In the past decade, one of the major breakthroughs in computer science theory is the first construction of fully homomorphic encryption (FHE) scheme introduced by Gentry. Using a FHE one may perform an arbitrary numbers of computations directly on the encrypted data without revealing of the secret key. Therefore, a practical FHE provides an invaluable security application for emerging technologies such as cloud computing and cloud-based storage. However, FHE is far from real life deployment due to serious efficiency impediments. The main part of this dissertation focuses on accelerating the existing FHE schemes using GPU and hardware design to make them more efficient and practical towards real-life applications. Another part of this dissertation is for the hardware design of the large key-size RSA cryptosystem. As the Moore law continues driving the computer technology, the key size of the Rivest-Shamir-Adelman (RSA) encryption is necessary to be upgraded to 2048, 4096 or even 8192 bits to provide higher level security. In this dissertation, the FFT multiplication is employed for the large-size RSA hardware design instead of using the traditional interleaved Montgomery multiplication to show the feasibility of the FFT multiplication for large-size RSA design.

No description available.

The RSA cryptosystem, named after its inventors, Rivest, Shamir and Adleman, is the most widely known and widely used public-key cryptosystem in the world today. Compared to other public-key cryptosystems, such as elliptic curve cryptography, RSA requires longer keylengths and is computationally more expensive. In order to address these shortcomings, many variants of RSA have been proposed over the years. While the security of RSA has been well studied since it was proposed in 1977, many of these variants have not. In this thesis, we investigate the security of five of these variants of RSA. In particular, we provide detailed analyses of the best known algebraic attacks (including some new attacks) on instances of RSA with certain special private exponents, multiple instances of RSA sharing a common small private exponent, Multi-prime RSA, Common Prime RSA and Dual RSA.

In this article we will deal with the prime numbers and its current use in encryption algorithms. Encryption algorithms make possible the exchange of sensible data in internet, such as bank transactions, email correspondence and other internet transactions where privacy is important.

Made available in DSpace on 2014-06-12T16:00:04Z (GMT). No. of bitstreams: 2 arquivo5842_1.pdf: 1043558 bytes, checksum: e8d7c13a9fafec9e1a4a7151ae67b6a0 (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2007 / Quando um esquema criptográfico é definido sobre um grupo, criptografar mensagens equivale a fazer com que variáveis de alguma equação tomem valores nesse grupo, enquanto que quebrar esse esquema significa descobrir quais valores as variáveis tomaram. Portanto, a segurança de tais esquemas está associada à dificuldade de se resolver equações sobre grupos. A utilização de grupos livres seria uma possível solução para esse problema; entretanto, apenas equações triviais podem ser resolvidas sobre grupos livres. Além disso, os grupos livres são infinitos, o que não é interessante do ponto de vista computacional. Uma alternativa foi proposta por Susan Hohenberger em 2003, dando origem à noção de grupos pseudo-livres , refinada posteriormente por R. Rivest. Informalmente, um grupo pseudo-livre caracteriza-se por não poder ser distinguido, de modo eficiente, de um grupo livre. Do ponto de vista computacional, isto significa que a probabilidade de que se resolva uma equação não trivial sobre um grupo pseudo-livre é desprezível. Dessa forma, encontramos um ambiente adequado para lidarmos com questões de segurança de esquemas criptográficos. Dois conceitos merecem destaque nesse contexto. O conceito de grupos pseudo-livres, como veremos a seguir, é de fundamental importância para a criptografia moderna, enquanto que o conceito de primos seguros tem sua relevância associada ao criptossistema RSA. Este trabalho tem três objetivos principais. Inicialmente estaremos interessados em estudar alguns dos chamados problemas computacionalmente difíceis e sua utilização na construção de esquemas criptográficos seguros. Um outro objetivo é o estudo detalhado do Teorema de Micciancio sobre grupos pseudo-livres. Finalmente, voltaremos nossas atenções para a geração de primos seguros, pois estes estão diretamente relacionados com a segurança do criptossistema RSA. Em particular, propomos um novo algoritmo para geração de primos seguros que, através de um teorema devido a Euler e Lagrange e da lei de reciprocidade quadrática de Gauss, evita em grande parte os testes de primalidade

RSA cryptography is one of the most widely used cryptosystems in the world. FPGA/ASIC implementations for the classic RSA cryptosystem have high resource utilization due to the use of the Extended Euclid's algorithm for MOD inverse generation, the MOD exponent operation for encryption and decryption, and through non finite-field arithmetic. This thesis translates the RSA cryptosystem into the finite-field domain of arithmetic which greatly increases the range of encryption and decryption keys and replaces the MOD exponent with a multiplication. A new algorithm, the SPX algorithm, is presented and shown to outperform Euclid's algorithm, which is the most widely used mechanism to compute the GCD in FPGA implementations of RSA. The SPX algorithm is then extended to support the computation of the MOD inverse and supply decryption keys. Lastly, a finite-field RSA system is created and shown to support character encryption and decryption while being designed to be integrated into any larger system.

No description available.

The capacity to estimate the number of objects (numerosity) in the environment is ontogenetically precocious and phylogenetically ancient. In animals, this ability holds significant adaptive advantages, directly influencing survival and reproductive success. In humans, it may serve an additional purpose by providing a start-up kit for the acquisition of symbolic numbers, thus making it a potential focus for mathematics education and intervention strategies. Behavioral, neurophysiological, and neuroimaging findings suggest that numerosity information is directly extracted from the environment. However, numerosity is inherently linked with other visual characteristics of sets (such as larger sets often occupy more space or are more densely spaced), making it challenging to determine the extent to which the observed response to numerosity is distinct from the response to other visual attributes. In my PhD research I provide experimental evidence through neuroimaging and computational modeling techniques elucidating where, when, and how numerical information is encoded in the human brain. This work therefore provides a threefold contribution. First, I show that numerosity is represented over and above nonnumeric visual features in a widespread network of areas starting from early visual areas and further amplified in associative areas along the dorsal but also notably the ventral stream, and that the neural representational geometries of regions across the two steams are substantially identical. Second, I showed that numerosity is represented at an early stage and seemingly in parallel across of a set of regions including early visual, parietal, and temporal, preceding the emergence of non-numeric features that could indirectly contribute to numerosity computation. Finally, by comparing the fMRI data with a convolutional neural network (CNN) to explore similarities and differences between the model and human brain data, I discovered that although the CNN can perform approximate numerosity comparisons and the structure of their representation in their hidden layers captures well numerosity representation in early visual areas of humans, it falls short of fully simulating the way in which associative brain regions represent numerosity. Taken together, the findings of this thesis provide experimental evidence supporting the notion that number is a primary visual feature, encoded independent from other visual features quickly and widely across the human brain. Furthermore, they emphasize the need for additional investigation to unravel the computational mechanisms underlying numerosity in the human brain.

Esta dissertação tem por objetivo apresentar a Criptografia RSA, que é o método de criptografia mais utilizado no mundo atualmente. Iniciamos a dissertação com um breve histórico sobre a criptografia e em seguida introduzimos a teoria matemática empregada no método pertencente a teoria dos números. Finalizamos a dissertação com a descrição de uma aplicação simples do método levado para uma sala de aula do ensino médio. Este texto pretende introduzir o tema de maneira simples e por esta razão, fazemos uso de muitos exemplos. Esperamos ainda que o leitor compreenda o que torna este método eficiente e seguro. / The main goal of this work is to introduce the RSA Criptography that is the most used method in Criptography nowadays. We begin the dissertation with a brief introduction about criptography and then we discuss concepts from number theory used in the method. Finally we present a description of a simple application of Criptography made in a High school classroom. This text intend to introduce the subject in a simple way for this reason we present several examples. We hope that the reader have the comprehension of the methods and of its security.

Sistemas criptográficos como o RSA e o Diffie-Hellman sobre Curvas Elípticas (DHCE) têm fundamento em problemas computacionais considerados difíceis, por exemplo, o problema do logaritmo (PLD) e o problema da fatoração de inteiros (PFI). Diversos trabalhos têm relacionado a segurança desses sistemas com os problemas subjacentes. Também é investigada a segurança do LSB (bit menos significativo) da chave secreta no DHCE (no RSA é o LSB da mensagem) com relação à segurança de toda a chave. Nesses trabalhos são apresentados algoritmos que conseguem inverter os sistemas criptográficos citados fazendo uso de oráculos que predizem o LSB. Nesta dissertação, fazemos a implementação de dois desses algoritmos. Identificamos parâmetros críticos e mudamos a amostragem do formato original. Com essa mudança na amostragem conseguimos uma melhora significativa nos tempos de execução. Um dos algoritmos (ACGS), para valores práticos do RSA, era mais lento que a solução para o PFI, com nosso resultado passou a ser mais veloz. Ainda, mostramos como provas teóricas podem não definir de maneira precisa o tempo de execução de um algoritmo. / Cryptographic systems like RSA and Elliptic Curve Diffie-Hellman (DHCE) is based on computational problems that are considered hard, e.g. the discrete logarithm (PLD) and integer factorization (PFI) problems. Many papers investigated the relationship between the security of these systems to the computational difficulty of the underlying problems. Moreover, they relate the bit security, actually the LSB (Least Significant Bit), of the secret key in the DHCE and the LSB of the message in the RSA, to the security of the whole key. In these papers, algorithms are presented to invert these cryptographic systems making use of oracles that predict the LSB. In this dissertation we implement two of them. Critical parameters are identified and the original sampling is changed. With the modified sampling we achieve an improvement in the execution times. For practical values of the RSA, the algorithm ACGS becomes faster than the PFI. Moreover, we show how theoretical proofs may lead to inaccurate timing estimates.