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.

Algebraic geometric codes over rings

The techniques of algebraic geometry have been widely and successfully applied to the study of linear codes over finite fields since the early 1980's. Recently, there has been an increased interest in the study of linear codes over finite rings. In this thesis, we combine these two approaches to coding theory by introducing and studying algebraic geometric codes over rings.

A Cryptosystem Using the Concepts of Algebraic Geometric Code

Cryptosystem using linear codes was developed in 1978 by Mc-Eliece. Later in 1985 Niederreiter and others developed a modified version of cryptosystem using concepts of linear codes. But these systems were not used frequently because of its larger key size. In this study we were designing a cryptosystem using the concepts of algebraic geometric codes with smaller key size. Error detection and correction can be done efficiently by simple decoding methods using the cryptosystem developed. Approach: Algebraic geometric codes are codes, generated using curves. The cryptosystem use basic concepts of elliptic curves cryptography and generator matrix. Decrypted information takes the form of a repetition code. Due to this complexity of decoding procedure is reduced. Error detection and correction can be carried out efficiently by solving a simple system of linear equations, there by imposing the concepts of security along with error detection and correction. Results: Implementation of the algorithm is done on MATLAB and comparative analysis is also done on various parameters of the system. Attacks are common to all cryptosystems. But by securely choosing curve, field and representation of elements in field, we can overcome the attacks and a stable system can be generated. Conclusion: The algorithm defined here protects the information from an intruder and also from the error in communication channel by efficient error correction methods.

Some problems in set topology relating group of homeomorphisms and order

In this thesis we investigate some problems in set theoretical topology related to the concepts of the group of homeomorphisms and order. Many problems considered are directly or indirectly related to the concept of the group of homeomorphisms of a topological space onto itself. Order theoretic methods are used extensively. Chapter-l deals with the group of homeomorphisms. This concept has been investigated by several authors for many years from different angles. It was observed that nonhomeomorphic topological spaces can have isomorphic groups of homeomorphisms. Many problems relating the topological properties of a space and the algebraic properties of its group of homeomorphisms were investigated. The group of isomorphisms of several algebraic, geometric, order theoretic and topological structures had also been investigated. A related concept of the semigroup of continuous functions of a topological space also received attention

The Dirichlet distribution with respect to the Aitchison measure on the simplex - a first approach

The algebraic-geometric structure of the simplex, known as Aitchison geometry, is used to look at the Dirichlet family of distributions from a new perspective. A classical Dirichlet density function is expressed with respect to the Lebesgue measure on real space. We propose here to change this measure by the Aitchison measure on the simplex, and study some properties and characteristic measures of the resulting density

From Dirac spinor fields to eigenspinoren des ladungskonjugationsoperators

Dual-helicity eigenspinors of the charge conjugation operator [eigenspinoren des ladungskonjugationsoperators (ELKO) spinor fields] belong-together with Majorana spinor fields-to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class (5), according to Lounesto spinor field classification based on the relations and values taken by their associated bilinear covariants. There exists only six such disjoint classes: the first three corresponding to Dirac spinor fields, and the other three, respectively, corresponding to flagpole, flag-dipole, and Weyl spinor fields. This paper is devoted to investigate and provide the necessary and sufficient conditions to map Dirac spinor fields to ELKO, in order to naturally extend the standard model to spinor fields possessing mass dimension 1. As ELKO is a prime candidate to describe dark matter, an adequate and necessary formalism is introduced and developed here, to better understand the algebraic, geometric, and physical properties of ELKO spinor fields, and their underlying relationship to Dirac spinor fields. (c) 2007 American Institute of Physics.

Implementación de algoritmos para la manipulación de curvas concoides en diseño geométrico

Este proyecto se enmarca dentro de la Computación Simbólica y de los fundamentos matemáticos del Diseño Geométrico Asistido por ordenador (CAGD). Se abordara uno de los problemas principales en el ámbito del CAGD y que es la manipulación de las Curvas Concoide. La importancia del avance en la manipulación de las curvas concoide radica en el papel fundamental que desempeñan en múltiples aplicaciones en la actualidad dentro de campos de diversa índole tales como la medicina, la óptica, el electromagnetismo, la construcción, etc. El objetivo principal de este proyecto es el diseño e implementación de algoritmos para el estudio, cálculo y manipulación de curvas concoides, utilizando técnicas propias del Calculo Simbólico. Esta implementación se ha programado utilizando el sistema de computación simbólica Maple. El proyecto consiste en dos partes bien diferenciadas, una parte teórica y otra más practica. La primera incluye la descripción geométrica y definición formal de curvas concoide, así como las ideas y propiedades básicas. De forma más precisa, se presenta un estudio matemático sobre el análisis de racionalidad de estas curvas, explicando los algoritmos que serán implementados en las segunda parte, y que constituye el objetivo principal de este proyecto. Para cerrar esta parte, se presenta una pequeña introducción al sistema y a la programación en Maple. Por otro lado, la segunda parte de este proyecto es totalmente original, y en ella el autor desarrolla las implementaciones en Maple de los algoritmos presentados en la parte anterior, así como la creación de un paquete Maple que las recoge. Por último, se crean las paginas de ayudas en el sistema Maple para la correcta utilización del paquete matemático anteriormente mencionado. Una vez terminada la parte de implementación, se aplican los algoritmos implementados a una colección de curvas clásicas conocidas, recogiendo los datos y resultados obtenidos en un atlas de curvas. Finalmente, se presenta una recopilación de las aplicaciones más destacadas en las que las concoides desempeñan un papel importante así como una breve reseña sobre las concoides de superficies, objeto de varios estudios en la actualidad y a los que se considera que el presente proyecto les puede resultar de gran utilidad. Abstract This project is set up in the framework of Symbolic Computation as well as in the implementation of algebraic-geometric problems that arise from Computer Aided Geometric Design (C.A.G.D.) applications. We address problems related to conchoid curves. The importance of these curves is the fundamental role that they play in current applications as medicine, optics, electromagnetism, construction, etc. The main goal of this project is to design and implement some algorithms to solve problems in studying, calculating and generating conchoid curves with symbolic computation techniques. For this purpose, we program our implementations in the symbolic system “Maple". The project consists of two differentiated parts, one more theoretical part and another part more practical. The first one includes the description of conchoid curves as well as the basic ideas about the concept and its basic properties. More precisely, we introduce in this part the mathematical analysis of the rationality of the conchoids, and we present the algorithms that will be implemented. Furthermore, the reader will be brie y introduced in Maple programming. On the other hand, the second part of this project is totally original. In this more practical part, the author presents the implemented algorithms and a Maple package that includes them, as well as their help pages. These implemented procedures will be check and illustrated with some classical and well known curves, collecting the main properties of the conchoid curves obtained in a brief atlas. Finally, a compilation of the most important applications where conchoids play a fundamental role, and a brief introduction to the conchoids of surfaces, subject of several studies today and where this project could be very useful, are presented.

Transmission of image data over digital networks involving mobile terminals

There is a growing demand for data transmission over digital networks involving mobile terminals. An important class of data required for transmission over mobile terminals is image information such as street maps, floor plans and identikit images. This sort of transmission is of particular interest to the service industries such as the Police force, Fire brigade, medical services and other services. These services cannot be applied directly to mobile terminals because of the limited capacity of the mobile channels and the transmission errors caused by the multipath (Rayleigh) fading. In this research, transmission of line diagram images such as floor plans and street maps, over digital networks involving mobile terminals at transmission rates of 2400 bits/s and 4800 bits/s have been studied. A low bit-rate source encoding technique using geometric codes is found to be suitable to represent line diagram images. In geometric encoding, the amount of data required to represent or store the line diagram images is proportional to the image detail. Thus a simple line diagram image would require a small amount of data. To study the effect of transmission errors due to mobile channels on the transmitted images, error sources (error files), which represent mobile channels under different conditions, have been produced using channel modelling techniques. Satisfactory models of the mobile channel have been obtained when compared to the field test measurements. Subjective performance tests have been carried out to evaluate the quality and usefulness of the received line diagram images under various mobile channel conditions. The effect of mobile transmission errors on the quality of the received images has been determined. To improve the quality of the received images under various mobile channel conditions, forward error correcting codes (FEC) with interleaving and automatic repeat request (ARQ) schemes have been proposed. The performance of the error control codes have been evaluated under various mobile channel conditions. It has been shown that a FEC code with interleaving can be used effectively to improve the quality of the received images under normal and severe mobile channel conditions. Under normal channel conditions, similar results have been obtained when using ARQ schemes. However, under severe mobile channel conditions, the FEC code with interleaving shows better performance.

Groups of unstable Adams operations on p-local compact groups

Peer reviewed

A Geometric Approach for Turbo Decoding of Reed-Solomon Codes in QAM Modulation Schemes

This work studies the turbo decoding of Reed-Solomon codes in QAM modulation schemes for additive white Gaussian noise channels (AWGN) by using a geometric approach. Considering the relations between the Galois field elements of the Reed-Solomon code and the symbols combined with their geometric dispositions in the QAM constellation, a turbo decoding algorithm, based on the work of Chase and Pyndiah, is developed. Simulation results show that the performance achieved is similar to the one obtained with the pragmatic approach with binary decomposition and analysis.

Lower Bounds on the State Complexity of Geometric Goppa Codes

We reinterpret the state space dimension equations for geometric Goppa codes. An easy consequence is that if deg G less than or equal to n-2/2 or deg G greater than or equal to n-2/2 + 2g then the state complexity of C-L(D, G) is equal to the Wolf bound. For deg G is an element of [n-1/2, n-3/2 + 2g], we use Clifford's theorem to give a simple lower bound on the state complexity of C-L(D, G). We then derive two further lower bounds on the state space dimensions of C-L(D, G) in terms of the gonality sequence of F/F-q. (The gonality sequence is known for many of the function fields of interest for defining geometric Goppa codes.) One of the gonality bounds uses previous results on the generalised weight hierarchy of C-L(D, G) and one follows in a straightforward way from first principles; often they are equal. For Hermitian codes both gonality bounds are equal to the DLP lower bound on state space dimensions. We conclude by using these results to calculate the DLP lower bound on state complexity for Hermitian codes.

Geometric and Algebraic Aspects of 3D Affine and Projective Structures from Perspective 2D Views

We investigate the differences --- conceptually and algorithmically --- between affine and projective frameworks for the tasks of visual recognition and reconstruction from perspective views. It is shown that an affine invariant exists between any view and a fixed view chosen as a reference view. This implies that for tasks for which a reference view can be chosen, such as in alignment schemes for visual recognition, projective invariants are not really necessary. We then use the affine invariant to derive new algebraic connections between perspective views. It is shown that three perspective views of an object are connected by certain algebraic functions of image coordinates alone (no structure or camera geometry needs to be involved).

ALGEBRAIC AND GEOMETRIC THEORY OF THE TOPOLOGICAL RING OF COLOMBEAU GENERALIZED FUNCTIONS

We continue the investigation of the algebraic and topological structure of the algebra of Colombeau generalized functions with the aim of building up the algebraic basis for the theory of these functions. This was started in a previous work of Aragona and Juriaans, where the algebraic and topological structure of the Colombeau generalized numbers were studied. Here, among other important things, we determine completely the minimal primes of (K) over bar and introduce several invariants of the ideals of 9(Q). The main tools we use are the algebraic results obtained by Aragona and Juriaans and the theory of differential calculus on generalized manifolds developed by Aragona and co-workers. The main achievement of the differential calculus is that all classical objects, such as distributions, become Cl-functions. Our purpose is to build an independent and intrinsic theory for Colombeau generalized functions and place them in a wider context.

Construction of Optimal Linear Codes by Geometric Puncturing

Dedicated to the memory of S.M. Dodunekov (1945–2012)Abstract. Geometric puncturing is a method to construct new codes. ACM Computing Classification System (1998): E.4.