Grupos abelianos e aritmética modular: fundamentos teóricos e aplicações em criptografia RSA

Abstract

Abelian groups form a fundamental class in modern algebra, characterized by the property of commutativity. They arise in profound mathematical contexts and are essential in areas such as Number Theory, Geometry, and Topology. In undergraduate Mathematics programs, the study of these groups offers an accessible introduction to abstract algebra, facilitating a moreintuitive understanding of structural properties. One of the main objectives of exploring this theory is to visualize the behavior of modular arithmetic when combined with group theory, understanding how this interaction naturally develops commutative groups in its constructions. For instance, congruence modulo 𝑛 defines equivalence relations on integers and forms the basis for structures such as the additive group Z/𝑛Z. These notions not only illustrate the structural beauty of group theory but also reveal deep connections with technology, especially in cryptography. When using modular arithmetic, depend directly naturally attributes commutative properties to operations on groups such that (Z/𝑛Z)*, the multiplicative group of integers modulo 𝑛. These Abelian groups are fundamental in practical applications, such as RSA encryption, where key exchange security and data encryption depend directly on their structure. RSA uses the properties of this group to perform essential modular operations in encoding and decoding messages.