Algebraic lattices and applications to codes and cryptography

Orientador: Sueli Irene Rodrigues CostaTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Neste trabalho estudamos a aplicação de reticulados algébricos a diferentes contextos. Ao todo, quatro objetivos norteiam esta tese. O primeiro objetivo consiste na construção de reticulados com boa densidade de centro via submódulos de anéis de inteiros de corpos de números algébricos. Nesse contexto, concluímos a construção algébrica do reticulado $D_n$, para qualquer $n$, e estudamos a sua distância produto mínima, bem como a de $\mathbb{Z}^n$. Além disso, calculamos a expressão da forma traço associada a corpos de números abelianos de grau primo ímpar ramificado, a qual está relacionada à densidade de centro de reticulados algébricos obtidos via o mergulho de Minkowski. O segundo objetivo se trata da análise de situações em que reticulados algébricos são bem arredondados. Provamos que em cada dimensão prima ímpar existem infinitos reticulados algébricos não equivalentes entre si que são bem arredondados. O terceiro objetivo é apresentar a aplicabilidade e a atualidade dos reticulados algébricos no contexto da criptografia pós-quântica. Além de resumir os avanços recentes da criptografia via reticulados, propomos a utilização de reticulados algébricos obtidos via o mergulho torcido a este contexto e provamos que a dificuldade de quebra da segurança do sistema proposto está associada à dificuldade de solucionar o problema anel-LWE. Por fim, o quarto objetivo trata do estudo dos reticulados logarítmicos, com especial destaque ao raio de cobertura através das unidades de qualquer corpo ciclotômico. Calculamos uma cota superior para o raio de cobertura de reticulados logarítmicos construídos através desses corpos. Nas abordagens dentro dos quatro propósitos acima fica ressaltado que ferramentas algébricas vêm contribuindo de forma eficaz para a produção de reticulados aplicáveis a diversos contextos em teoria de códigos e criptografiaAbstract: In this work we study applications of algebraic lattices in different contexts. Four goals guide this PhD thesis. The first goal is the construction of lattices with great center density via submodules of the ring of integers of algebraic number fields. In this approach, we obtain the algebraic construction of the lattice $D_n$, for all $n$, and study its minimum product distance, as well as of the lattice $\mathbb{Z}^n$. Besides, we calculate the expression of the trace form associated with abelian number fields of ramified odd prime degree, which is related to the center density of algebraic lattices obtained via the Minkowski embedding. The second goal is the analysis of cases which provide well rounded algebraic lattices. We prove that for each odd prime dimension there exist infinitely many non-equivalent algebraic lattices which are well rounded. The third goal is to present the application of algebraic lattices in the context of the so called post-quantum cryptography. We resume recent advances of lattice cryptography, propose the use of algebraic lattices coming from twisted embedding in this context and prove that the hardness of broking the security of the proposed system is related to the hardness to solve the ring-LWE problem. The fourth goal is the study of logarithmic lattices, specially the analysis of the covering radius of those obtained from units of cyclotomic number fields. We calculate an upper bound of the covering radius of the logarithmic lattices constructed from these fields. In the four objectives described above it is stressed that algebraic tools have good contributions to produce lattices used in coding theory and cryptographyDoutoradoMatematicaDoutor em MatemáticaCAPE

Minimal group codes over alternating groups

In this work we show that every minimal code in a semisimple group algebra $\mathbb{F}_qG$ is essential if $G$ is a simple group. Since the alternating group $A_n$ is simple if $n=3$ or $n\geq 5$, we present some examples of minimal codes in $\mathbb{F}_qA_n$. For this purpose, if $char(\mathbb{F}_q)> n$, we present the Wedderburn-Artin decomposition of $\mathbb{F}_qS_n$ and $\mathbb{F}_qA_n$ and explicit some of the centrally primitive idempotents of $\mathbb{F}_qS_n$ and $\mathbb{F}_qA_n$.Comment: 16 page

Trace Forms of Certain Subfields of Cyclotomic Fields and Applications

In this work, we present a explicit trace forms for maximal real subfields of cyclotomic fields as tools for constructing algebraic lattices in Euclidean space with optimal center density. We also obtain a closed formula for the Gram matrix of algebraic lattices obtained from these subfields. The obtained lattices are rotated versions of the lattices Λ9,Λ10\Lambda_9, \Lambda_{10}Λ​9​​,Λ​10​​ and Λ11\Lambda_{11}Λ​11​​ and they are images of Z\mathbb{Z}Z-submodules of rings of integers under the twisted homomorphism, and these constructions, as algebraic lattices, are new in the literature. We also obtain algebraic lattices in odd dimensions up to 777 over real subfields, calculate their minimum product distance and compare with those known in literatura, since lattices constructed over real subfields have full diversity

In Praise of Twisted Embeddings

Our main result in this work is the extension of the Ring-LWE problem in lattice-based cryptography to include algebraic lattices, realized through twisted embeddings. We define the class of problems Twisted Ring-LWE, which replaces the canonical embedding by an extended form. We prove that our generalization for Ring-LWE is secure by providing a security reduction from Ring-LWE to Twisted Ring-LWE in both search and decision forms. It is also shown that the addition of a new parameter, the torsion factor defining the twisted embedding, does not affect the asymptotic approximation factors in the worst-case to average-case reductions. Thus, Twisted Ring-LWE maintains the consolidated hardness guarantee of Ring-LWE and increases the existing scope of algebraic lattices that can be considered for cryptographic applications. Additionally, we expand on the results of Ducas and Durmus (Public-Key Cryptography, 2012) on spherical Gaussian distributions to the proposed class of lattices under certain restrictions. Thus, sampling from a spherical Gaussian distribution can be done directly in the respective number field, while maintaining its shape and standard deviation when seen in $\mathbb{R}^n$ via twisted embeddings

Pervasive gaps in Amazonian ecological research

Biodiversity loss is one of the main challenges of our time,1,2 and attempts to address it require a clear un derstanding of how ecological communities respond to environmental change across time and space.3,4 While the increasing availability of global databases on ecological communities has advanced our knowledge of biodiversity sensitivity to environmental changes,5–7 vast areas of the tropics remain understudied.8–11 In the American tropics, Amazonia stands out as the world’s most diverse rainforest and the primary source of Neotropical biodiversity,12 but it remains among the least known forests in America and is often underrepre sented in biodiversity databases.13–15 To worsen this situation, human-induced modifications16,17 may elim inate pieces of the Amazon’s biodiversity puzzle before we can use them to understand how ecological com munities are responding. To increase generalization and applicability of biodiversity knowledge,18,19 it is thus crucial to reduce biases in ecological research, particularly in regions projected to face the most pronounced environmental changes. We integrate ecological community metadata of 7,694 sampling sites for multiple or ganism groups in a machine learning model framework to map the research probability across the Brazilian Amazonia, while identifying the region’s vulnerability to environmental change. 15%–18% of the most ne glected areas in ecological research are expected to experience severe climate or land use changes by 2050. This means that unless we take immediate action, we will not be able to establish their current status, much less monitor how it is changing and what is being lostinfo:eu-repo/semantics/publishedVersio

Impact of COVID-19 on cardiovascular testing in the United States versus the rest of the world

Objectives: This study sought to quantify and compare the decline in volumes of cardiovascular procedures between the United States and non-US institutions during the early phase of the coronavirus disease-2019 (COVID-19) pandemic. Background: The COVID-19 pandemic has disrupted the care of many non-COVID-19 illnesses. Reductions in diagnostic cardiovascular testing around the world have led to concerns over the implications of reduced testing for cardiovascular disease (CVD) morbidity and mortality. Methods: Data were submitted to the INCAPS-COVID (International Atomic Energy Agency Non-Invasive Cardiology Protocols Study of COVID-19), a multinational registry comprising 909 institutions in 108 countries (including 155 facilities in 40 U.S. states), assessing the impact of the COVID-19 pandemic on volumes of diagnostic cardiovascular procedures. Data were obtained for April 2020 and compared with volumes of baseline procedures from March 2019. We compared laboratory characteristics, practices, and procedure volumes between U.S. and non-U.S. facilities and between U.S. geographic regions and identified factors associated with volume reduction in the United States. Results: Reductions in the volumes of procedures in the United States were similar to those in non-U.S. facilities (68% vs. 63%, respectively; p = 0.237), although U.S. facilities reported greater reductions in invasive coronary angiography (69% vs. 53%, respectively; p < 0.001). Significantly more U.S. facilities reported increased use of telehealth and patient screening measures than non-U.S. facilities, such as temperature checks, symptom screenings, and COVID-19 testing. Reductions in volumes of procedures differed between U.S. regions, with larger declines observed in the Northeast (76%) and Midwest (74%) than in the South (62%) and West (44%). Prevalence of COVID-19, staff redeployments, outpatient centers, and urban centers were associated with greater reductions in volume in U.S. facilities in a multivariable analysis. Conclusions: We observed marked reductions in U.S. cardiovascular testing in the early phase of the pandemic and significant variability between U.S. regions. The association between reductions of volumes and COVID-19 prevalence in the United States highlighted the need for proactive efforts to maintain access to cardiovascular testing in areas most affected by outbreaks of COVID-19 infection

Search for dark matter produced in association with bottom or top quarks in √s = 13 TeV pp collisions with the ATLAS detector

A search for weakly interacting massive particle dark matter produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and miss- ing transverse momentum are considered. The analysis uses 36.1 fb−1 of proton–proton collision data recorded by the ATLAS experiment at √s = 13 TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are in- terpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour- neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross- section of 300 times the predicted rate for mediators with masses between 10 and 50 GeV and assuming a dark-matter mass of 1 GeV and unitary coupling. Constraints on colour- charged scalar simplified models are also presented. Assuming a dark-matter particle mass of 35 GeV, mediator particles with mass below 1.1 TeV are excluded for couplings yielding a dark-matter relic density consistent with measurements

ATLANTIC-PRIMATES: a dataset of communities and occurrences of primates in the Atlantic Forests of South America

Primates play an important role in ecosystem functioning and offer critical insights into human evolution, biology, behavior, and emerging infectious diseases. There are 26 primate species in the Atlantic Forests of South America, 19 of them endemic. We compiled a dataset of 5,472 georeferenced locations of 26 native and 1 introduced primate species, as hybrids in the genera Callithrix and Alouatta. The dataset includes 700 primate communities, 8,121 single species occurrences and 714 estimates of primate population sizes, covering most natural forest types of the tropical and subtropical Atlantic Forest of Brazil, Paraguay and Argentina and some other biomes. On average, primate communities of the Atlantic Forest harbor 2 ± 1 species (range = 1–6). However, about 40% of primate communities contain only one species. Alouatta guariba (N = 2,188 records) and Sapajus nigritus (N = 1,127) were the species with the most records. Callicebus barbarabrownae (N = 35), Leontopithecus caissara (N = 38), and Sapajus libidinosus (N = 41) were the species with the least records. Recorded primate densities varied from 0.004 individuals/km 2 (Alouatta guariba at Fragmento do Bugre, Paraná, Brazil) to 400 individuals/km 2 (Alouatta caraya in Santiago, Rio Grande do Sul, Brazil). Our dataset reflects disparity between the numerous primate census conducted in the Atlantic Forest, in contrast to the scarcity of estimates of population sizes and densities. With these data, researchers can develop different macroecological and regional level studies, focusing on communities, populations, species co-occurrence and distribution patterns. Moreover, the data can also be used to assess the consequences of fragmentation, defaunation, and disease outbreaks on different ecological processes, such as trophic cascades, species invasion or extinction, and community dynamics. There are no copyright restrictions. Please cite this Data Paper when the data are used in publications. We also request that researchers and teachers inform us of how they are using the data. © 2018 by the The Authors. Ecology © 2018 The Ecological Society of Americ

Pervasive gaps in Amazonian ecological research

Biodiversity loss is one of the main challenges of our time,1,2 and attempts to address it require a clear understanding of how ecological communities respond to environmental change across time and space.3,4 While the increasing availability of global databases on ecological communities has advanced our knowledge of biodiversity sensitivity to environmental changes,5,6,7 vast areas of the tropics remain understudied.8,9,10,11 In the American tropics, Amazonia stands out as the world's most diverse rainforest and the primary source of Neotropical biodiversity,12 but it remains among the least known forests in America and is often underrepresented in biodiversity databases.13,14,15 To worsen this situation, human-induced modifications16,17 may eliminate pieces of the Amazon's biodiversity puzzle before we can use them to understand how ecological communities are responding. To increase generalization and applicability of biodiversity knowledge,18,19 it is thus crucial to reduce biases in ecological research, particularly in regions projected to face the most pronounced environmental changes. We integrate ecological community metadata of 7,694 sampling sites for multiple organism groups in a machine learning model framework to map the research probability across the Brazilian Amazonia, while identifying the region's vulnerability to environmental change. 15%–18% of the most neglected areas in ecological research are expected to experience severe climate or land use changes by 2050. This means that unless we take immediate action, we will not be able to establish their current status, much less monitor how it is changing and what is being lost