en día todavía existe un gran interés en crear nuevas primitivas criptográficas o mejorar las ya existentes. Algunas de las razones son las siguientes:• Primero, debido el desarrollo de las tecnologías de la comunicación, la cantidad de información que se transmite está constantemente incrementándose. En este contexto, existen numerosas aplicaciones que requieren encriptar una gran cantidad de datos en tiempo real o en un intervalo de tiempo muy reducido. Un ejemplo de ello puede ser el encriptado de videos de alta resolución en tiempo real. Desafortunadamente, la mayoría de los algoritmos de encriptado usados hoy en día no son capaces de encriptar una gran cantidad de datos a alta velocidad mientras mantienen altos estándares de seguridad.• Debido al gran aumento de la potencia de cálculo de los ordenadores, muchos algoritmos que tradicionalmente se consideraban seguros, actualmente pueden ser atacados por métodos de “fuerza bruta” en una cantidad de tiempo razonable. Por ejemplo, cuando el algoritmo de encriptado DES (Data Encryption Standard) fue lanzado por primera vez, el tamaño de la clave era sólo de 56 bits mientras que, hoy en día, el NIST (National Institute of Standards and Technology) recomienda que los algoritmos de encriptado simétricos tengan una clave de, al menos, 112 bits. Por otro lado, actualmente se está investigando y logrando avances significativos en el campo de la computación cuántica y se espera que, en el futuro, se desarrollen ordenadores cuánticos a gran escala. De ser así, se ha demostrado que algunos algoritmos que se usan actualmente como el RSA (Rivest Shamir Adleman) podrían ser atacados con éxito.• Junto al desarrollo en el campo de la criptografía, también ha habido un gran desarrollo en el campo del criptoanálisis. Por tanto, se están encontrando nuevas vulnerabilidades y proponiendo nuevos ataques constantemente. Por consiguiente, es necesario buscar nuevos algoritmos que sean robustos frente a todos los ataques conocidos para sustituir a los algoritmos en los que se han encontrado vulnerabilidades. En este aspecto, cabe destacar que algunos algoritmos como el RSA y ElGamal están basados en la suposición de que algunos problemas como la factorización del producto de dos números primos o el cálculo de logaritmos discretos son difíciles de resolver. Sin embargo, no se ha descartado que, en el futuro, se puedan desarrollar algoritmos que resuelvan estos problemas de manera rápida (en tiempo polinomial).• Idealmente, las claves usadas para encriptar los datos deberían ser generadas de manera aleatoria para ser completamente impredecibles. Dado que las secuencias generadas por generadores pseudoaleatorios, PRNGs (Pseudo Random Number Generators) son predecibles, son potencialmente vulnerables al criptoanálisis. Por tanto, las claves suelen ser generadas usando generadores de números aleatorios verdaderos, TRNGs (True Random Number Generators). Desafortunadamente, los TRNGs normalmente generan los bits a menor velocidad que los PRNGs y, además, las secuencias generadas suelen tener peores propiedades estadísticas, lo que hace necesario que pasen por una etapa de post-procesado. El usar un TRNG de baja calidad para generar claves, puede comprometer la seguridad de todo el sistema de encriptado, como ya ha ocurrido en algunas ocasiones. Por tanto, el diseño de nuevos TRNGs con buenas propiedades estadísticas es un tema de gran interés.En resumen, es claro que existen numerosas líneas de investigación en el ámbito de la criptografía de gran importancia. Dado que el campo de la criptografía es muy amplio, esta tesis se ha centra en tres líneas de investigación: el diseño de nuevos TRNGs, el diseño de nuevos cifradores de flujo caóticos rápidos y seguros y, finalmente, la implementación de nuevos criptosistemas para comunicaciones ópticas Gigabit Ethernet a velocidades de 1 Gbps y 10 Gbps. Dichos criptosistemas han estado basados en los algoritmos caóticos propuestos, pero se han adaptado para poder realizar el encriptado en la capa física, manteniendo el formato de la codificación. De esta forma, se ha logrado que estos sistemas sean capaces no sólo de encriptar los datos sino que, además, un atacante no pueda saber si se está produciendo una comunicación o no. Los principales aspectos cubiertos en esta tesis son los siguientes:• Estudio del estado del arte, incluyendo los algoritmos de encriptado que se usan actualmente. En esta parte se analizan los principales problemas que presentan los algoritmos de encriptado standard actuales y qué soluciones han sido propuestas. Este estudio es necesario para poder diseñar nuevos algoritmos que resuelvan estos problemas.• Propuesta de nuevos TRNGs adecuados para la generación de claves. Se exploran dos diferentes posibilidades: el uso del ruido generado por un acelerómetro MEMS (Microelectromechanical Systems) y el ruido generado por DNOs (Digital Nonlinear Oscillators). Ambos casos se analizan en detalle realizando varios análisis estadísticos a secuencias obtenidas a distintas frecuencias de muestreo. También se propone y se implementa un algoritmo de post-procesado simple para mejorar la aleatoriedad de las secuencias generadas. Finalmente, se discute la posibilidad de usar estos TRNGs como generadores de claves. • Se proponen nuevos algoritmos de encriptado que son rápidos, seguros y que pueden implementarse usando una cantidad reducida de recursos. De entre todas las posibilidades, esta tesis se centra en los sistemas caóticos ya que, gracias a sus propiedades intrínsecas como la ergodicidad o su comportamiento similar al comportamiento aleatorio, pueden ser una buena alternativa a los sistemas de encriptado clásicos. Para superar los problemas que surgen cuando estos sistemas son digitalizados, se proponen y estudian diversas estrategias: usar un sistema de multi-encriptado, cambiar los parámetros de control de los sistemas caóticos y perturbar las órbitas caóticas.• Se implementan los algoritmos propuestos. Para ello, se usa una FPGA Virtex 7. Las distintas implementaciones son analizadas y comparadas, teniendo en cuenta diversos aspectos tales como el consumo de potencia, uso de área, velocidad de encriptado y nivel de seguridad obtenido. Uno de estos diseños, se elige para ser implementado en un ASIC (Application Specific Integrate Circuit) usando una tecnología de 0,18 um. En cualquier caso, las soluciones propuestas pueden ser también implementadas en otras plataformas y otras tecnologías.• Finalmente, los algoritmos propuestos se adaptan y aplican a comunicaciones ópticas Gigabit Ethernet. En particular, se implementan criptosistemas que realizan el encriptado al nivel de la capa física para velocidades de 1 Gbps y 10 Gbps. Para realizar el encriptado en la capa física, los algoritmos propuestos en las secciones anteriores se adaptan para que preserven el formato de la codificación, 8b/10b en el caso de 1 Gb Ethernet y 64b/10b en el caso de 10 Gb Ethernet. En ambos casos, los criptosistemas se implementan en una FPGA Virtex 7 y se diseña un set experimental, que incluye dos módulos SFP (Small Form-factor Pluggable) capaces de transmitir a una velocidad de hasta 10.3125 Gbps sobre una fibra multimodo de 850 nm. Con este set experimental, se comprueba que los sistemas de encriptado funcionan correctamente y de manera síncrona. Además, se comprueba que el encriptado es bueno (pasa todos los test de seguridad) y que el patrón del tráfico de datos está oculto.<br /

Analog Photonics Computing for Information Processing, Inference and Optimisation

This review presents an overview of the current state-of-the-art in photonics computing, which leverages photons, photons coupled with matter, and optics-related technologies for effective and efficient computational purposes. It covers the history and development of photonics computing and modern analogue computing platforms and architectures, focusing on optimization tasks and neural network implementations. The authors examine special-purpose optimizers, mathematical descriptions of photonics optimizers, and their various interconnections. Disparate applications are discussed, including direct encoding, logistics, finance, phase retrieval, machine learning, neural networks, probabilistic graphical models, and image processing, among many others. The main directions of technological advancement and associated challenges in photonics computing are explored, along with an assessment of its efficiency. Finally, the paper discusses prospects and the field of optical quantum computing, providing insights into the potential applications of this technology.Comment: Invited submission by Journal of Advanced Quantum Technologies; accepted version 5/06/202

Nonlinear effects in low-dimensional systems: graphene membrane and electron transport in semiconductor superlattices

Mención Internacional en el título de doctorThis PhD dissertation deals with two different topics: Mechanics of graphene from a statistical mechanics approach, where internal interactions and effects due to temperature are considered. And electron dynamics and chaos in semiconductor superlattices, where we aim at enhancing the chaotic behavior, with its applicability to random number generation in mind. It is not our purpose to bridge these two different topics. But we do believe that with the rise of nanotechnology and the ever-increasing interdisciplinary of science, studies where different topics are approached and discussed are highly desirable. Nanotechnology already rules our life. However, it is still surprising how much progress has been achieved without a fully understanding of the physics governing these structures. In particular, out-of-equilibrium behavior and non-linear responses are present in every nanostructure, but, sometimes, it is possible to avoid their effects at large time scales or small interactions. However, the increasing demand of better and/or new performances makes them sometimes unavoidable, or even, desirable. Micro-metric samples of graphene or semiconductor superlattices cannot be studied taking into account every microscopic interaction, which makes it necessary to use mesoscopic models with a certain range of validity. Throughout this work, we have tried to improve our understanding of the topics stated above, using mesoscopic physical models and techniques from statistical mechanics and dynamical systems. We hope that the obtained results will help the scientific community to gain insight into these fascinating topics and will motivate new research in this direction.Spanish Ministerio de Economía y Competitividad (MINECO) Grant No. MTM2014-56948-C2-2-P and FIS2011-28838-C02-01 and Comunidad de Madrid Grant No. P2009/ENE-1597(HYSYCOMB)Programa Oficial de Doctorado en Ciencia e Ingeniería de MaterialesPresidente: Francisco Guinea López.- Secretario: Jesús Salas Martínez.- Vocal: Beatriz Olmos Sánche

Experiments on networks of coupled opto-electronic oscillators and physical random number generators

In this thesis, we report work in two areas: synchronization in networks of coupled oscillators and the evaluation of physical random number generators. A ``chimera state'' is a dynamical pattern that occurs in a network of coupled identical oscillators when the symmetry of the oscillator population is spontaneously broken into coherent and incoherent parts. We report a study of chimera states and cluster synchronization in two different opto-electronic experiments. The first experiment is a traditional network of four opto-electronic oscillators coupled by optical fibers. We show that the stability of the observed chimera state can be determined using the same group-theoretical techniques recently developed for the study of cluster synchrony. We present three novel results: (i) chimera states can be experimentally observed in small networks, (ii) chimera states can be stable, and (iii) at least some types of chimera states (those with identically synchronized coherent regions) are closely related to cluster synchronization. The second experiment uses a single opto-electronic feedback loop to investigate the dynamics of oscillators coupled in large complex networks with arbitrary topology. Recent work has demonstrated that an opto-electronic feedback loop can be used to realize ring networks of coupled oscillators. We significantly extend these capabilities and implement networks with arbitrary topologies by using field programmable gate arrays (FPGAs) to design appropriate digital filters and time delays. With this system, we study (i) chimeras in a five-node globally-coupled network, (ii) synchronization of clusters that are not predicted by network symmetries, and (iii) optimal networks for cluster synchronization. The field of random number generation is currently undergoing a fundamental shift from relying solely on pseudo-random algorithms to employing physical entropy sources. The standard evaluation practices, which were designed for pseudo-random number generators, are ill-suited to quantify the entropy that underlies physical random number generation. We review the state of the art in the evaluation of physical random number generation and recommend a new paradigm: quantifying entropy generation and understanding the physical limits for harvesting entropy from sources of randomness. As an illustration of our recommendations, we evaluate three common optical entropy sources: single photon time-of-arrival detection, chaotic lasers, and amplified spontaneous emission