Modular Arithmetic: Foundations and Applications

This lecture covers the principles of modular arithmetic, a fundamental concept in number theory essential for cryptography and channel coding. It begins with an introduction to congruence classes and the notation Z/mZ, explaining how integers can be represented in modular systems. The instructor provides definitions and examples, illustrating how to perform operations such as addition and multiplication within these systems. The properties of modular arithmetic, including the existence of additive and multiplicative identities, are discussed in detail. The lecture also explores the concept of multiplicative inverses and their significance in solving equations in modular arithmetic. Various examples are presented to clarify these concepts, including exercises that challenge students to identify elements with multiplicative inverses. The session concludes with a discussion on the practical applications of modular arithmetic in fields such as cryptography, emphasizing its importance in modern computational systems.