Reticulados q-ários
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Data
2024-03-22
Autores
Gonçalves, Lucas Eduardo Nogueira [UNIFESP]
Orientadores
Jorge, Grasiele Cristiane [UNIFESP]
Tipo
Dissertação de mestrado
Título da Revista
ISSN da Revista
Título de Volume
Resumo
Um reticulado em R^n é caracterizado como um conjunto discreto de pontos, obtido através de todas as combinações lineares inteiras de um conjunto de vetores linearmente independentes sobre R. Problemas envolvendo reticulados têm intrigado matemáticos desde, pelo menos, o século XVII. A partir da década de 1970, reticulados encontraram aplicação na Teoria dos Códigos Corretores de Erros, que busca por sistemas de comunicações eficientes em canais ruidosos. Na década de 1990, a criptografia baseada em reticulados emergiu como uma nova abordagem para construções de sistemas criptográficos e, desde então, ganhou destaque sob o nome de Criptografia Pós-Quântica. A proposta central desta dissertação é realizar um estudo de famílias de reticulados obtidos através de códigos corretores de erros contidos em Z_q^n. Para isso, exploraremos as Construções A, B, C, C^{*}, D' e D\. Tais construções associam códigos a constelações de sinais em R^n, que sob certas condições são reticulados.
A lattice in R^n is characterized as a discrete set of points, obtained through all the integer linear combinations of a set of linearly independent vectors over R. Problems involving lattices have intrigued mathematicians since at least the 17th century. From the 1970s, lattices were applied to the Theory of Error-Correcting Codes, which searches for efficient communication systems in noisy channels. In the 1990s, lattice-based cryptography emerged as a new approach of constructing cryptographic systems and, since then, has gained prominence in the so Post-Quantum Cryptography. The central proposal of this dissertation is to conduct a study of families of lattices obtained through error-correcting codes contained in Z_q^n. For this, we will explore Constructions A, B, C, C^{*}, D' e D\. Such constructions associate codes with signal constellations in R^n, which under certain conditions are lattices.
A lattice in R^n is characterized as a discrete set of points, obtained through all the integer linear combinations of a set of linearly independent vectors over R. Problems involving lattices have intrigued mathematicians since at least the 17th century. From the 1970s, lattices were applied to the Theory of Error-Correcting Codes, which searches for efficient communication systems in noisy channels. In the 1990s, lattice-based cryptography emerged as a new approach of constructing cryptographic systems and, since then, has gained prominence in the so Post-Quantum Cryptography. The central proposal of this dissertation is to conduct a study of families of lattices obtained through error-correcting codes contained in Z_q^n. For this, we will explore Constructions A, B, C, C^{*}, D' e D\. Such constructions associate codes with signal constellations in R^n, which under certain conditions are lattices.