Teoria dos números e criptografia RSA: uma proposta de ensino para alunos de matemática olímpica

Abstract

This work presents a teaching proposal of RSA Cryptography for a group of 8th and 9th grade students of Elementary School, volunteers in a Mathematics Club from Colégio Militar de Porto Alegre (CMPA). Most of the students in this class are frequent medalists in Mathematical Olympiads, but they all share the ease and curiosity in solving challenges.These students, even without any aversion to Mathematics, and presenting above-average reasoning, questioned the practical use of Number Theory tools, as well as the study of prime numbers and Modular Arithmetic, unlike other areas such as Geometry and Algebra. In order to present an application suitable for the class level, was created a sequence of thirteen classes on the subject of RSA Cryptography which included an overview of history and the use of simple cryptographic methods, up to the prerequisites necessary to understand the operation of the RSA, where the Modular Arithmetic, the calculation of multiplicative inverses and the Euler function are fundamental part of this process. It should be noted that RSA is a cryptographic method used in online shopping and banking transactions which, together with the historical character and the challenge of deciphering messages, has become naturally attractive to students. During all classes, were used questions Mathematical Olympiad and other competitions, aggregating in-depth discussions, conjecturing results and allowing small demonstrations that aided in the development of thinking and mathematical reasoning. At appropriate times, it was also made use of applications for smartphones and software associated with Cryptography. A satisfactory result was obtained in view of the students' return (registered in shape of questionnaire) and the new possibilities that will contribute even more to the formation of this teacher.