We define new metrics for linear codes over the ring Fq[u]/(ut) via an Fq-module monomorphism on linear codes over Fq. The construction generalizes the Gray map, Gray weight, and Lee weight; and the technique allows us to find some new optimal linear codes and their weight enumerator polynomial.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-09T18:02:46Z No. of bitstreams: 1 LINO MARCOS DA SILVA - DISSERTAÇÃO PPGMAT 2006..pdf: 606293 bytes, checksum: f6729428e1a4d16d1b38704fe9b418a4 (MD5) / Made available in DSpace on 2018-07-09T18:02:46Z (GMT). No. of bitstreams: 1 LINO MARCOS DA SILVA - DISSERTAÇÃO PPGMAT 2006..pdf: 606293 bytes, checksum: f6729428e1a4d16d1b38704fe9b418a4 (MD5) Previous issue date: 2006-03 / Capes / Neste trabalho apresentamos algumas relações entre matróides e códigos lineares. Estudamos vários invariantes numéricos de matróides e vemos que este é um dos muitos aspectos de teoria das matróides que tiveram origem em teoria dos grafos. Analisamos uma classe especial de tais invariantes: os invariantes Tutte-Grothendieck. Mostramos que o polinômio de Tutte é o invariante T-Guniversal (Brilawski,1972) e o relacionamos à teoria dos códigos mostrando que a distribuição de pesos de palavras-código em um código linear é um invariante T-G generalizado (Greene,1976). / In this work we present a relation between matroid and linear codes. Numericals invariants for matroids is one the many topics of matroid theory having its origins graph theory. The Tutte Polynomial of the matroid play a role very important in various problems concerned with such invariants. In 1972 Brylawski showed that the Tutte Polynomial is a T-G invariant. In 1976, Greene established a relation among linear codes and the Tutte Polynomial showing that the distribuition of codeweigths in a linear codes is a generalized T-G invariant.

In this thesis, we review basic properties of linear codes and lattices with a certain focus on their interplay. In particular, we focus on the analogous con- structions of cyclic codes and cyclic lattices. We start out with a brief overview of the basic theory and properties of linear codes. We then demonstrate the construction of cyclic codes and emphasize their importance in error-correcting coding theory. Next we survey properties of lattices, focusing on algorithmic lattice problems, exhibit the construction of cyclic lattices and discuss their applications in cryptography. We emphasize the similarity and common prop- erties of the two cyclic constructions.

No description available.

We provide a quantum algorithm for the exact evaluation of the Potts partition function for a certain class of restricted instances of graphs that correspond to irreducible cyclic codes. We use the same approach to demonstrate that quantum computers can provide an exponential speed up over the best classical algorithms for the exact evaluation of the weight enumerator polynomial for a family of classical cyclic codes. In addition to this we also provide an efficient quantum approximation algorithm for a function (signed-Euler generating function) closely related to the Ising partition function and demonstrate that this problem is BQP-complete. We accomplish the above for the Potts partition function by using a series of links between Gauss sums, classical coding theory, graph theory and the partition function. We exploit the fact that there exists an efficient approximation algorithm for Gauss sums and the fact that this problem is equivalent in complexity to evaluating discrete log. A theorem of McEliece allows one to turn the Gauss sum approximation into an exact evaluation of the Potts partition function. Stripping the physics from this result leaves one with the result for the weight enumerator polynomial. The result for the approximation of the signed-Euler generating function was accomplished by fashioning a new mapping between quantum circuits and graphs. The mapping provided us with a way of relating the cycle structure of graphs with quantum circuits. Using a slight variant of this mapping, we present the final result of this thesis which presents a way of testing families of quantum circuits for their classical simulatability. We thus provide an efficient way of deciding whether a quantum circuit provides any additional computational power over classical computation and this is achieved by exploiting the fact that planar instances of the Ising partition function (with no external magnetic field) can be efficiently classically computed.

We provide a quantum algorithm for the exact evaluation of the Potts partition function for a certain class of restricted instances of graphs that correspond to irreducible cyclic codes. We use the same approach to demonstrate that quantum computers can provide an exponential speed up over the best classical algorithms for the exact evaluation of the weight enumerator polynomial for a family of classical cyclic codes. In addition to this we also provide an efficient quantum approximation algorithm for a function (signed-Euler generating function) closely related to the Ising partition function and demonstrate that this problem is BQP-complete. We accomplish the above for the Potts partition function by using a series of links between Gauss sums, classical coding theory, graph theory and the partition function. We exploit the fact that there exists an efficient approximation algorithm for Gauss sums and the fact that this problem is equivalent in complexity to evaluating discrete log. A theorem of McEliece allows one to turn the Gauss sum approximation into an exact evaluation of the Potts partition function. Stripping the physics from this result leaves one with the result for the weight enumerator polynomial. The result for the approximation of the signed-Euler generating function was accomplished by fashioning a new mapping between quantum circuits and graphs. The mapping provided us with a way of relating the cycle structure of graphs with quantum circuits. Using a slight variant of this mapping, we present the final result of this thesis which presents a way of testing families of quantum circuits for their classical simulatability. We thus provide an efficient way of deciding whether a quantum circuit provides any additional computational power over classical computation and this is achieved by exploiting the fact that planar instances of the Ising partition function (with no external magnetic field) can be efficiently classically computed.

[pt] Começamos estudando uma classe particular de códigos lineares, os chamados códigos de Goppa que são obtidos calculando o valor de certas funções em pontos de Kn, onde K é um corpo finito. Apresentamos uma generalização desta construção e definimos códigos de avaliação sobre K- ágebras satisfazendo certas propriedades. Para estes códigos, descrevemos um algoritmo de decodificação e mostramos que se considerarmos os códigos de Goppa em um ponto como exemplo desta nova construção, o algoritmo corrige mais erros do que o algoritmo clássico para os códigos de Goppa. / [en] We begin studying a certain type of linear code the so-called Goppa codes. These codes are constructed by taking the evaluation of certain functions at points in Kn, where K is a finite field. As a generalization of this construction, we introduce the so-called evaluation codes defined over K-algebras satisfying some properties. For these codes, we describe a decoding algorithm and we show that if we consider classical one-point Goppa codes as an example of the new construction, this algorithm correct more errors that the classical algorithm for Goppa codes.

Este trabalho, elaborado por meio de pesquisa bibliográfica, apresenta um apanhado sobre os dígitos verificadores presentes no Cadastro de Pessoas Físicas (CPF), no código de barras, e no sistema ISBN; faz uma introdução sobre a métrica de Hamming e os códigos corretores de erros; cita a classe de códigos mais utilizada, que são os códigos lineares, e deixa a sugestão de uma proposta pedagógica para professores de matemática aplicarem no Ensino Fundamental, podendo ser ajustada também para o Ensino Médio. No apêndice A, são propostos alguns exercícios que podem ser trabalhados com os alunos em sala de aula. / This work, based on the attached references, presents an overview of the check digits that appear in the Brazilian document CPF, in the bar code and the ISBN system. Moreover, it makes an introduction to the Hamming metric and error-correcting codes. In particular, some considerations about linear codes are done and it makes a suggestion of a pedagogical approach to apply it in middle school and can also be adjusted to high school. In the Appendix A are proposed some exercises to students.

Depuis la fin des années 90, les cryptosystèmes implantés sur carte à puce doivent faire face à deux grandes catégories d'attaques : les attaques par canaux cachés et les attaques par injection de fautes. Pour s'en prémunir, des contre-mesures sont élaborées, puis validées en considérant un modèle d'attaquant bien défini. Les travaux réalisés dans cette thèse se concentrent sur la protection des cryptosystèmes symétriques contre les attaques par canaux cachés. Plus précisément, on s'intéresse aux contre-mesures de masquage permettant de se prémunir des attaques statistiques d'ordre supérieur pour lesquelles un attaquant est capable de cibler t valeurs intermédiaires. Après avoir rappelé l'analogie entre les contre-mesures de masquage et les schémas de partage de secret, on présente la construction des schémas de partage de secret à partir de codes linéaires, introduite par James L. Massey en 1993. En adaptant cette construction et des outils issus du calcul multi-parties, on propose une méthode générique de contre-mesure de masquage résistante aux attaques statistiques d'ordre supérieur. De plus, en fonction des cryptosystèmes à protéger et donc des opérations à effectuer, cette solution permet d'optimiserle coût induit par les contre-mesures en sélectionnant les codes les plus adéquats. Dans cette optique, on propose deux contre-mesures de masquage pour implanter le cryptosystème AES. La première est basée sur une famille de code d'évaluation proche de celle utilisée pour le schéma de partage de secret de Shamir, tandis que la seconde considéré la famille des codes auto-duaux et faiblement auto-duaux ayant leur matrice génératrice à coefficient sur F2 ou F4. Ces deux alternatives se révèlent plus efficaces que les contremesures de masquage publiées en 2011 et basées sur le schéma de partage de secret de Shamir. De plus la seconde s'avère compétitive pour t=1 comparée aux solutions usuelles. / Since the late 90s, the implementation of cryptosystems on smart card faces two kinds of attacks : side-channel attacks and fault injection attacks. Countermeasures are then developed and validated by considering a well-defined attacker model. This thesis focuses on the protection of symmetric cryptosystems against side-channel attacks. Specifically, we are interested in masking countermeasures in order to tackle high-order attacks for which an attacker is capable of targeting t intermediate values. After recalling the analogy between masking countermeasures and secret sharing schemes, the construction of secret sharing schemes from linear codes introduced by James L. Massey in 1993 is presented.By adapting this construction together with tools from the field of Multi-Party Computation, we propose a generic masking countermeasure resistant to high-order attacks. Furthermore, depending on the cryptosystem to protect, this solution optimizes the cost of the countermeasure by selecting the most appropriate code. In this context, we propose two countermeasures to implement the AES cryptosystem. The first is based on a family of evaluation codes similar to the Reed Solomon code used in the secret sharing scheme of Shamir. The second considers the family of self-dual and self-orthogonal codes generated by a matrix defined over GF(2) or GF(4). These two alternatives are more effective than masking countermeasures from 2011 based on Shamir's secret sharing scheme. Moreover, for t=1, the second solution is competitive with usual solutions.

Este trabalho, elaborado por meio de pesquisa bibliográfica, apresenta um apanhado sobre os dígitos verificadores presentes no Cadastro de Pessoas Físicas (CPF), no código de barras, e no sistema ISBN; faz uma introdução sobre a métrica de Hamming e os códigos corretores de erros; cita a classe de códigos mais utilizada, que são os códigos lineares, e deixa a sugestão de uma proposta pedagógica para professores de matemática aplicarem no Ensino Fundamental, podendo ser ajustada também para o Ensino Médio. No apêndice A, são propostos alguns exercícios que podem ser trabalhados com os alunos em sala de aula. / This work, based on the attached references, presents an overview of the check digits that appear in the Brazilian document CPF, in the bar code and the ISBN system. Moreover, it makes an introduction to the Hamming metric and error-correcting codes. In particular, some considerations about linear codes are done and it makes a suggestion of a pedagogical approach to apply it in middle school and can also be adjusted to high school. In the Appendix A are proposed some exercises to students.