[pt] Curvas elípticas são objeto de estudo pelos matemáticos há mais de 200 anos. Por si só, é uma teoria bastante interessante por estar relacionada com diversas áreas da matemática: álgebra, equações diofantinas e geometria algébrica, dentre outras. Recentemente, diversos pesquisadores sugeriram o uso de curvas elípticas para resolver problemas práticos; como exemplos, podemos citar a criptografia, algoritmos para fatoração de números inteiros e testes de primalidade. Uma curva elíptica é definida sobre um corpo (no sentido algébrico). Essa dissertação tem por objetivo apresentar os primeiros elementos da teoria das curvas elípticas sobre corpos finitos. Como veremos, o desenvolvimento do tema aborda diversos tópicos da educação básica. Para isso, iniciaremos o trabalho com uma introdução utilizando o corpo dos números reais e, em seguida, incluiremos a teoria mais geral sobre essas curvas algébricas. Concluiremos então com algumas propriedades e resultados de curvas elípticas sobre corpos finitos, incluindo alguns exemplos e a interpretação geométrica da soma de dois pontos de curvas sobre corpos finitos específicos. / [en] Elliptic curves have been studied by mathematicians for over 200 years. By itself, it is a remarkably interesting theory as it is related to several areas of mathematics: algebra, Diophantine equations and algebraic geometry, among others. Recently, several researchers have suggested the use of elliptic curves to solve practical problems; as examples, we can mention cryptography, integer factorization algorithms and primality tests. An elliptic curve is defined over a field (in algebraic sense). This dissertation aims to present the first elements in the theory of elliptic curves on finite fields. As we will see, the development of the subject addresses a number of topics covered in basic education. In order to accomplish this, we will start the work with an introduction using the field of real numbers and then we will include the more general theory about these algebraic curves. Finally, we will present some properties and results on elliptic curves over finite fields, including some examples and a geometric interpretation of the sum of two points over specific finite fields.

Let X be the affine flag manifold of Lie type A_n-1⁽¹⁾ where n ≥ 3 and let W_aff be the associated affine Weyl group. The moment graph for X encodes the torus fixed points (which are elements of the affine Weyl group W_aff and the torus stable curves in X. Given a fixed point u ∈ W_aff and a degree d = (d₀, d₁, ..., d_n−1) ∈ ℤ_≥0ⁿ, the combinatorial curve neighborhood is the set of maximal elements in the moment graph of X which can be reached from u′ ≤ u by a chain of curves of total degree ≤ d. In this thesis we give combinatorial formulas and algorithms for calculating these elements. / Doctor of Philosophy / The study of curves on flag manifolds is motivated by questions in enumerative geometry and physics. To a space of curves and incidence conditions one can associate a combinatorial object called the ‘combinatorial curve neighborhood’. For a fixed degree d and a (Schubert) cycle, the curve neighborhood consists of the set of all elements in the Weyl group which can be reached from the given cycle by a path of fixed degree in the moment graph of the flag manifold, and are also Bruhat maximal with respect to this property. For finite dimensional flag manifolds, a description of the curve neighborhoods helped answer questions related to the quantum cohomology and quantum K theory rings, and ultimately about enumerative geometry of the flag manifolds. In this thesis we study the situation of the affine flag manifolds, which are infinite dimensional. We obtain explicit combinatorial formulas and recursions which characterize the curve neighborhoods for flag manifolds of affine Lie type A. Among the conclusions, we mention that, unlike in the finite dimensional case, the curve neighborhoods have more than one component, although all components have the same length. In general, calculations reflect a close connection between the curve neighborhoods and the Lie combinatorics of the affine root system, especially the imaginary roots.

This dissertation emphasizes the security aspects of real-time media. The problems of existing real-time media protections are identified in this research, and viable solutions are proposed. First, the security of real-time media depends on the Secure Real-time Transport Protocol (SRTP) mechanism. We identified drawbacks of the existing SRTP Systems, which use symmetric key encryption schemes, which can be exploited by attackers. Elliptic Curve Cryptography (ECC), an asymmetric key cryptography scheme, is proposed to resolve these problems. Second, the ECC encryption scheme is based on elliptic curves. This dissertation explores the weaknesses of a widely used elliptic curve in terms of security and describes a more secure elliptic curve suitable for real-time media protection. Eighteen elliptic curves had been tested in a real-time video transmission system, and fifteen elliptic curves had been tested in a real-time audio transmission system. Based on the performance, X9.62 standard 256-bit prime curve, NIST-recommended 256-bit prime curves, and Brainpool 256-bit prime curves were found to be suitable for real-time audio encryption. Again, X9.62 standard 256-bit prime and 272-bit binary curves, and NIST-recommended 256-bit prime curves were found to be suitable for real-time video encryption.The weaknesses of NIST-recommended elliptic curves are discussed and a more secure new elliptic curve is proposed which can be used for real-time media encryption. The proposed curve has fulfilled all relevant security criteria, but the corresponding NIST curve could not fulfill two such criteria. The research is applicable to strengthen the security of the Internet of Things (IoT) devices, especially VoIP cameras. IoT devices have resource constraints and thus need lightweight encryption schemes for security. ECC could be a better option for these devices. VoIP cameras use a similar methodology to traditional real-time video transmission, so this research could be useful to find a better security solution for these devices.

Cette thèse présente la construction de courbes tropicales réelles dans R^3 dont la projectivisation, qui est un entrelacs projectif dans IRP^3, est constituée de 2 composantes, I'une étant isotope à un noeud donné au départ. Dans le cas de certains noeuds toriques, il est possible de modifier cette construction pour que I'entrelacs projectif correspondant ait une seule composante isotope au noeud torique considéré. Pour chacune de ces courbes tropicales réelles, nous faisons appel au théorème récent de G. Mikhalkin, qui affirme l'existence d'une algébrique réelle non singulière dans IRP^3, de même genre et degré que la courbe tropicale réelle considérée, et qui est isotope à l'entrelacs projectif correspondant. / In this thesis, we construct real tropical curves in R^3 whose projectivization - which is a projective link in RP^3 - has 2connected components, one of them being isotopic to a given knot. For some torus knots, it is possible to modify thetropical construction such that the corresponding projective link is a knot (with a single component) isotopic to the giventorus knot. For each of these real tropical curve, we use a recent result of G. Mikhalkin, asserting the existence of a realnon singular algebraic curve in RP^3, of the same genus and degree as the real tropical curve, and isotopic to thecorresponding projective link.

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Given an elliptic curve E over Fp and an integer e ≥ 1, we define a new object, called “elliptic loop”, as the set of plane projective points over Z/p^e Z lying over E, endowed with an operation inherited by the curve addition. This object is proved to be a power-associative abelian algebraic loop. Its substructures are investigated by means of other algebraic cubics defined over the same ring, which we named “shadow curve” and “layers”. When E has trace 1, a distinctive behavior is detected and employed for producing an isomorphism attack to the discrete logarithm on this family of curves. Stronger properties are derived for small values of e, which lead to an explicit description of the infinity part and to characterizing the geometry of rational |E|-torsion points. / Data una curva ellittica E su Fp ed un intero e ≥ 1, definiamo un nuovo oggetto, chiamato "loop ellittico", come l'insieme dei punti nel piano proiettivo su Z/p^e Z che stanno sopra ad E, dotato di una operazione ereditata dalla somma di punti sulla curva. Questo oggetto si prova essere un loop algebrico con associatività delle potenze. Le sue sotto-strutture sono investigate utilizzando altre cubiche definite sullo stesso anello, che abbiamo chiamato "curva ombra" e "strati". Quando E ha traccia 1, un comportamento speciale viene notato e sfruttato per produrre un attacco di isomorfismo al problema del logaritmo discreto su questa famiglia di curve. Migliori proprietà vengono trovate per bassi valori di e, che portano ad una descrizione esplicita della parte all'infinito e alla caratterizzazione della geometria dei punti razionali di |E|-torsione.

Doctor of Philosophy / Department of Physics / Nobel S. Rebello / This research project investigates the difficulties students encounter when solving physics problems involving the integral and the area under the curve concepts and the strategies to facilitate students learning to solve those types of problems. The research contexts of this project are calculus-based physics courses covering mechanics and electromagnetism. In phase I of the project, individual teaching/learning interviews were conducted with 20 students in mechanics and 15 students from the same cohort in electromagnetism. The students were asked to solve problems on several topics of mechanics and electromagnetism. These problems involved calculating physical quantities (e.g. velocity, acceleration, work, electric field, electric resistance, electric current) by integrating or finding the area under the curve of functions of related quantities (e.g. position, velocity, force, charge density, resistivity, current density). Verbal hints were provided when students made an error or were unable to proceed. A total number of 140 one-hour interviews were conducted in this phase, which provided insights into students’ difficulties when solving the problems involving the integral and the area under the curve concepts and the hints to help students overcome those difficulties. In phase II of the project, tutorials were created to facilitate students’ learning to solve physics problems involving the integral and the area under the curve concepts. Each tutorial consisted of a set of exercises and a protocol that incorporated the helpful hints to target the difficulties that students expressed in phase I of the project. Focus group learning interviews were conducted to test the effectiveness of the tutorials in comparison with standard learning materials (i.e. textbook problems and solutions). Overall results indicated that students learning with our tutorials outperformed students learning with standard materials in applying the integral and the area under the curve concepts to physics problems. The results of this project provide broader and deeper insights into students’ problem solving with the integral and the area under the curve concepts and suggest strategies to facilitate students’ learning to apply these concepts to physics problems. This study also has significant implications for further research, curriculum development and instruction.

International Telemetering Conference Proceedings / October 20-23, 2003 / Riviera Hotel and Convention Center, Las Vegas, Nevada / SSL protocol is one industrial standard to protect data transferred securely on Internet. Firstly SSL is analyzed, according to its characteristics, one solution plan on mobile bank based on SSL is proposed and presented, in which GPRS technology is adopted and elliptic curve algorithm is used for the session key, finally several functional modules of mobile bank are designed in details and its security is analyzed.

This thesis is devoted to investigating some properties of the sequence (Wn) of the denominators. This is a divisibility sequence; that is, Wm | Wn whenever m | n. Our task here is to examine a conjecture on the number of prime terms in (Wn), well known as the Primality conjecture. We will prove that there is a uniform lower bound on n beyond such that all terms Wn have at least two distinct prime factors. In some cases, the bound is as low as n = 2.

Synchronized neural activity, in which the firing of neurons is coordinated in time, is an observed phenomenon in many neural functions. The conditions that promote synchrony and the dynamics of synchronized activity are active areas of investigation because they are incompletely understood. In addition, variability is intrinsic to biological systems, but the effect of neuron spike time variability on synchronization dynamics is a question that merits more attention. Previous experiments using a hybrid circuit of one biological neuron coupled to one computational neuron revealed that irregularity in biological neuron spike timing could change synchronization in the circuit, transitioning the activity between phase-locked and phase slipping. Simulations of this circuit could not replicate the transitions in network activity if neuron period was represented as a Gaussian process, but could if a process with history and a stochastic component were used. The phase resetting curve (PRC), which describes how neuron cycles change in response to input, can be used to construct a map that predicts if synchronization will occur in hybrid circuits. Without modification, these maps did not always capture observed network activity. I conducted long-term recordings of invertebrate neurons and show that interspike interval (ISI) can be represented as an autoregressive integrated moving average process, where ISI is dependent on past history and a stochastic component with history. Using integrate and fire model simulations, I suggest that stochastic activity in adaptation channels could be responsible for the history dependence and correlational structure observed in these neurons. This evidence for stochastic, history-dependent noise in neural systems indicates that our understanding of network dynamics could be enhanced by including more complex, but relevant, forms of noise. I show that cycle-by-cycle dynamics of the coupled system can be used to infer features of the dynamic map, even if it cannot be measured or is changing over time. Using this method, stable fixed points can be distinguished from ghost attractors in the presence of noise, networks with similar phase but different underlying dynamics can be resolved, and the movement of stable fixed points can be observed. The time-series vector method is a valuable tool for distinguishing dynamics and describing robustness. It can be adapted for use in larger populations and non-reciprocal circuits. Finally, some larger implications of neuroscience research, specifically the use of neural interfaces for national security, are discussed. Neural interfaces for human enhancement in a national security context raise a number of unique ethical and policy concerns not common to dual use research of concern or traditional human subjects research. Guidelines about which technologies should be developed are lacking. We discuss a two-step framework with 1) an initial screen to prioritize technologies that should be reviewed immediately, and 2) a comprehensive ethical review regarding concerns for the enhanced individual, operational norms, and multi-use applications in the case of transfer to civilian contexts.