Cryptography fascinates people of all generations and is increasingly presented as an example for the relevance and application of the mathematical sciences. Indeed, many principles of modern cryptography can be described at a secondary school level. In this context, the mathematical background is often only sparingly shown. In the worst case, giving mathematics this character of a tool reduces the application of mathematical insights to the message ”cryptography contains math”. This paper examines the question as to what else cryptography can offer to mathematics education. Using the RSA cryptosystem and related content, specific mathematical competencies are highlighted that complement standard teaching, can be taught with cryptography as an example, and extend and deepen key mathematical concepts.

Applications of finite-field wavelets, paraunitary matrices, and multivariate polynomials in the design of efficient cryptographic algorithms for resource-limited devices and wireless sensor nodes is the main topic of this thesis. In this research, multivariate paraunitary matrices over fields of characteristic two are of special importance. Therefore, the factorization of their bivariate counterpart into the product of fully-parameterized building blocks was studied. Result were a two-level factorization algorithm and new building blocks over the ring of polynomials that allow a complete first-level factorization. One of the contributions in this thesis was a completely new design for self-synchronizing stream ciphers based on wavelets over fields of characteristic two. Since these wavelets can be efficiently designed and implemented using paraunitary matrices, the designed cipher is highly efficient in terms of encryption and decryption complexities. The cryptanalysis of the proposed cipher did not reveal any vulnerabilities to the current state of the art attacks developed for stream ciphers. A completely novel framework for the design of multivariate asymmetric cryptosystems (based on paraunitary matrices) is a main contribution in this thesis. Using algebraic properties of paraunitary matrices, the computational security of systems designed based on this framework was studied. It was proved, for the first time, that breaking any instance of such systems provides a positive answer to an algebraic longstanding (non- computational) open problem. Therefore, the proposed framework certainly is an improvement toward the design of provably secure multivariate cryptosystems. Using this approach, a public-key cryptosystem and a digital signature scheme was proposed. Considering the attractiveness of algebraic techniques, their applications in the design of cryptographic algorithms for wireless sensor networks was investigated. A novel key pre-distribution scheme for data confidentiality in sensor networks was proposed. This scheme outperforms all previous designs in terms of network resiliency against the node capture. Theoretical analysis showed improvement over previous schemes and also robustness in design. In addition to key pre-distribution, a location-aware scheme was proposed that provides authenticity and availability for sensor networks. Main ingredients of this scheme are node collaboration for entity authenticity, hash tree for data authenticity, and random network coding for data availability. This scheme is the first one in its category that provides a practical solution to all the aforementioned security services.

From the time of the Greeks, primality testing and factoring have fascinated mathematicians, and for centuries following the Greeks primality testing and factorization were pursued by enthusiasts and professional mathematicians for their intrisic value. There was little practical application. One example application was to determine whether or not the Fermat numbers, that is, numbers of the form F;, = 2'" + 1 were prime. Fermat conjectured that for all n they were prime. For n = 1,2,3,4, the Fermat numbers are prime, but Euler showed that F; was not prime and to date no F,, n 2 5 has been found to be prime. Thus, for nearly 2000 years primality testing and factorization was largely pure mathematics. This all changed in the mid 1970's with the advent of public key cryptography. Large prime numbers are used in generating keys in many public key cryptosystems and the security of many of these cryptosystems depends on the difficulty of factoring numbers with large prime factors. Thus, the race was on to develop new algorithms to determine the primality or otherwise of a given large integer and to determine the factors of given large integers. The development of such algorithms continues today. This thesis develops both of these themes. The first part of this thesis deals with primality testing and after a brief introduction to primality testing a new probabilistic primality algorithm, ALI, is introduced. It is analysed in detail and compared to Fermat and Miller-Rabin primality tests. It is shown that the ALI algorithm is more efficient than the Miller-Rabin algorithm in some aspects. The second part of the thesis deals with factoring and after looking closely at various types of algorithms a new algorithm, RAK, is presented. It is analysed in detail and compared with Fermat factorization. The RAK algorithm is shown to be significantly more efficient than the Fermat factoring algorithm. A number of enhancements is made to the basic RAK algorithm in order to improve its performance. The RAK algorithm with its enhancements is known as IMPROVEDRAK. In conjunction with this work on factorization an improvement to Shor's factoring algorithm is presented. For many integers Shor's algorithm uses a quantum computer multiple times to factor a composite number into its prime factors. It is shown that Shor's alorithm can be modified in a way such that the use of a quantum computer is required just once. The common thread throughout this thesis is the application of factoring and primality testing techniques to integer types which commonly occur in public key cryptosystems. Thus, this thesis contributes not only in the area of pure mathematics but also in the very contemporary area of cryptology.

Zugl.: Bochum, Univ., Diss., 2009

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-20T20:19:30Z No. of bitstreams: 1 JEAN CAMINHA - DISSERTAÇÃO PPGEE 2006..pdf: 1518338 bytes, checksum: e7ba2faff350c19009f55c5eec73fc4a (MD5) / Made available in DSpace on 2018-08-20T20:19:30Z (GMT). No. of bitstreams: 1 JEAN CAMINHA - DISSERTAÇÃO PPGEE 2006..pdf: 1518338 bytes, checksum: e7ba2faff350c19009f55c5eec73fc4a (MD5) Previous issue date: 2006-12-22 / A identificação de objetos que participam de uma arquitetura de processamento de dados é uma preocupação relevante para a segurança das informações e fator decisivo para a utilização de serviços que utilizam dados sensíveis. Este trabalho propõe uma arquitetura para a autenticação de dispositivos móveis utilizando a Infra-estrutura de Chave Pública (PKI) e o Protocolo de Iniciação de Sessões (SIP) de modo a minimizar as limitações de armazenamento e processamento destes aparelhos. / The identification of objects that participates of architecture of data processing is a concern for information security and decisive factor for services that need manipulate sensible data. This work considers architecture for mobile devices authentication through in a Public Key Infrastructure and the Session Initiation Protocol (SIP) as a way to minimize storage and processing limitations of those devices.

Made available in DSpace on 2016-12-01T19:18:49Z (GMT). No. of bitstreams: 1 Fabricio1.pdf: 69759 bytes, checksum: 0fdd39ba777606ce0670212bb86298e2 (MD5) Previous issue date: 2010-06-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The Brazilian Public Key Infrastructure is a set of techniques, architectures, organization, practices and procedures defined by brazilian government and private organizations that support, in overall, the implementation and operation of a certification system. The product of the organizations that form this structure is the digital certificate, a particular electronic document of a person or organization which guarantees the authenticity, integrity and legal validity of documents in electronic form and enables safes electronic transactions. The entities that are part of Brazilian Public Key Infrastructure receive constant audits to verify and control the perfect follow all the rules that are submitted in order to ensure quality and reliability of services offered. Although the structure of entities exist on the basis of digital certification and trust in the electronic documents and digital transactions, their process of auditing and supervision are not using and enjoying fully all the advantages and facilities that the digital certificate, electronic document and information technology and communication provides. This work focuses on the modernization and automation of these processes. In the survey was used a theoretical-empirical approach using the literature review on the subject and research in normative documents of Brazilian Public Key Infrastructure to understand the current status of their audit procedures and be able to develop and propose a new methodology to achieve automated auditing and supervision of entities of the structure, using digital certification and electronic document transactions between the system and its users. Research shows that the new methodology proposal is feasible and can be implemented, but needs more studies to be able to deploy it / A Infraestrutura de Chaves Públicas Brasileira é um conjunto de técnicas, arquiteturas, organização, práticas e procedimentos, definidos pelas organizações governamentais e privadas brasileiras que suportam, em conjunto, a implementação e a operação de um sistema de certificação. O produto das organizações que formam essa estrutura é o certificado digital, um documento eletrônico particular de uma pessoa física ou jurídica que garante a autenticidade, integridade e validade jurídica de documentos em forma eletrônica e possibilita as transações eletrônicas seguras. As entidades que fazem parte da Infraestrutura de Chaves Públicas Brasileira recebem auditorias constantes a fim de verificar e controlar o perfeito seguimento de todas as normas a que estão submetidas, de modo a garantir a qualidade e confiabilidade dos serviços oferecidos. Apesar das entidades da estrutura existirem em função da certificação digital e da confiança nos documentos e nas transações eletrônicas, seus processos de auditoria e fiscalização não utilizam plenamente todas as vantagens e facilidades que o certificado digital, o documento eletrônico e a tecnologia da informação e comunicação proporcionam. O presente trabalho foca na modernização e automação destes processos. Na pesquisa foi utilizada uma abordagem teórico-empírica, utilizando a revisão da literatura sobre o assunto e a pesquisa nos documentos de normatização da Infraestrutura de Chaves Públicas Brasileira para entender a atual situação dos seus processos de auditoria e poder desenvolver e propor uma nova metodologia para a realização automatizada da auditoria e fiscalização das entidades da estrutura, utilizando a certificação digital e o documento eletrônico nas transações entre o sistema e seus usuários. A pesquisa mostra que a nova metodologia proposta é viável e possível de ser implementada, porém necessita de mais estudos para ser possível implantá-la

Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-05-19T15:20:50Z No. of bitstreams: 2 arquivototal.pdf: 1103922 bytes, checksum: fee5e8830b60905917fc3ab1fb8c2aae (MD5) license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) / Approved for entry into archive by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-05-19T15:21:56Z (GMT) No. of bitstreams: 2 arquivototal.pdf: 1103922 bytes, checksum: fee5e8830b60905917fc3ab1fb8c2aae (MD5) license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) / Made available in DSpace on 2015-05-19T15:21:56Z (GMT). No. of bitstreams: 2 arquivototal.pdf: 1103922 bytes, checksum: fee5e8830b60905917fc3ab1fb8c2aae (MD5) license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) Previous issue date: 2013-08-13 / The encryption algorithm written by Egyptian Taher ElGamal computes discrete logarithms with elements of a finite group G Cyclical. These elements have properties that during the study Chapter 1. Knowing the definitions and some properties studied, we can define and compute discrete logarithms, using knowledge of arithmetic and congruence of Remains and Theorem Remainder of Chinese. We will study public key algorithms, in particular the algorithm written by ElGamal, seeking to understand the diffculties presented by it and show its applications in the field of cryptography. We present a sequence of activities, aimed at students of the first grade of high school, targeting the learning of some subjects covered at work. / O algoritmo de criptografia escrito pelo egípcio Taher ElGamal calcula logaritmos discretos com elementos de um Grupo Cíclico finito G. Esses elementos possuem propriedades que estudaremos no decorrer do capítulo 1. Conhecendo as definições e algumas propriedades estudadas, poderemos definir e calcular logaritmos discretos, utilizando conhecimentos da Aritmética dos Restos e Congruências, bem como o Teorema Chinês dos Restos. Vamos estudar algoritmos de chave pública, em particular o algoritmo escrito por ElGamal, buscando entender as dificuldades apresentadas por ele e mostrar suas aplicações no campo da Criptografia. Apresentaremos uma sequencia de atividades, voltadas para estudantes do primeiro ano do Ensino Médio, visando o aprendizado de alguns assuntos abordados no trabalho.

Orientador: Marco Aurelio Amaral Henriques / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-16T22:16:21Z (GMT). No. of bitstreams: 1 Oliveira_MatheusFernandesde_M.pdf: 924070 bytes, checksum: b0355f2150875c0a6c636bf2da2ea8a9 (MD5) Previous issue date: 2010 / Resumo: Emparelhamentos bilineares sobre curvas elípticas são funções matemáticas que viabilizam o desenvolvimento de uma série de novos protocolos criptográficos, entre eles, os criptossistemas baseados em identidades. Esses criptossistemas representam uma nova forma de se implementar criptografia de chaves públicas na qual são atenuadas ou completamente retiradas as restrições relativas ao tipo, tamanho e formato das chaves públicas. Este trabalho apresenta um estudo sobre a implementação de criptossistemas baseados em emparelhamentos bilineares sobre curvas elípticas em cartões inteligentes de oito bits. O trabalho mostra ainda o desenvolvimento de equações específicas baseadas no método conhecido como Montgomery's Ladder para multiplicação escalar de curvas elípticas supersingulares em corpos binários. Estas novas equações tornam o algoritmo mais rápido sem perder suas características de segurança originais. O trabalho apresenta também a técnica de delegação segura de emparelhamentos, na qual um dispositivo computacionalmente restrito, como um cartão inteligente, delega o cálculo do emparelhamento para um dispositivo com maior poder computacional. É proposta uma modificação nesta técnica que diminui o número de operações executadas pelo cartão inteligente / Abstract: Bilinear pairings over elliptic curves are mathematical functions that enable the development of a set of new cryptographic protocols, including the so called identity based cryptosystems. These cryptosystems represent a new way to implement public- key cryptography in such a way that the restrictions related to public keys type, size and format are reduced or completely removed. This work presents a study about implementation of pairing based cryptosystems in 8-bit smart cards. It also presents new equations to be used in Montgomery's Ladder algorithm for scalar multiplication of supersingular ellipitic curves over binary fields. These equations make the algorithm faster without compromising its security characteristics. Finally, it discusses the secure delegation of pairing computation, that enables a computationally limited device, like a smart card, to delegate the computation of pairings to a more powerful device. It is proposed a modification in this technique to decrease the number of operations executed by the smart card / Mestrado / Engenharia de Computação / Mestre em Engenharia Elétrica

Orientador: Ricardo Dahab / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-08T18:08:23Z (GMT). No. of bitstreams: 1 Pedroso_EdsonTessarini_M.pdf: 1601998 bytes, checksum: a37e50088eb3b06977e4d812a3951695 (MD5) Previous issue date: 2006 / Resumo: Grade computacional é um conceito que explora as potencialidades das redes de computadores, com o objetivo específico de disponibilizar camadas virtuais que permitem a um usuário ter acesso a aplicações altamente exigentes, bem como aderir a comunidades virtuais de grande escala, com uma grande diversidade de recursos de computação e de repositórios de informações. Grades computacionais são sistemas de suporte à execução de aplicações paralelas que acoplam recursos heterogêneos distribuídos, oferecendo acesso consistente e barato aos recursos, independente de sua posição geográfica. As tecnologias de grades computacionais possibilitam agregar recursos computacionais variados e dispersos, acelerando a execução de vários processos computacionais. Para melhor entendimento das questões de segurança, principal foco deste trabalho, um estudo geral sobre a grade computacional envolvendo assuntos como arquitetura, funcionalidades, aplicações e serviços, foi realizado com o objetivo de identificar e demonstrar a complexidade existente por trás destes cenários. As exigências de segurança são fundamentais a um projeto de grade computacional. Os componentes de segurança devem fornecer os mecanismos corretos para uma comunicação segura em um ambiente de grade. Sem estes mecanismos, as informações processadas dentro da grade tornam-se vulneráveis. O propósito deste trabalho é a realização de um levantamento sobre as questões de segurança em grade computacional, identificando problemas existentes, soluções, arquiteturas, ferramentas e técnicas aplicadas. Com base nessas informações é possível entender como funcionam os mecanismos de segurança em grade, identificando o que já existe de efetivo e quais as necessidades para que a maturidade e popularidade neste ambiente possam ocorrer / Abstract: Grid computing is a concept that exploits the power of computer networks, with the specific aim of making virtual layers available that allow users to have access to highly demanding applications, as well as to adhere to large scale, highly diverse, virtual communities. Grid computing provides support for the execution of parallel applications, grouping together distributed heterogeneous resources, offering consistent and inexpensive access to them, independently of their geographical location. The technology of grid computing allow the gathering of different and disperse computer resources, accelerating the execution of various computer processes. To better understand security issues in grids, the principal focus of this work, a general study of grid computing including architecture, functionalities, applications and services was done, with the goal of identifying and demonstrating the existent complexity behind this scenery. Security is fundamental to a grid project. Security components must supply the correct mechanisms for secure communication, without which processed information inside a grid becomes vulnerable. The purpose of this work is to survey security issues in grid computing, identifying existent problems, solutions, architectures, tools and techniques. Based on this information it is possible to understand how security mechanisms in grids work, identifying those mechanisms already in place and working, as well as what is needed for the full development of grid computing / Mestrado / Mestre Profissional em Computação

Orientador: Júlio César Lopez Hernández / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-19T05:47:42Z (GMT). No. of bitstreams: 1 Aranha_DiegodeFreitas_D.pdf: 2545815 bytes, checksum: b630a80d0f8be161e6cb7519072882ed (MD5) Previous issue date: 2011 / Resumo: O advento da criptografia assimétrica ou de chave pública possibilitou a aplicação de criptografia em novos cenários, como assinaturas digitais e comércio eletrônico, tornando-a componente vital para o fornecimento de confidencialidade e autenticação em meios de comunicação. Dentre os métodos mais eficientes de criptografia assimétrica, a criptografia de curvas elípticas destaca-se pelos baixos requisitos de armazenamento para chaves e custo computacional para execução. A descoberta relativamente recente da criptografia baseada em emparelhamentos bilineares sobre curvas elípticas permitiu ainda sua flexibilização e a construção de sistemas criptográficos com propriedades inovadoras, como sistemas baseados em identidades e suas variantes. Porém, o custo computacional de criptossistemas baseados em emparelhamentos ainda permanece significativamente maior do que os assimétricos tradicionais, representando um obstáculo para sua adoção, especialmente em dispositivos com recursos limitados. As contribuições deste trabalho objetivam aprimorar o desempenho de criptossistemas baseados em curvas elípticas e emparelhamentos bilineares e consistem em: (i) implementação eficiente de corpos binários em arquiteturas embutidas de 8 bits (microcontroladores presentes em sensores sem fio); (ii) formulação eficiente de aritmética em corpos binários para conjuntos vetoriais de arquiteturas de 64 bits e famílias mais recentes de processadores desktop dotadas de suporte nativo à multiplicação em corpos binários; (iii) técnicas para implementação serial e paralela de curvas elípticas binárias e emparelhamentos bilineares simétricos e assimétricos definidos sobre corpos primos ou binários. Estas contribuições permitiram obter significativos ganhos de desempenho e, conseqüentemente, uma série de recordes de velocidade para o cálculo de diversos algoritmos criptográficos relevantes em arquiteturas modernas que vão de sistemas embarcados de 8 bits a processadores com 8 cores / Abstract: The development of asymmetric or public key cryptography made possible new applications of cryptography such as digital signatures and electronic commerce. Cryptography is now a vital component for providing confidentiality and authentication in communication infra-structures. Elliptic Curve Cryptography is among the most efficient public-key methods because of its low storage and computational requirements. The relatively recent advent of Pairing-Based Cryptography allowed the further construction of flexible and innovative cryptographic solutions like Identity-Based Cryptography and variants. However, the computational cost of pairing-based cryptosystems remains significantly higher than traditional public key cryptosystems and thus an important obstacle for adoption, specially in resource-constrained devices. The main contributions of this work aim to improve the performance of curve-based cryptosystems, consisting of: (i) efficient implementation of binary fields in 8-bit microcontrollers embedded in sensor network nodes; (ii) efficient formulation of binary field arithmetic in terms of vector instructions present in 64-bit architectures, and on the recently-introduced native support for binary field multiplication in the latest Intel microarchitecture families; (iii) techniques for serial and parallel implementation of binary elliptic curves and symmetric and asymmetric pairings defined over prime and binary fields. These contributions produced important performance improvements and, consequently, several speed records for computing relevant cryptographic algorithms in modern computer architectures ranging from embedded 8-bit microcontrollers to 8-core processors / Doutorado / Ciência da Computação / Doutor em Ciência da Computação