Number Theory: Foundations and Applications in Cryptography

This lecture covers the fundamentals of number theory, emphasizing its critical role in public-key cryptography. The instructor begins by discussing the operations within the set of integers, particularly focusing on addition, subtraction, and multiplication, while highlighting the limitations of division. The concept of Euclidean division is introduced, explaining how unique integers can be derived from any two integers. The lecture progresses to congruences, defining when two integers are congruent modulo a given integer and exploring equivalence relations. The instructor illustrates these concepts with practical examples, including the modulo operation and its applications in programming languages. The importance of number theory in creating secure communication systems is underscored, as the lecture aims to equip students with the necessary mathematical tools to understand finite fields and their applications in cryptography. The session concludes with a discussion on the relevance of these mathematical principles in real-world scenarios, particularly in the context of digital security and data integrity.