Analysis of nonlinear sequences and streamciphers

Streamciphers are common cryptographic algorithms used to protect the confidentiality of frame-based communications like mobile phone conversations and Internet traffic. Streamciphers are ideal cryptographic algorithms to encrypt these types of traffic as they have the potential to encrypt them quickly and securely, and have low error propagation. The main objective of this thesis is to determine whether structural features of keystream generators affect the security provided by stream ciphers.These structural features pertain to the state-update and output functions used in keystream generators. Using linear sequences as keystream to encrypt messages is known to be insecure. Modern keystream generators use nonlinear sequences as keystream.The nonlinearity can be introduced through a keystream generator's state-update function, output function, or both. The first contribution of this thesis relates to nonlinear sequences produced by the well-known Trivium stream cipher. Trivium is one of the stream ciphers selected in a final portfolio resulting from a multi-year project in Europe called the ecrypt project. Trivium's structural simplicity makes it a popular cipher to cryptanalyse, but to date, there are no attacks in the public literature which are faster than exhaustive keysearch. Algebraic analyses are performed on the Trivium stream cipher, which uses a nonlinear state-update and linear output function to produce keystream. Two algebraic investigations are performed: an examination of the sliding property in the initialisation process and algebraic analyses of Trivium-like streamciphers using a combination of the algebraic techniques previously applied separately by Berbain et al. and Raddum. For certain iterations of Trivium's state-update function, we examine the sets of slid pairs, looking particularly to form chains of slid pairs. No chains exist for a small number of iterations.This has implications for the period of keystreams produced by Trivium. Secondly, using our combination of the methods of Berbain et al. and Raddum, we analysed Trivium-like ciphers and improved on previous on previous analysis with regards to forming systems of equations on these ciphers. Using these new systems of equations, we were able to successfully recover the initial state of Bivium-A.The attack complexity for Bivium-B and Trivium were, however, worse than exhaustive keysearch. We also show that the selection of stages which are used as input to the output function and the size of registers which are used in the construction of the system of equations affect the success of the attack. The second contribution of this thesis is the examination of state convergence. State convergence is an undesirable characteristic in keystream generators for stream ciphers, as it implies that the effective session key size of the stream cipher is smaller than the designers intended. We identify methods which can be used to detect state convergence. As a case study, theMixer streamcipher, which uses nonlinear state-update and output functions to produce keystream, is analysed. Mixer is found to suffer from state convergence as the state-update function used in its initialisation process is not one-to-one. A discussion of several other streamciphers which are known to suffer from state convergence is given. From our analysis of these stream ciphers, three mechanisms which can cause state convergence are identified.The effect state convergence can have on stream cipher cryptanalysis is examined. We show that state convergence can have a positive effect if the goal of the attacker is to recover the initial state of the keystream generator. The third contribution of this thesis is the examination of the distributions of bit patterns in the sequences produced by nonlinear filter generators (NLFGs) and linearly filtered nonlinear feedback shift registers. We show that the selection of stages used as input to a keystream generator's output function can affect the distribution of bit patterns in sequences produced by these keystreamgenerators, and that the effect differs for nonlinear filter generators and linearly filtered nonlinear feedback shift registers. In the case of NLFGs, the keystream sequences produced when the output functions take inputs from consecutive register stages are less uniform than sequences produced by NLFGs whose output functions take inputs from unevenly spaced register stages. The opposite is true for keystream sequences produced by linearly filtered nonlinear feedback shift registers.

A Lagrangian approach to nonlinear modeling of anti-roll tanks

This paper presents two novel nonlinear models of u-shaped anti-roll tanks for ships, and their linearizations. In addition, a third simplified nonlinear model is presented. The models are derived using Lagrangian mechanics. This formulation not only simplifies the modeling process, but also allows one to obtain models that satisfy energy-related physical properties. The proposed nonlinear models and their linearizations are validated using model-scale experimental data. Unlike other models in the literature, the nonlinear models in this paper are valid for large roll amplitudes. Even at moderate roll angles, the nonlinear models have three orders of magnitude lower mean square error relative to experimental data than the linear models.

A Crank--Nicolson ADI spectral method for a two-dimensional riesz space fractional nonlinear reaction-diffusion equation

In this paper, a new alternating direction implicit Galerkin--Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank--Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order $2$ in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh--Nagumo model. Numerical results are provided to verify the theoretical analysis.

Study on tracheal collapsibility, compliance, and stress by considering nonlinear mechanical property of cartilage

Tracheal cartilage has been widely regarded as a linear elastic material either in experimental studies or in analytic and numerical models. However, it has been recently demonstrated that, like other fiber-oriented biological tissues, tracheal cartilage is a nonlinear material, which displays higher strength in compression than in extension. Considering the nonlinearity requires a more complex theoretical frame work and costs more to simulate. This study aims to quantify the deviation due to the simplified treatment of the tracheal cartilage as a linear material. It also evaluates the improved accuracy gained by considering the nonlinearity. Pig tracheal rings were used to exam the mechanical properties of cartilage and muscular membrane. By taking into account the asymmetric shape of tracheal cartilage, the collapse behavior of complete rings was simulated, and the compliance of airway and stress in the muscular membrane were discussed. The results obtained were compared with those assuming linear mechanical properties. The following results were found: (1) Models based on both types of material properties give a small difference in representing collapse behavior; (2) regarding compliance, the relative difference is big, ranging from 10 to 40% under negative pressure conditions; and (3) the difference in determining stress in the muscular membrane is small too: <5%. In conclusion, treating tracheal cartilage as a linear material will not cause big deviations in representing the collapse behavior, and mechanical stress in the muscular part, but it will induce a big deviation in predicting the compliance, particularly when the transmural pressure is lower than -0.5 kPa. The results obtained in this study may be useful in both understanding the collapse behavior of trachea and in evaluating the error induced by the simplification of treating the tracheal cartilage as a linear elastic material.

Nonlinear mixture autoregressive model: economic applications

The aim of this dissertation is to model economic variables by a mixture autoregressive (MAR) model. The MAR model is a generalization of linear autoregressive (AR) model. The MAR -model consists of K linear autoregressive components. At any given point of time one of these autoregressive components is randomly selected to generate a new observation for the time series. The mixture probability can be constant over time or a direct function of a some observable variable. Many economic time series contain properties which cannot be described by linear and stationary time series models. A nonlinear autoregressive model such as MAR model can a plausible alternative in the case of these time series. In this dissertation the MAR model is used to model stock market bubbles and a relationship between inflation and the interest rate. In the case of the inflation rate we arrived at the MAR model where inflation process is less mean reverting in the case of high inflation than in the case of normal inflation. The interest rate move one-for-one with expected inflation. We use the data from the Livingston survey as a proxy for inflation expectations. We have found that survey inflation expectations are not perfectly rational. According to our results information stickiness play an important role in the expectation formation. We also found that survey participants have a tendency to underestimate inflation. A MAR model has also used to model stock market bubbles and crashes. This model has two regimes: the bubble regime and the error correction regime. In the error correction regime price depends on a fundamental factor, the price-dividend ratio, and in the bubble regime, price is independent of fundamentals. In this model a stock market crash is usually caused by a regime switch from a bubble regime to an error-correction regime. According to our empirical results bubbles are related to a low inflation. Our model also imply that bubbles have influences investment return distribution in both short and long run.

A single network adaptive critic (SNAC) architecture for optimal control synthesis for a class of nonlinear systems

Even though dynamic programming offers an optimal control solution in a state feedback form, the method is overwhelmed by computational and storage requirements. Approximate dynamic programming implemented with an Adaptive Critic (AC) neural network structure has evolved as a powerful alternative technique that obviates the need for excessive computations and storage requirements in solving optimal control problems. In this paper, an improvement to the AC architecture, called the �Single Network Adaptive Critic (SNAC)� is presented. This approach is applicable to a wide class of nonlinear systems where the optimal control (stationary) equation can be explicitly expressed in terms of the state and costate variables. The selection of this terminology is guided by the fact that it eliminates the use of one neural network (namely the action network) that is part of a typical dual network AC setup. As a consequence, the SNAC architecture offers three potential advantages: a simpler architecture, lesser computational load and elimination of the approximation error associated with the eliminated network. In order to demonstrate these benefits and the control synthesis technique using SNAC, two problems have been solved with the AC and SNAC approaches and their computational performances are compared. One of these problems is a real-life Micro-Electro-Mechanical-system (MEMS) problem, which demonstrates that the SNAC technique is applicable to complex engineering systems.

{C}^0$ penalty methods for the fully nonlinear Monge-Ampère equation

In this paper, we develop and analyze C(0) penalty methods for the fully nonlinear Monge-Ampere equation det(D(2)u) = f in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the well-posedness of the penalty method as well as quasi-optimal error estimates using the Banach fixed-point theorem as our main tool. Numerical experiments are presented which support the theoretical results.

Active Vibration Suppression of One-dimensional Nonlinear Structures Using Optimal Dynamic Inversion

A flexible robot arm can be modeled as an Euler-Bernoulli beam which are infinite degrees of freedom (DOF) system. Proper control is needed to track the desired motion of a robotic arm. The infinite number of DOF of beams are reduced to finite number for controller implementation, which brings in error (due to their distributed nature). Therefore, to represent reality better distributed parameter systems (DPS) should be controlled using the systems partial differential equation (PDE) directly. In this paper, we propose to use a recently developed optimal dynamic inversion technique to design a controller to suppress nonlinear vibration of a beam. The method used in this paper determines control forces directly from the PDE model of the system. The formulation has better practical significance, because it leads to a closed form solution of the controller (hence avoids computational issues).

Estimation of heavy rainfall during cyclonic storms from microwave observations using nonlinear approach over Indian Ocean

In this study, an effort has been made to study heavy rainfall events during cyclonic storms over Indian Ocean. This estimate is based on microwave observations from tropical rainfall measuring mission (TRMM) Microwave Imager (TMI). Regional scattering index (SI) developed for Indian region based on measurements at 19-, 21- and 85-GHz brightness temperature and polarization corrected temperature (PCT) at 85 GHz have been utilized in this study. These PCT and SI are collocated against Precipitation Radar (PR) onboard TRMM to establish a relationship between rainfall rate, PCT and SI. The retrieval technique using both linear and nonlinear regressions has been developed utilizing SI, PCT and the combination of SI and PCT. The results have been compared with the observations from PR. It was found that a nonlinear algorithm using combination of SI and PCT is more accurate than linear algorithm or nonlinear algorithm using either SI or PCT. Statistical comparison with PR exhibits the correlation coefficients (CC) of 0.68, 0.66 and 0.70, and root mean square error (RMSE) of 1.78, 1.96 and 1.68 mm/h from the observations of SI, PCT and combination of SI and PCT respectively using linear regressions. When nonlinear regression is used, the CC of 0.73, 0.71, 0.79 and RMSE of 1.64, 1.95, 1.54 mm/h are observed from the observations of SI, PCT and combination of SI and PCT, respectively. The error statistics for high rain events (above 10 mm/h) shows the CC of 0.58, 0.59, 0.60 and RMSE of 5.07, 5.47, 5.03 mm/h from the observations of SI, PCT and combination of SI and PCT, respectively, using linear regression, and on the other hand, use of nonlinear regression yields the CC of 0.66, 0.64, 0.71 and RMSE of 4.68, 5.78 and 4.02 mm/h from the observations of SI, PCT and combined SI and PCT, respectively.

Signal Detection in Generalized Gaussian Noise by Nonlinear Wavelet Denoising

In this paper, a nonlinear suboptimal detector whose performance in heavy-tailed noise is significantly better than that of the matched filter is proposed. The detector consists of a nonlinear wavelet denoising filter to enhance the signal-to-noise ratio, followed by a replica correlator. Performance of the detector is investigated through an asymptotic theoretical analysis as well as Monte Carlo simulations. The proposed detector offers the following advantages over the optimal (in the Neyman-Pearson sense) detector: it is easier to implement, and it is more robust with respect to error in modeling the probability distribution of noise.

A Kushner-Stratonovich Monte Carlo filter applied to nonlinear dynamical system identification

A Monte Carlo filter, based on the idea of averaging over characteristics and fashioned after a particle-based time-discretized approximation to the Kushner-Stratonovich (KS) nonlinear filtering equation, is proposed. A key aspect of the new filter is the gain-like additive update, designed to approximate the innovation integral in the KS equation and implemented through an annealing-type iterative procedure, which is aimed at rendering the innovation (observation prediction mismatch) for a given time-step to a zero-mean Brownian increment corresponding to the measurement noise. This may be contrasted with the weight-based multiplicative updates in most particle filters that are known to precipitate the numerical problem of weight collapse within a finite-ensemble setting. A study to estimate the a-priori error bounds in the proposed scheme is undertaken. The numerical evidence, presently gathered from the assessed performance of the proposed and a few other competing filters on a class of nonlinear dynamic system identification and target tracking problems, is suggestive of the remarkably improved convergence and accuracy of the new filter. (C) 2013 Elsevier B.V. All rights reserved.

A Matroidal Framework for Network-Error Correcting Codes

Matroidal networks were introduced by Dougherty et al. and have been well studied in the recent past. It was shown that a network has a scalar linear network coding solution if and only if it is matroidal associated with a representable matroid. A particularly interesting feature of this development is the ability to construct (scalar and vector) linearly solvable networks using certain classes of matroids. Furthermore, it was shown through the connection between network coding and matroid theory that linear network coding is not always sufficient for general network coding scenarios. The current work attempts to establish a connection between matroid theory and network-error correcting and detecting codes. In a similar vein to the theory connecting matroids and network coding, we abstract the essential aspects of linear network-error detecting codes to arrive at the definition of a matroidal error detecting network (and similarly, a matroidal error correcting network abstracting from network-error correcting codes). An acyclic network (with arbitrary sink demands) is then shown to possess a scalar linear error detecting (correcting) network code if and only if it is a matroidal error detecting (correcting) network associated with a representable matroid. Therefore, constructing such network-error correcting and detecting codes implies the construction of certain representable matroids that satisfy some special conditions, and vice versa. We then present algorithms that enable the construction of matroidal error detecting and correcting networks with a specified capability of network-error correction. Using these construction algorithms, a large class of hitherto unknown scalar linearly solvable networks with multisource, multicast, and multiple-unicast network-error correcting codes is made available for theoretical use and practical implementation, with parameters, such as number of information symbols, number of sinks, number of coding nodes, error correcting capability, and so on, being arbitrary but for computing power (for the execution of the algorithms). The complexity of the construction of these networks is shown to be comparable with the complexity of existing algorithms that design multicast scalar linear network-error correcting codes. Finally, we also show that linear network coding is not sufficient for the general network-error correction (detection) problem with arbitrary demands. In particular, for the same number of network errors, we show a network for which there is a nonlinear network-error detecting code satisfying the demands at the sinks, whereas there are no linear network-error detecting codes that do the same.

Nonlinear Differential Games-Based Impact-Angle-Constrained Guidance Law

The problem of intercepting a maneuvering target at a prespecified impact angle is posed in nonlinear zero-sum differential games framework. A feedback form solution is proposed by extending state-dependent Riccati equation method to nonlinear zero-sum differential games. An analytic solution is obtained for the state-dependent Riccati equation corresponding to the impact-angle-constrained guidance problem. The impact-angle-constrained guidance law is derived using the states line-of-sight rate and projected terminal impact angle error. Local asymptotic stability conditions for the closed-loop system corresponding to these states are studied. Time-to-go estimation is not explicitly required to derive and implement the proposed guidance law. Performance of the proposed guidance law is validated using two-dimensional simulation of the relative nonlinear kinematics as well as a thrust-driven realistic interceptor model.

Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems

This paper deals with the adaptive mesh generation for singularly perturbed nonlinear parameterized problems with a comparative research study on them. We propose an a posteriori error estimate for singularly perturbed parameterized problems by moving mesh methods with fixed number of mesh points. The well known a priori meshes are compared with the proposed one. The comparison results show that the proposed numerical method is highly effective for the generation of layer adapted a posteriori meshes. A numerical experiment of the error behavior on different meshes is carried out to highlight the comparison of the approximated solutions. (C) 2015 Elsevier B.V. All rights reserved.

Weakly corrected numerical solutions to stochastically driven nonlinear dynamical systems

A method to weakly correct the solutions of stochastically driven nonlinear dynamical systems, herein numerically approximated through the Eule-Maruyama (EM) time-marching map, is proposed. An essential feature of the method is a change of measures that aims at rendering the EM-approximated solution measurable with respect to the filtration generated by an appropriately defined error process. Using Ito's formula and adopting a Monte Carlo (MC) setup, it is shown that the correction term may be additively applied to the realizations of the numerically integrated trajectories. Numerical evidence, presently gathered via applications of the proposed method to a few nonlinear mechanical oscillators and a semi-discrete form of a 1-D Burger's equation, lends credence to the remarkably improved numerical accuracy of the corrected solutions even with relatively large time step sizes. (C) 2015 Elsevier Inc. All rights reserved.