99 research outputs found
Anticodes and error-correcting for digital data transmission
The work reported in this thesis is an investigation in the field of error-control coding. This subject is concerned with increasing the reliability of digital data transmission through a noisy medium, by coding the transmitted data. In this respect, an extension and development of a method for finding optimum and near-optimum codes, using N.m digital arrays known as anticodes, is established and described. The anticodes, which have opposite properties to their complementary related error-control codes, are disjoined fron the original maximal-length code, known as the parent anticode, to leave good linear block codes. The mathematical analysis of the parent anticode and as a result the mathematical analysis of its related anticodes has given some useful insight into the construction of a large number of optimum and near-optimum anticodes resulting respectively in a large number of optimum and near-optimum codes. This work has been devoted to the construction of anticodes from unit basic (small dimension) anticodes by means of various systematic construction and refinement techniques, which simplifies the construction of the associated linear block codes over a wide range of parameters. An extensive list of these anticodes and codes is given in the thesis. The work also has been extended to the construction of anticodes in which the symbols have been chosen from the elements of the finite field GF(q), and, in particular, a large number of optimum and near-optimum codes over GF(3) have been found. This generalises the concept of anticodes into the subject of multilevel codes
Eisenstein field BCH codes construction and decoding
First, we will go through the theory behind the Eisenstein field (EF) and its extension field. In contrast, we provide a detailed framework for building BCH codes over the EF in the second stage. BCH codes over the EF are decoded using the Berlekamp-Massey algorithm (BMA) in this article. We investigate the error-correcting capabilities of these codes and provide expressions for minimal distance. We provide researchers and engineers creating and implementing robust error-correcting codes for digital communication systems with detailed information on building, decoding and performance assessment
Easily decoded error correcting codes
This thesis is concerned with the decoding aspect of linear block error-correcting codes. When, as in most practical situations, the decoder cost is limited an optimum code may be inferior in performance to a longer sub-optimum code' of the same rate. This consideration is a central theme of the thesis.
The best methods available for decoding short optimum codes and long B.C.H. codes are discussed, in some cases new decoding algorithms for the codes are introduced.
Hashim's "Nested" codes are then analysed. The method of nesting codes which was given by Hashim is shown to be optimum - but it is seen that the codes are less easily decoded than was previously thought.
"Conjoined" codes are introduced. It is shown how two codes with identical numbers of information bits may be "conjoined" to give a code with length and minimum distance equal to the sum of the respective parameters of the constituent codes but with the same number of information bits. A very simple decoding algorithm is given for the codes whereby each constituent codeword is decoded and then a decision is made as to the correct decoding. A technique is given for adding more codewords to conjoined codes without unduly increasing the decoder complexity.
Lastly, "Array" codes are described. They are formed by making parity checks over carefully chosen patterns of information bits arranged in a two-dimensional array. Various methods are given for choosing suitable patterns. Some of the resulting codes are self-orthogonal and certain of these have parameters close to the optimum for such codes. A method is given for adding more codewords to array codes, derived from a process of augmentation known for product codes
Modern Cryptography Volume 1
This open access book systematically explores the statistical characteristics of cryptographic systems, the computational complexity theory of cryptographic algorithms and the mathematical principles behind various encryption and decryption algorithms. The theory stems from technology. Based on Shannon's information theory, this book systematically introduces the information theory, statistical characteristics and computational complexity theory of public key cryptography, focusing on the three main algorithms of public key cryptography, RSA, discrete logarithm and elliptic curve cryptosystem. It aims to indicate what it is and why it is. It systematically simplifies and combs the theory and technology of lattice cryptography, which is the greatest feature of this book. It requires a good knowledge in algebra, number theory and probability statistics for readers to read this book. The senior students majoring in mathematics, compulsory for cryptography and science and engineering postgraduates will find this book helpful. It can also be used as the main reference book for researchers in cryptography and cryptographic engineering areas
Understanding binary-Goppa decoding
This paper reviews, from bottom to top, a polynomial-time algorithm to correct t errors in classical binary Goppa codes defined by squarefree degree-t polynomials. The proof is factored through a proof of a simple Reed--Solomon decoder, and the algorithm is simpler than Patterson\u27s algorithm. All algorithm layers are expressed as Sage scripts. The paper also covers the use of decoding inside the Classic McEliece cryptosystem, including reliable recognition of valid inputs
Chaves mais pequenas para criptossistemas de McEliece usando codificadores convolucionais
The arrival of the quantum computing era is a real threat to the confidentiality
and integrity of digital communications. So, it is urgent to develop alternative
cryptographic techniques that are resilient to quantum computing. This is the
goal of pos-quantum cryptography. The code-based cryptosystem called
Classical McEliece Cryptosystem remains one of the most promising postquantum
alternatives. However, the main drawback of this system is that the
public key is much larger than in the other alternatives. In this thesis we study
the algebraic properties of this type of cryptosystems and present a new variant
that uses a convolutional encoder to mask the so-called Generalized Reed-
Solomon code. We conduct a cryptanalysis of this new variant to show that
high levels of security can be achieved using significant smaller keys than in
the existing variants of the McEliece scheme. We illustrate the advantages of
the proposed cryptosystem by presenting several practical examples.A chegada da era da computação quântica é uma ameaça real à
confidencialidade e integridade das comunicações digitais. É, por isso, urgente
desenvolver técnicas criptográficas alternativas que sejam resilientes à
computação quântica. Este é o objetivo da criptografia pós-quântica. O
Criptossistema de McEliece continua a ser uma das alternativas pós-quânticas
mais promissora, contudo, a sua principal desvantagem é o tamanho da chave
pública, uma vez que é muito maior do que o das outras alternativas. Nesta
tese estudamos as propriedades algébricas deste tipo de criptossistemas e
apresentamos uma nova variante que usa um codificador convolucional para
mascarar o código de Generalized Reed-Solomon. Conduzimos uma
criptoanálise dessa nova variante para mostrar que altos níveis de segurança
podem ser alcançados usando uma chave significativamente menor do que as
variantes existentes do esquema de McEliece. Ilustramos, assim, as vantagens
do criptossistema proposto apresentando vários exemplos práticos.Programa Doutoral em Matemátic
Decoding and constructions of codes in rank and Hamming metric
As coding theory plays an important role in data transmission, decoding algorithms for new families of error correction codes are of great interest. This dissertation is dedicated to the decoding algorithms for new families of maximum rank distance (MRD) codes including additive generalized twisted Gabidulin (AGTG) codes and Trombetti-Zhou (TZ) codes, decoding algorithm for Gabidulin codes beyond half the minimum distance and also encoding and decoding algorithms for some new optimal rank metric codes with restrictions.
We propose an interpolation-based decoding algorithm to decode AGTG codes where the decoding problem is reduced to the problem of solving a projective polynomial equation of the form q(x) = xqu+1 +bx+a = 0 for a,b ∈ Fqm. We investigate the zeros of q(x) when gcd(u,m)=1 and proposed a deterministic algorithm to solve a linearized polynomial equation which has a close connection to the zeros of q(x).
An efficient polynomial-time decoding algorithm is proposed for TZ codes. The interpolation-based decoding approach transforms the decoding problem of TZ codes to the problem of solving a quadratic polynomial equation. Two new communication models are defined and using our models we manage to decode Gabidulin codes beyond half the minimum distance by one unit. Our models also allow us to improve the complexity for decoding GTG and AGTG codes.
Besides working on MRD codes, we also work on restricted optimal rank metric codes including symmetric, alternating and Hermitian rank metric codes. Both encoding and decoding algorithms for these optimal families are proposed. In all the decoding algorithms presented in this thesis, the properties of Dickson matrix and the BM algorithm play crucial roles.
We also touch two problems in Hamming metric. For the first problem, some cryptographic properties of Welch permutation polynomial are investigated and we use these properties to determine the weight distribution of a binary linear codes with few weights. For the second one, we introduce two new subfamilies for maximum weight spectrum codes with respect to their weight distribution and then we investigate their properties.Doktorgradsavhandlin
Variable Redundancy Coding for Adaptive Error Control
This thesis is concerned with variable redundancy(VR) error control coding. VR coding is proposed as one method of providing efficient adaptive error control for time-varying digital data transmission links. The VR technique involves using a set of short, easy to implement, block codes; rather than the one code of a fixed redundancy system which is usually inefficient, and complex to decode. With a VR system, efficient data-rate low-power codes are used when channel conditions are good, and very high-power inefficient codes are used when the channel is noisy. The decoder decides which code is required to cope with current conditions, and communicates this decision to the encoder by means of a feedback link. This thesis presents a theoretical and practical investigation of the VR technique, and aims to show that when compared with a fixed redundancy system one or more of the advantages of increased average data throughput, decreased maximum probability of erroneous decoding, and decreased complexity can be realised. This is confirmed by the practical results presented in the thesis, which were obtained from field trials of an experimental VR system operating over the HE’ radio channel, and from computer simulations. One consequence of the research has been the inception of a study of codes with disjoint code books and mutual Hamming distance (initially considered for combatting feedback errors), and this topic is introduced in the thesis
Versatile Error-Control Coding Systems
$NC research reported in this thesis is in the field of error-correcting codes, which has evolved as a very important branch of information theory. The main use of error-correcting codes is to increase the reliability of digital data transmitted through a noisy environment. There are, sometimes, alternative ways of increasing the reliability of data transmission, but coding methods are now competitive in cost and complexity in many cases because of recent advances in technology. The first two chapters of this thesis introduce the subject of error-correcting codes, review some of the published literature in this field and discuss the advantages of various coding techniques. After presenting linear block codes attention is from then on concentrated on cyclic codes, which is the subject of Chapter 3. The first part of Chapter 3 presents the mathematical background necessary for the study of cyclic codes and examines existing methods of encoding and their practical implementation. In the second part of Chapter 3 various ways of decoding cyclic codes are studied and from these considerations, a general decoder for cyclic codes is devised and is presented in Chapter 4. Also, a review of the principal classes of cyclic codes is presented. Chapter 4 describes an experimental system constructed for measuring the performance of cyclic codes initially RC5GI5SCD by random errors and then by bursts of errors. Simulated channels are used both for random and burst errors. A computer simulation of the whole system was made in order to verify the accuracy of the experimental results obtained. Chapter 5 presents the various results obtained with the experimental system and by computer simulation, which allow a comparison of the efficiency of various cyclic codes to be made. Finally, Chapter 6 summarises and discusses the main results of the research and suggests interesting points for future investigation in the area. The main objective of this research is to contribute towards the solution of a fairly wide range of problems arising in the design of efficient coding schemes for practical applications; i.e. a study of coding from an engineering point of view
Algorithms for Linearly Recurrent Sequences of Truncated Polynomials
Linear recurrent sequences are those whose elements are defined as linear combinations of preceding elements, and finding recurrence relations is a fundamental problem in computer algebra. In this paper, we focus on sequences whose elements are vectors over the ring A = K[x]/ of truncated polynomials. Finding the ideal of their recurrence relations has applications such as the computation of minimal polynomials and determinants of sparse matrices over A. We present three methods for finding this ideal: a Berlekamp-Massey-like approach due to Kurakin, one which computes the kernel of some block-Hankel matrix over A via a minimal approximant basis, and one based on bivariate Pade approximation. We propose complexity improvements for the first two methods, respectively by avoiding the computation of redundant relations and by exploiting the Hankel structure to compress the approximation problem. Then we confirm these improvements empirically through a C++ implementation, and we discuss the above-mentioned applications
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