41 research outputs found
Convolutional Goppa Codes
We define Convolutional Goppa Codes over algebraic curves and construct their
corresponding dual codes. Examples over the projective line and over elliptic
curves are described, obtaining in particular some Maximum-Distance Separable
(MDS) convolutional codes.Comment: 8 pages, submitted to IEEE Trans. Inform. Theor
Algebraic geometry constructions of convolutional codes
Las técnicas de geometría algebraica para construir códigos lineales pueden ser aplicados a la construcción de códigos convolucionales, usando curvas algebraicas sobre los campos de función. En este sentido se construyen códigos Goppa convolucionales y se provee un sistema para construir códigos convolucionales con propiedades prescritas.Algebraic-geometric techniques to construct linear codes can be appliedto construct convolutional codes, using algebraic curves over functionfields. In this way we construct convolutional Goppa codes and providea systematic way for constructing convolutional codes with prescribedproperties. We study convolutional Goppa codes defined by the projec-tive line and elliptic curves in detail
Variations of the McEliece Cryptosystem
Two variations of the McEliece cryptosystem are presented. The first one is
based on a relaxation of the column permutation in the classical McEliece
scrambling process. This is done in such a way that the Hamming weight of the
error, added in the encryption process, can be controlled so that efficient
decryption remains possible. The second variation is based on the use of
spatially coupled moderate-density parity-check codes as secret codes. These
codes are known for their excellent error-correction performance and allow for
a relatively low key size in the cryptosystem. For both variants the security
with respect to known attacks is discussed
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
Error Correcting Codes on Algebraic Surfaces
Error correcting codes are defined and important parameters for a code are
explained. Parameters of new codes constructed on algebraic surfaces are
studied. In particular, codes resulting from blowing up points in \proj^2 are
briefly studied, then codes resulting from ruled surfaces are covered. Codes
resulting from ruled surfaces over curves of genus 0 are completely analyzed,
and some codes are discovered that are better than direct product Reed Solomon
codes of similar length. Ruled surfaces over genus 1 curves are also studied,
but not all classes are completely analyzed. However, in this case a family of
codes are found that are comparable in performance to the direct product code
of a Reed Solomon code and a Goppa code. Some further work is done on surfaces
from higher genus curves, but there remains much work to be done in this
direction to understand fully the resulting codes. Codes resulting from blowing
points on surfaces are also studied, obtaining necessary parameters for
constructing infinite families of such codes.
Also included is a paper giving explicit formulas for curves with more
\field{q}-rational points than were previously known for certain combinations
of field size and genus. Some upper bounds are now known to be optimal from
these examples.Comment: This is Chris Lomont's PhD thesis about error correcting codes from
algebriac surface
Recommended from our members
Contemporary Coding Theory
Coding Theory naturally lies at the intersection of a large number
of disciplines in pure and applied mathematics. A multitude of
methods and means has been designed to construct, analyze, and
decode the resulting codes for communication. This has suggested to
bring together researchers in a variety of disciplines within
Mathematics, Computer Science, and Electrical Engineering, in order
to cross-fertilize generation of new ideas and force global
advancement of the field. Areas to be covered are Network Coding,
Subspace Designs, General Algebraic Coding Theory, Distributed
Storage and Private Information Retrieval (PIR), as well as
Code-Based Cryptography