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 Title
 ACCURATE HIGH ORDER COMPUTATION OF INVARIANT MANIFOLDS FOR LONG PERIODIC ORBITS OF MAPS AND EQUILIBRIUM STATES OF PDE.
 Creator
 Gonzalez, Jorge L., MirelesJames, Jason, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

The study of the long time behavior of nonlinear systems is not effortless, but it is very rewarding. The computation of invariant objects, in particular manifolds provide the scientist with the ability to make predictions at the frontiers of science. However, due to the presence of strong nonlinearities in many important applications, understanding the propagation of errors becomes necessary in order to quantify the reliability of these predictions, and to build sound foundations for future...
Show moreThe study of the long time behavior of nonlinear systems is not effortless, but it is very rewarding. The computation of invariant objects, in particular manifolds provide the scientist with the ability to make predictions at the frontiers of science. However, due to the presence of strong nonlinearities in many important applications, understanding the propagation of errors becomes necessary in order to quantify the reliability of these predictions, and to build sound foundations for future discoveries. This dissertation develops methods for the accurate computation of highorder polynomial approximations of stable/unstable manifolds attached to long periodic orbits in discrete time dynamical systems. For this purpose a multiple shooting scheme is applied to invariance equations for the manifolds obtained using the Parameterization Method developed by Xavier Cabre, Ernest Fontich and Rafael De La Llave in [CFdlL03a, CFdlL03b, CFdlL05].
Show less  Date Issued
 2020
 PURL
 http://purl.flvc.org/fau/fd/FA00013468
 Subject Headings
 Invariant manifolds, Nonlinear systems, Diffeomorphisms, Parabolic partial differential equations, Differential equations, Partial
 Format
 Document (PDF)
 Title
 THE CHANGE POINT PROBLEM FOR TWO CLASSES OF STOCHASTIC PROCESSES.
 Creator
 Ball, Cory, Long, Hongwei, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

The change point problem is a problem where a process changes regimes because a parameter changes at a point in time called the change point. The objective of this problem is to estimate the change point and each of the parameters of the stochastic process. In this thesis, we examine the change point problem for two classes of stochastic processes. First, we consider the volatility change point problem for stochastic diffusion processes driven by Brownian motions. Then, we consider the drift...
Show moreThe change point problem is a problem where a process changes regimes because a parameter changes at a point in time called the change point. The objective of this problem is to estimate the change point and each of the parameters of the stochastic process. In this thesis, we examine the change point problem for two classes of stochastic processes. First, we consider the volatility change point problem for stochastic diffusion processes driven by Brownian motions. Then, we consider the drift change point problem for OrnsteinUhlenbeck processes driven by _stable Levy motions. In each problem, we establish the consistency of the estimators, determine asymptotic behavior for the changing parameters, and finally, we perform simulation studies to computationally assess the convergence of parameters.
Show less  Date Issued
 2020
 PURL
 http://purl.flvc.org/fau/fd/FA00013462
 Subject Headings
 Stochastic processes, Changepoint problems, Brownian motion processes, OrnsteinUhlenbeck process, Computer simulation
 Format
 Document (PDF)
 Title
 ALGORITHMS IN LATTICEBASED CRYPTANALYSIS.
 Creator
 Miller, Shaun, Bai, Shi, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

An adversary armed with a quantum computer has algorithms[66, 33, 34] at their disposal, which are capable of breaking our current methods of encryption. Even with the birth of postquantum cryptography[52, 62, 61], some of best cryptanalytic algorithms are still quantum [45, 8]. This thesis contains several experiments on the efficacy of lattice reduction algorithms, BKZ and LLL. In particular, the difficulty of solving Learning With Errors is assessed by reducing the problem to an instance...
Show moreAn adversary armed with a quantum computer has algorithms[66, 33, 34] at their disposal, which are capable of breaking our current methods of encryption. Even with the birth of postquantum cryptography[52, 62, 61], some of best cryptanalytic algorithms are still quantum [45, 8]. This thesis contains several experiments on the efficacy of lattice reduction algorithms, BKZ and LLL. In particular, the difficulty of solving Learning With Errors is assessed by reducing the problem to an instance of the Unique Shortest Vector Problem. The results are used to predict the behavior these algorithms may have on actual cryptographic schemes with security based on hard lattice problems. Lattice reduction algorithms require several floatingpoint operations including multiplication. In this thesis, I consider the resource requirements of a quantum circuit designed to simulate floatingpoint multiplication with high precision.
Show less  Date Issued
 2020
 PURL
 http://purl.flvc.org/fau/fd/FA00013543
 Subject Headings
 Cryptanalysis, Cryptography, Algorithms, Lattices, Quantum computing
 Format
 Document (PDF)
 Title
 INFECTION AGE STRUCTURED VECTOR BORNE DISEASE MODEL WITH DIRECT TRANSMISSION.
 Creator
 Giri, Sunil, Tuncer, Necibe, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

Mathematical modeling is a powerful tool to study and analyze the disease dynamics prevalent in the community. This thesis studies the dynamics of two time since infection structured vector borne models with direct transmission. We have included disease induced death rate in the first model to form the second model. The aim of this thesis is to analyze whether these two models have same or different disease dynamics. An explicit expression for the reproduction number denoted by R0 is derived....
Show moreMathematical modeling is a powerful tool to study and analyze the disease dynamics prevalent in the community. This thesis studies the dynamics of two time since infection structured vector borne models with direct transmission. We have included disease induced death rate in the first model to form the second model. The aim of this thesis is to analyze whether these two models have same or different disease dynamics. An explicit expression for the reproduction number denoted by R0 is derived. Dynamical analysis reveals the forward bifurcation in the first model. That is when the threshold value R0 < 1, disease freeequilibrium is stable locally implying that if there is small perturbation of the system, then after some time, the system will return to the disease free equilibrium. When R0 > 1 the unique endemic equilibrium is locally asymptotically stable. For the second model, analysis of the existence and stability of equilibria reveals the existence of backward bifurcation i.e. where the disease free equilibrium coexists with the endemic equilibrium when the reproduction number R02 is less than unity. This aspect shows that in order to control vector borne disease, it is not sufficient to have reproduction number less than unity although necessary. Thus, the infection can persist in the population even if the reproduction number is less than unity. Numerical simulation is presented to see the bifurcation behaviour in the model. By taking the reproduction number as the bifurcation parameter, we find the system undergoes backward bifurcation at R02 = 1. Thus, the model has backward bifurcation and have two positive endemic equilibrium when R02 < 1 and unique positive endemic equilibrium whenever R02 > 1. Stability analysis shows that disease free equilibrium is locally asymptotically stable when R02 < 1 and unstable when R02 > 1. When R02 < 1, lower endemic equilibrium in backward bifurcation is locally unstable.
Show less  Date Issued
 2020
 PURL
 http://purl.flvc.org/fau/fd/FA00013552
 Subject Headings
 Vector Borne Diseases, Mathematical models, Simulations, DynamicsMathematical models
 Format
 Document (PDF)
 Title
 ANNIHILATORS AND A + B RINGS.
 Creator
 Epstein, Alexandra Nicole, Klingler, Lee, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

A + B rings are constructed from a ring A and nonempty set of prime ideals of A. Initially, these rings were created to provide examples of reduced rings which satisfy certain annihilator conditions. We describe precisely when A + B rings have these properties, based on the ring A and set of prime ideals of A. We continue by giving necessary and su cient conditions for A + B rings to have various other properties. We also consider annihilators in the context of frames of ideals of reduced rings.
 Date Issued
 2020
 PURL
 http://purl.flvc.org/fau/fd/FA00013588
 Subject Headings
 Rings (Algebra)
 Format
 Document (PDF)
 Title
 PREDICTING TROPICAL CYCLONE INTENSITY FROM GEOSYNCHRONOUS SATELLITE IMAGES USING DEEP NEURAL NETWORKS.
 Creator
 Udumulla, Niranga Mahesh, Motta, Francis, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

Tropical cyclones are among the most devastating natural disasters for human beings and the natural and manmade assets near to Atlantic basin. Estimating the current and future intensity of these powerful storms is crucial to protect life and property. Many methods and models exist for predicting the evolution of Atlantic basin cyclones, including numerical weather prediction models that simulate the dynamics of the atmosphere which require accurate measurements of the current state of the...
Show moreTropical cyclones are among the most devastating natural disasters for human beings and the natural and manmade assets near to Atlantic basin. Estimating the current and future intensity of these powerful storms is crucial to protect life and property. Many methods and models exist for predicting the evolution of Atlantic basin cyclones, including numerical weather prediction models that simulate the dynamics of the atmosphere which require accurate measurements of the current state of the atmosphere (NHC, 2019). Often these models fail to capture dangerous aspects of storm evolution, such as rapid intensification (RI), in which a storm undergoes a steep increase in intensity over a short time. To improve prediction of these events, scientists have turned to statistical models to predict current and future intensity using readily collected satellite image data (Pradhan, 2018). However, even the currentintensity prediction models have shown limited success in generalizing to unseen data, a result we confirm in this study. Therefore, building models for the estimating the current and future intensity of hurricanes is valuable and challenging. In this study we focus on to estimating cyclone intensity using Geostationary Operational Environmental Satellite images. These images represent five spectral bands covering the visible and infrared spectrum. We have built and compared various types of deep neural models, including convolutional networks based on long short term memory models and convolutional regression models that have been trained to predict the intensity, as measured by maximum sustained wind speed.
Show less  Date Issued
 2020
 PURL
 http://purl.flvc.org/fau/fd/FA00013626
 Subject Headings
 Tropical cyclones, CyclonesTropicsForecasting, Geosynchronous satellites, Neural networks (Computer science)
 Format
 Document (PDF)
 Title
 DEVELOPING A DEEP LEARNING PIPELINE TO AUTOMATICALLY ANNOTATE GOLD PARTICLES IN IMMUNOELECTRON MICROSCOPY IMAGES.
 Creator
 Jerez, Diego Alejandro, Hahn, William, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

Machine learning has been utilized in bioimaging in recent years, however as it is relatively new and evolving, some researchers who wish to utilize machine learning tools have limited access because of a lack of programming knowledge. In electron microscopy (EM), immunogold labeling is commonly used to identify the target proteins, however the manual annotation of the gold particles in the images is a timeconsuming and laborious process. Conventional image processing tools could provide...
Show moreMachine learning has been utilized in bioimaging in recent years, however as it is relatively new and evolving, some researchers who wish to utilize machine learning tools have limited access because of a lack of programming knowledge. In electron microscopy (EM), immunogold labeling is commonly used to identify the target proteins, however the manual annotation of the gold particles in the images is a timeconsuming and laborious process. Conventional image processing tools could provide semiautomated annotation, but those require that users make manual adjustments for every step of the analysis. To create a new highthroughput image analysis tool for immunoEM, I developed a deep learning pipeline that was designed to deliver a completely automated annotation of immunogold particles in EM images. The program was made accessible for users without prior programming experience and was also expanded to be used on different types of immunoEM images.
Show less  Date Issued
 2020
 PURL
 http://purl.flvc.org/fau/fd/FA00013628
 Subject Headings
 Electron microscopy, Immunogold labeling, Image analysis, Deep learning
 Format
 Document (PDF)
 Title
 PARAMETER ESTIMATION FOR GEOMETRIC L EVY PROCESSES WITH STOCHASTIC VOLATILITY.
 Creator
 Chhetri, Sher B., Long, Hongwei, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

In finance, various stochastic models have been used to describe the price movements of financial instruments. After Merton's [38] seminal work, several jump diffusion models for option pricing and risk management have been proposed. In this dissertation, we add alphastable Levy motion to the process related to dynamics of logreturns in the BlackScholes model where the volatility is assumed to be constant. We use the sample characteristic function approach in order to study parameter...
Show moreIn finance, various stochastic models have been used to describe the price movements of financial instruments. After Merton's [38] seminal work, several jump diffusion models for option pricing and risk management have been proposed. In this dissertation, we add alphastable Levy motion to the process related to dynamics of logreturns in the BlackScholes model where the volatility is assumed to be constant. We use the sample characteristic function approach in order to study parameter estimation for discretely observed stochastic differential equations driven by Levy noises. We also discuss the consistency and asymptotic properties of the proposed estimators. Simulation results of the model are also presented to show the validity of the estimators. We then propose a new model where the volatility is not a constant. We consider generalized alphastable geometric Levy processes where the stochastic volatility follows the CoxIngersollRoss (CIR) model in Cox et al. [9]. A number of methods have been proposed for estimating parameters for stable laws. However, a complication arises in estimation of the parameters in our model because of the presence of the unobservable stochastic volatility. To combat this complication we use the sample characteristic function method proposed by Press [48] and the conditional least squares method as mentioned in Overbeck and Ryden [47] to estimate all the parameters. We then discuss the consistency and asymptotic properties of the proposed estimators and establish a Central Limit Theorem. We perform simulations to assess the validity of the estimators. We also present several tables to show the comparison of estimators using different choices of arguments ui's. We conclude that all the estimators converge as expected regardless of the choice of ui's.
Show less  Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013294
 Subject Headings
 Stochastic models, Lévy processes, Parameter estimation, Finance, Simulations
 Format
 Document (PDF)
 Title
 Norm Inequalities for the Fourier Coefficients of Some Almost Periodic Functions.
 Creator
 Boryshchak, Yarema, Sagher, Yoram, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Using C. Fefferman's embedding of a charge space in a measure space allows us to apply standard interpolation theorems to the establishment of norm inequalities for Besicovitch almost periodic functions. This yields a significant improvement to the results of A. Avantaggiati, G. Bruno and R. Iannacci.
 Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013191
 Subject Headings
 Fourier series, Almost periodic functions, Norm
 Format
 Document (PDF)
 Title
 CHARACTERIZATIONS OF LINEAR ISOMETRIES ON COMPLEX SEQUENCE SPACES.
 Creator
 Babun Codorniu, Omar, Zhang, XiaoDong, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

An operator acting on a Banach space is called an isometry if it preserves the norm of the space. An interesting problem is to determine the form or structure of linear isometries on Banach spaces. This can be done in some instances. This dissertation presents several theorems that provide necessary and sufficient conditions for some linear operators acting on finite and infinite dimensional sequence spaces of complex numbers to be isometries. In all cases, the linear isometries have the form...
Show moreAn operator acting on a Banach space is called an isometry if it preserves the norm of the space. An interesting problem is to determine the form or structure of linear isometries on Banach spaces. This can be done in some instances. This dissertation presents several theorems that provide necessary and sufficient conditions for some linear operators acting on finite and infinite dimensional sequence spaces of complex numbers to be isometries. In all cases, the linear isometries have the form of a permutation of the elements of the sequences in the given space, with each component of each sequence multiplied by a complex number of absolute value 1.
Show less  Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013354
 Subject Headings
 Banach spaces, Isometrics (Mathematics), Matrices, Linear operators, Normed linear spaces
 Format
 Document (PDF)
 Title
 CONTRIBUTIONS TO QUANTUMSAFE CRYPTOGRAPHY: HYBRID ENCRYPTION AND REDUCING THE T GATE COST OF AES.
 Creator
 Pham, Hai, Steinwandt, Rainer, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Quantum cryptography offers a wonderful source for current and future research. The idea started in the early 1970s, and it continues to inspire work and development toward a popular goal, largescale communication networks with strong security guarantees, based on quantummechanical properties. Quantum cryptography builds on the idea of exploiting physical properties to establish secure cryptographic operations. A particular quantumbased protocol has gathered interest in recent years for...
Show moreQuantum cryptography offers a wonderful source for current and future research. The idea started in the early 1970s, and it continues to inspire work and development toward a popular goal, largescale communication networks with strong security guarantees, based on quantummechanical properties. Quantum cryptography builds on the idea of exploiting physical properties to establish secure cryptographic operations. A particular quantumbased protocol has gathered interest in recent years for its use of mesoscopic coherent states. The AlphaEta protocol has been designed to exploit properties of coherent states of light to transmit data securely over an optical channel. AlphaEta aims to draw security from the uncertainty of any measurement of the transmitted coherent states due to intrinsic quantum noise. We propose a framework to combine this protocol with classical preprocessing, taking into account errorcorrection for the optical channel and establishing a strong provable security guarantee. Integrating a stateoftheart solution for fast authenticated encryption is straightforward, but in this case the security analysis requires heuristic reasoning.
Show less  Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013339
 Subject Headings
 Cryptography, Quantum computing, Algorithms, Mesoscopic coherent states
 Format
 Document (PDF)
 Title
 An Algorithmic Approach to Tran Van Trung's Basic Recursive Construction of tDesigns.
 Creator
 Lopez, Oscar A., Magliveras, Spyros S., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

It was not until the 20th century that combinatorial design theory was studied as a formal subject. This field has many applications, for example in statistical experimental design, coding theory, authentication codes, and cryptography. Major approaches to the problem of discovering new tdesigns rely on (i) the construction of large sets of t designs, (ii) using prescribed automorphism groups, (iii) recursive construction methods. In 2017 and 2018, Tran Van Trung introduced new recursive...
Show moreIt was not until the 20th century that combinatorial design theory was studied as a formal subject. This field has many applications, for example in statistical experimental design, coding theory, authentication codes, and cryptography. Major approaches to the problem of discovering new tdesigns rely on (i) the construction of large sets of t designs, (ii) using prescribed automorphism groups, (iii) recursive construction methods. In 2017 and 2018, Tran Van Trung introduced new recursive techniques to construct t – (v, k, λ) designs. These methods are of purely combinatorial nature and require using "ingredient" tdesigns or resolutions whose parameters satisfy a system of nonlinear equations. Even after restricting the range of parameters in this new method, the task is computationally intractable. In this work, we enhance Tran Van Trung's "Basic Construction" by a robust and efficient hybrid computational apparatus which enables us to construct hundreds of thousands of new t – (v, k, Λ) designs from previously known ingredient designs. Towards the end of the dissertation we also create a new family of 2resolutions, which will be infinite if there are infinitely many Sophie Germain primes.
Show less  Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013233
 Subject Headings
 Combinatorial designs and configurations, Algorithms, tdesigns
 Format
 Document (PDF)
 Title
 DETERMINANTS OF WOMEN'S ATTITUDE TOWARDS INTIMATE PARTNER VIOLENCE: EVIDENCE FROM BANGLADESH.
 Creator
 Khan, Md Tareq Ferdous, Qian, Lianfen, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

This thesis uses Bangladesh Demographic and Health Survey 2014 data to identify the important determinants due to which women justification towards intimate partner violence (IPV) varies. Statistical analyses reveal that among the individuallevel independent variables age at first marriage, respondent's education, decision score, religion, NGO membership, access to information, husband's education, normalized wealth score, and division indicator have significant effects on the women's...
Show moreThis thesis uses Bangladesh Demographic and Health Survey 2014 data to identify the important determinants due to which women justification towards intimate partner violence (IPV) varies. Statistical analyses reveal that among the individuallevel independent variables age at first marriage, respondent's education, decision score, religion, NGO membership, access to information, husband's education, normalized wealth score, and division indicator have significant effects on the women's attitude towards IPV. It shows that other than religion, NGO membership, and division indicator, the higher the value of the variable, the lower the likelihood of justifying IPV. However, being a Muslim, NGO member, and resident of other divisions, women are found more tolerant of IPV from their respective counterparts. Among the three communitylevel variables, only the mean decision score is found significant in lowering the likelihood. The thesis concludes with some policy recommendations and a proposal for future research.
Show less  Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013325
 Subject Headings
 Intimate partner violence, Bangladesh, Women
 Format
 Document (PDF)
 Title
 HLOCAL RINGS.
 Creator
 Omairi, Akeel, Klingler, Lee, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

We say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any Rmodule which decomposes into a _nite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains and in a 2011 paper Ay and Klingler obtain similar results for Noetherian reduced rings. We characterize the UDI property for Noetherian...
Show moreWe say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any Rmodule which decomposes into a _nite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains and in a 2011 paper Ay and Klingler obtain similar results for Noetherian reduced rings. We characterize the UDI property for Noetherian rings in general.
Show less  Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013336
 Subject Headings
 Noetherian rings, Prüfer rings, Local rings
 Format
 Document (PDF)
 Title
 The Covering Numbers of Some Finite Simple Groups.
 Creator
 Epstein, Michael, Magliveras, Spyros S., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

A finite cover C of a group G is a finite collection of proper subgroups of G such that G is equal to the union of all of the members of C. Such a cover is called minimal if it has the smallest cardinality among all finite covers of G. The covering number of G, denoted by σ(G), is the number of subgroups in a minimal cover of G. Here we determine the covering numbers of the projective special unitary groups U3(q) for q ≤ 5, and give upper and lower bounds for the covering number of U3(q) when...
Show moreA finite cover C of a group G is a finite collection of proper subgroups of G such that G is equal to the union of all of the members of C. Such a cover is called minimal if it has the smallest cardinality among all finite covers of G. The covering number of G, denoted by σ(G), is the number of subgroups in a minimal cover of G. Here we determine the covering numbers of the projective special unitary groups U3(q) for q ≤ 5, and give upper and lower bounds for the covering number of U3(q) when q > 5. We also determine the covering number of the McLaughlin sporadic simple group, and verify previously known results on the covering numbers of the HigmanSims and Held groups.
Show less  Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013203
 Subject Headings
 Finite simple groups, Covering numbers
 Format
 Document (PDF)
 Title
 Quantum Circuits for Symmetric Cryptanalysis.
 Creator
 Langenberg, Brandon Wade, Steinwandt, Rainer, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Quantum computers and quantum computing is a reality of the near feature. Companies such as Google and IBM have already declared they have built a quantum computer and tend to increase their size and capacity moving forward. Quantum computers have the ability to be exponentially more powerful than classical computers today. With this power modeling behavior of atoms or chemical reactions in unusual conditions, improving weather forecasts and traffic conditions become possible. Also, their...
Show moreQuantum computers and quantum computing is a reality of the near feature. Companies such as Google and IBM have already declared they have built a quantum computer and tend to increase their size and capacity moving forward. Quantum computers have the ability to be exponentially more powerful than classical computers today. With this power modeling behavior of atoms or chemical reactions in unusual conditions, improving weather forecasts and traffic conditions become possible. Also, their ability to exponentially speed up some computations makes the security of todays data and items a major concern and interest. In the area of cryptography, some encryption schemes (such as RSA) are already deemed broken by the onset of quantum computing. Some encryption algorithms have already been created to be quantum secure and still more are being created each day. While these algorithms in use today are considered quantumsafe not much is known of what a quantum attack would look like on these algorithms. Specifically, this paper discusses how many quantum bits, quantum gates and even the depth of these gates that would be needed for such an attack. The research below was completed to shed light on these areas and offer some concrete numbers of such an attack.
Show less  Date Issued
 2018
 PURL
 http://purl.flvc.org/fau/fd/FA00013010
 Subject Headings
 Quantum computing, Cryptography, Cryptanalysis, Data encryption (Computer science), Computer algorithms
 Format
 Document (PDF)
 Title
 QuantumResistant Key Agreement and Key Encapsulation.
 Creator
 Robinson, Angela, Steinwandt, Rainer, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

We explore quantumresistant key establishment and hybrid encryption. We nd that while the discrete logarithm problem is e ciently solved by a quantum computer using Shor's algorithm, some instances are insecure even using classical computers. The discrete logarithm problem based on a symmetric group Sn is e  ciently solved in polynomial time. We design a PUFbased 4round group key establishment protocol, adjusting the model to include a physical channel capable of PUF transmission, and...
Show moreWe explore quantumresistant key establishment and hybrid encryption. We nd that while the discrete logarithm problem is e ciently solved by a quantum computer using Shor's algorithm, some instances are insecure even using classical computers. The discrete logarithm problem based on a symmetric group Sn is e  ciently solved in polynomial time. We design a PUFbased 4round group key establishment protocol, adjusting the model to include a physical channel capable of PUF transmission, and modify adversarial capabilities with respect to the PUFs. The result is a novel group key establishment protocol which avoids computational hardness assumptions and achieves key secrecy. We contribute a hybrid encryption scheme by combining a key encapsulation mechanism (KEM) with a symmetric key encryption scheme by using two hash functions. We require only oneway security in the quantum random oracle model (QROM) of the KEM and onetime security of the symmetric encryption scheme in the QROM. We show that this hybrid scheme is INDCCA secure in the QROM. We rely on a powerful theorem by Unruh that provides an upper bound on indistinguishability between the output of a random oracle and a random string, when the oracle can be accessed in quantum superposition. Our result contributes to the available INDCCA secure encryption schemes in a setting where quantum computers are under adversarial control. Finally, we develop a framework and describe biometric visual cryptographic schemes generically under our framework. We formalize several security notions and de nitions including sheet indistinguishability, perfect indistinguishability, index recovery, perfect index privacy, and perfect resistance against false authentication. We also propose new and generic strategies for attacking eBVC schemes such as new distinguishing attack, new index recovery, and new authentication attack. Our quantitative analysis veri es the practical impact of our framework and o ers concrete upper bounds on the security of eBVC.
Show less  Date Issued
 2018
 PURL
 http://purl.flvc.org/fau/fd/FA00013023
 Subject Headings
 Quantum computing, Data encryption (Computer science), Cryptography
 Format
 Document (PDF)
 Title
 On the Study of the Aizawa System.
 Creator
 Fleurantin, Emmanuel, MirelesJames, Jason D., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

In this report we study the Aizawa field by first computing a Taylor series expansion for the solution of an initial value problem. We then look for singularities (equilibrium points) of the field and plot the set of solutions which lie in the linear subspace spanned by the eigenvectors. Finally, we use the Parameterization Method to compute one and two dimensional stable and unstable manifolds of equilibria for the system.
 Date Issued
 2018
 PURL
 http://purl.flvc.org/fau/fd/FA00005994
 Subject Headings
 Series, Mathematics, Eigenvectors, Aizawa field
 Format
 Document (PDF)
 Title
 Various Approaches on Parameter Estimation in Mixture and NonMixture Cure Models.
 Creator
 Kutal, Durga Hari, Qian, Lianfen, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Analyzing lifetime data with longterm survivors is an important topic in medical application. Cure models are usually used to analyze survival data with the proportion of cure subjects or longterm survivors. In order to include the propor tion of cure subjects, mixture and nonmixture cure models are considered. In this dissertation, we utilize both maximum likelihood and Bayesian methods to estimate model parameters. Simulation studies are carried out to verify the nite sample per...
Show moreAnalyzing lifetime data with longterm survivors is an important topic in medical application. Cure models are usually used to analyze survival data with the proportion of cure subjects or longterm survivors. In order to include the propor tion of cure subjects, mixture and nonmixture cure models are considered. In this dissertation, we utilize both maximum likelihood and Bayesian methods to estimate model parameters. Simulation studies are carried out to verify the nite sample per formance of the estimation methods. Real data analyses are reported to illustrate the goodnessof t via Fr echet, Weibull and Exponentiated Exponential susceptible distributions. Among the three parametric susceptible distributions, Fr echet is the most promising. Next, we extend the nonmixture cure model to include a change point in a covariate for right censored data. The smoothed likelihood approach is used to address the problem of a loglikelihood function which is not di erentiable with respect to the change point. The simulation study is based on the nonmixture change point cure model with an exponential distribution for the susceptible subjects. The simulation results revealed a convincing performance of the proposed method of estimation.
Show less  Date Issued
 2018
 PURL
 http://purl.flvc.org/fau/fd/FA00013083
 Subject Headings
 Survival Analysis., Bayesian statistical decision theory., Parameter estimation., Weibull distribution.
 Format
 Document (PDF)
 Title
 Algorithms in Elliptic Curve Cryptography.
 Creator
 Hutchinson, Aaron, Karabina, Koray, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Elliptic curves have played a large role in modern cryptography. Most notably, the Elliptic Curve Digital Signature Algorithm (ECDSA) and the Elliptic Curve Di eHellman (ECDH) key exchange algorithm are widely used in practice today for their e ciency and small key sizes. More recently, the Supersingular Isogenybased Di eHellman (SIDH) algorithm provides a method of exchanging keys which is conjectured to be secure in the postquantum setting. For ECDSA and ECDH, e cient and secure...
Show moreElliptic curves have played a large role in modern cryptography. Most notably, the Elliptic Curve Digital Signature Algorithm (ECDSA) and the Elliptic Curve Di eHellman (ECDH) key exchange algorithm are widely used in practice today for their e ciency and small key sizes. More recently, the Supersingular Isogenybased Di eHellman (SIDH) algorithm provides a method of exchanging keys which is conjectured to be secure in the postquantum setting. For ECDSA and ECDH, e cient and secure algorithms for scalar multiplication of points are necessary for modern use of these protocols. Likewise, in SIDH it is necessary to be able to compute an isogeny from a given nite subgroup of an elliptic curve in a fast and secure fashion. We therefore nd strong motivation to study and improve the algorithms used in elliptic curve cryptography, and to develop new algorithms to be deployed within these protocols. In this thesis we design and develop dMUL, a multidimensional scalar multiplication algorithm which is uniform in its operations and generalizes the well known 1dimensional Montgomery ladder addition chain and the 2dimensional addition chain due to Dan J. Bernstein. We analyze the construction and derive many optimizations, implement the algorithm in software, and prove many theoretical and practical results. In the nal chapter of the thesis we analyze the operations carried out in the construction of an isogeny from a given subgroup, as performed in SIDH. We detail how to e ciently make use of parallel processing when constructing this isogeny.
Show less  Date Issued
 2018
 PURL
 http://purl.flvc.org/fau/fd/FA00013113
 Subject Headings
 Curves, Elliptic, Cryptography, Algorithms
 Format
 Document (PDF)