Analysis discrete function theory

There is a recent trend to describe physical phenomena without the use of infinitesimals or infinites. This has been accomplished replacing differential calculus by the finite difference theory. Discrete function theory was first introduced in l94l. This theory is concerned with a study of functions defined on a discrete set of points in the complex plane. The theory was extensively developed for functions defined on a Gaussian lattice. In 1972 a very suitable lattice H: {Ci qmxO,I qnyo), X0) 0, X3) 0, O < q < l, m, n 5 Z} was found and discrete analytic function theory was developed. Very recently some work has been done in discrete monodiffric function theory for functions defined on H. The theory of pseudoanalytic functions is a generalisation of the theory of analytic functions. When the generator becomes the identity, ie., (l, i) the theory of pseudoanalytic functions reduces to the theory of analytic functions. Theugh the theory of pseudoanalytic functions plays an important role in analysis, no discrete theory is available in literature. This thesis is an attempt in that direction. A discrete pseudoanalytic theory is derived for functions defined on H.

Algebraic Geometric Codes and their relation to Cryptography using Elliptic Curves

Communication is the process of transmitting data across channel. Whenever data is transmitted across a channel, errors are likely to occur. Coding theory is a stream of science that deals with finding efficient ways to encode and decode data, so that any likely errors can be detected and corrected. There are many methods to achieve coding and decoding. One among them is Algebraic Geometric Codes that can be constructed from curves. Cryptography is the science ol‘ security of transmitting messages from a sender to a receiver. The objective is to encrypt message in such a way that an eavesdropper would not be able to read it. A eryptosystem is a set of algorithms for encrypting and decrypting for the purpose of the process of encryption and decryption. Public key eryptosystem such as RSA and DSS are traditionally being prel‘en‘ec| for the purpose of secure communication through the channel. llowever Elliptic Curve eryptosystem have become a viable altemative since they provide greater security and also because of their usage of key of smaller length compared to other existing crypto systems. Elliptic curve cryptography is based on group of points on an elliptic curve over a finite field. This thesis deals with Algebraic Geometric codes and their relation to Cryptography using elliptic curves. Here Goppa codes are used and the curves used are elliptic curve over a finite field. We are relating Algebraic Geometric code to Cryptography by developing a cryptographic algorithm, which includes the process of encryption and decryption of messages. We are making use of fundamental properties of Elliptic curve cryptography for generating the algorithm and is used here to relate both.

Transcendental Brauer groups of products of CM elliptic curves

Let L be a number field and let E/L be an elliptic curve with complex multiplication by the ring of integers O_K of an imaginary quadratic field K. We use class field theory and results of Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the abelian surface ExE. The results for the odd order torsion also apply to the Brauer group of the K3 surface Kum(ExE). We describe explicitly the elliptic curves E/Q with complex multiplication by O_K such that the Brauer group of ExE contains a transcendental element of odd order. We show that such an element gives rise to a Brauer-Manin obstruction to weak approximation on Kum(ExE), while there is no obstruction coming from the algebraic part of the Brauer group.

Shadow lines in the arithmetic of elliptic curves

Let E/Q be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field K/Q satisfying the Heegner hypothesis for E we have a corresponding line in E(K)\otimes Q_p, known as a shadow line. When E/Q has analytic rank 2 and E/K has analytic rank 3, shadow lines are expected to lie in E(Q)\otimes Qp. If, in addition, p splits in K/Q, then shadow lines can be determined using the anticyclotomic p-adic height pairing. We develop an algorithm to compute anticyclotomic p-adic heights which we then use to provide an algorithm to compute shadow lines. We conclude by illustrating these algorithms in a collection of examples.

Criptografia de chave pública, criptografia RSA

Pós-graduação em Matemática Universitária - IGCE

A study of the errors of the averaged models in the restricted three-body problem in a short time scale

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Euler characteristics and elliptic curves

Let E be a modular elliptic curve over ℚ, without complex multiplication; let p be a prime number where E has good ordinary reduction; and let F∞ be the field obtained by adjoining to ℚ all p-power division points on E. Write G∞ for the Galois group of F∞ over ℚ. Assume that the complex L-series of E over ℚ does not vanish at s = 1. If p ⩾ 5, we make a precise conjecture about the value of the G∞-Euler characteristic of the Selmer group of E over F∞. If one makes a standard conjecture about the behavior of this Selmer group as a module over the Iwasawa algebra, we are able to prove our conjecture. The crucial local calculations in the proof depend on recent joint work of the first author with R. Greenberg.

Complex Hyperbolic Surfaces of Abelian Type

2000 Mathematics Subject Classification: 11G15, 11G18, 14H52, 14J25, 32L07.

Deniable Authenticated Key Establishment for Internet Protocols

We propose two public-key schemes to achieve “deniable authentication” for the Internet Key Exchange (IKE). Our protocols can be implemented using different concrete mechanisms and we discuss different options; in particular we suggest solutions based on elliptic curve pairings. The protocol designs use the modular construction method of Canetti and Krawczyk which provides the basis for a proof of security. Our schemes can, in some situations, be more efficient than existing IKE protocols as well as having stronger deniability properties.

Twisted Edwards curves revisited

This paper introduces fast algorithms for performing group operations on twisted Edwards curves, pushing the recent speed limits of Elliptic Curve Cryptography (ECC) forward in a wide range of applications. Notably, the new addition algorithm uses for suitably selected curve constants. In comparison, the fastest point addition algorithms for (twisted) Edwards curves stated in the literature use . It is also shown that the new addition algorithm can be implemented with four processors dropping the effective cost to . This implies an effective speed increase by the full factor of 4 over the sequential case. Our results allow faster implementation of elliptic curve scalar multiplication. In addition, the new point addition algorithm can be used to provide a natural protection from side channel attacks based on simple power analysis (SPA).

Jacobi quartic curves revisited

This paper provides new results about efficient arithmetic on Jacobi quartic form elliptic curves, y 2 = d x 4 + 2 a x 2 + 1. With recent bandwidth-efficient proposals, the arithmetic on Jacobi quartic curves became solidly faster than that of Weierstrass curves. These proposals use up to 7 coordinates to represent a single point. However, fast scalar multiplication algorithms based on windowing techniques, precompute and store several points which require more space than what it takes with 3 coordinates. Also note that some of these proposals require d = 1 for full speed. Unfortunately, elliptic curves having 2-times-a-prime number of points, cannot be written in Jacobi quartic form if d = 1. Even worse the contemporary formulae may fail to output correct coordinates for some inputs. This paper provides improved speeds using fewer coordinates without causing the above mentioned problems. For instance, our proposed point doubling algorithm takes only 2 multiplications, 5 squarings, and no multiplication with curve constants when d is arbitrary and a = ±1/2.

Optimal control of distributed generators and capacitors by hybrid DPSO

In this paper, a comprehensive planning methodology is proposed that can minimize the line loss, maximize the reliability and improve the voltage profile in a distribution network. The injected active and reactive power of Distributed Generators (DG) and the installed capacitor sizes at different buses and for different load levels are optimally controlled. The tap setting of HV/MV transformer along with the line and transformer upgrading is also included in the objective function. A hybrid optimization method, called Hybrid Discrete Particle Swarm Optimization (HDPSO), is introduced to solve this nonlinear and discrete optimization problem. The proposed HDPSO approach is a developed version of DPSO in which the diversity of the optimizing variables is increased using the genetic algorithm operators to avoid trapping in local minima. The objective function is composed of the investment cost of DGs, capacitors, distribution lines and HV/MV transformer, the line loss, and the reliability. All of these elements are converted into genuine dollars. Given this, a single-objective optimization method is sufficient. The bus voltage and the line current as constraints are satisfied during the optimization procedure. The IEEE 18-bus test system is modified and employed to evaluate the proposed algorithm. The results illustrate the unavoidable need for optimal control on the DG active and reactive power and capacitors in distribution networks.