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.

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.

Spectral Analysis of Bounded Self-adjoint operators - A Linear Algebraic Approach

This thesis Entitled Spectral theory of bounded self-adjoint operators -A linear algebraic approach.The main results of the thesis can be classified as three different approaches to the spectral approximation problems. The truncation method and its perturbed versions are part of the classical linear algebraic approach to the subject. The usage of block Toeplitz-Laurent operators and the matrix valued symbols is considered as a particular example where the linear algebraic techniques are effective in simplifying problems in inverse spectral theory. The abstract approach to the spectral approximation problems via pre-conditioners and Korovkin-type theorems is an attempt to make the computations involved, well conditioned. However, in all these approaches, linear algebra comes as the central object. The objective of this study is to discuss the linear algebraic techniques in the spectral theory of bounded self-adjoint operators on a separable Hilbert space. The usage of truncation method in approximating the bounds of essential spectrum and the discrete spectral values outside these bounds is well known. The spectral gap prediction and related results was proved in the second chapter. The discrete versions of Borg-type theorems, proved in the third chapter, partly overlap with some known results in operator theory. The pure linear algebraic approach is the main novelty of the results proved here.

Development of Techniques for the Automatic Extraction and Grade Detection of Glioma Tumors from Conventional Brain Magnetic Resonant Images

Cerebral glioma is the most prevalent primary brain tumor, which are classified broadly into low and high grades according to the degree of malignancy. High grade gliomas are highly malignant which possess a poor prognosis, and the patients survive less than eighteen months after diagnosis. Low grade gliomas are slow growing, least malignant and has better response to therapy. To date, histological grading is used as the standard technique for diagnosis, treatment planning and survival prediction. The main objective of this thesis is to propose novel methods for automatic extraction of low and high grade glioma and other brain tissues, grade detection techniques for glioma using conventional magnetic resonance imaging (MRI) modalities and 3D modelling of glioma from segmented tumor slices in order to assess the growth rate of tumors. Two new methods are developed for extracting tumor regions, of which the second method, named as Adaptive Gray level Algebraic set Segmentation Algorithm (AGASA) can also extract white matter and grey matter from T1 FLAIR an T2 weighted images. The methods were validated with manual Ground truth images, which showed promising results. The developed methods were compared with widely used Fuzzy c-means clustering technique and the robustness of the algorithm with respect to noise is also checked for different noise levels. Image texture can provide significant information on the (ab)normality of tissue, and this thesis expands this idea to tumour texture grading and detection. Based on the thresholds of discriminant first order and gray level cooccurrence matrix based second order statistical features three feature sets were formulated and a decision system was developed for grade detection of glioma from conventional T2 weighted MRI modality.The quantitative performance analysis using ROC curve showed 99.03% accuracy for distinguishing between advanced (aggressive) and early stage (non-aggressive) malignant glioma. The developed brain texture analysis techniques can improve the physician’s ability to detect and analyse pathologies leading to a more reliable diagnosis and treatment of disease. The segmented tumors were also used for volumetric modelling of tumors which can provide an idea of the growth rate of tumor; this can be used for assessing response to therapy and patient prognosis.

Automatic Detection and Classification of Glioma Tumors using Statistical Features

The characterization and grading of glioma tumors, via image derived features, for diagnosis, prognosis, and treatment response has been an active research area in medical image computing. This paper presents a novel method for automatic detection and classification of glioma from conventional T2 weighted MR images. Automatic detection of the tumor was established using newly developed method called Adaptive Gray level Algebraic set Segmentation Algorithm (AGASA).Statistical Features were extracted from the detected tumor texture using first order statistics and gray level co-occurrence matrix (GLCM) based second order statistical methods. Statistical significance of the features was determined by t-test and its corresponding p-value. A decision system was developed for the grade detection of glioma using these selected features and its p-value. The detection performance of the decision system was validated using the receiver operating characteristic (ROC) curve. The diagnosis and grading of glioma using this non-invasive method can contribute promising results in medical image computing