NTRU is a fast public-key cryptosystem that is constructed using polynomial rings with integer coefficients. We present ETRU, an NTRU-like cryptosystem based on the Eisenstein integers. We discuss parameter selection and develop a model for the probabilty of decryption failure. We also provide an implementation of ETRU. We use theoretical and experimental data to compare the security and efficiency of ETRU to NTRU with comparable parameter sets and show that ETRU is an improvement over NTRU in terms of security.

NTRU is a fast public-key cryptosystem that is constructed using polynomial rings with integer coefficients. We present ETRU, an NTRU-like cryptosystem based on the Eisenstein integers. We discuss parameter selection and develop a model for the probabilty of decryption failure. We also provide an implementation of ETRU. We use theoretical and experimental data to compare the security and efficiency of ETRU to NTRU with comparable parameter sets and show that ETRU is an improvement over NTRU in terms of security.

NTRU is a fast public-key cryptosystem that is constructed using polynomial rings with integer coefficients. We present ETRU, an NTRU-like cryptosystem based on the Eisenstein integers. We discuss parameter selection and develop a model for the probabilty of decryption failure. We also provide an implementation of ETRU. We use theoretical and experimental data to compare the security and efficiency of ETRU to NTRU with comparable parameter sets and show that ETRU is an improvement over NTRU in terms of security.

NTRU is a fast public-key cryptosystem that is constructed using polynomial rings with integer coefficients. We present ETRU, an NTRU-like cryptosystem based on the Eisenstein integers. We discuss parameter selection and develop a model for the probabilty of decryption failure. We also provide an implementation of ETRU. We use theoretical and experimental data to compare the security and efficiency of ETRU to NTRU with comparable parameter sets and show that ETRU is an improvement over NTRU in terms of security.

Este trabalho descreve as soluções de algumas equações diofantinas em duas e três variáveis. O objetivo é apresentar a análise de alguns casos simples e de outros mais difíceis relativos ao Último Teorema de Fermat. Primeiramente são apresentados os pré-requisitos necessários dentre os quais incluímos a noção de número primo, máximo divisor comum, congruência, o Algoritmo de Euclides e o Teorema Fundamental da Aritmética. Este material é desenvolvido primeiramente no anel dos inteiros racionais e posteriormente em duas extensões algébricas conhecidas como os inteiros de Gauss e de Eisenstein. A estrutura dos últimos é indispensável na resolução do primeiro caso não trivial do Último Teorema de Fermat, a saber, da equação diofantina x3 + y3 = z3. O último capítulo apresenta algumas aplicações de problemas diofantinos e do Algoritmo de Euclides que podem ser desenvolvidos em sala de aula com alunos do sexto e do oitavo ano. / This work describes the solutions to some diophantine equations in two and three variables. The objective is to present the analysis of some simple and other more difficult cases related to Fermats Last Theorem. First, we present the necessary prerequisites which include the notion of a prime number, the maximum common divisor, congruences, Euclids Algorithm and the Fundamental Theorem of Arithmetic. This material is first developed by using the rational integers and then presented for two algebraic extensions known as Gauss and Eisenstein integers. The structure of the latter is indispensable for the first non-trivial case of Fermats Last Theorem, namely, the diophantine equation x3 + y3 = z3. The last chapter presents some applications of simple diophantine equations and Euclids algorithm which can be developed in the classroom with sixth and eight grade students.