Commutative Groups: Foundations for Cryptography

This lecture introduces the concept of commutative groups, essential for understanding cryptographic algorithms such as Diffie-Hellman, RSA, and ElGamal. The instructor begins by discussing the importance of finite groups and fields, emphasizing their roles in cryptography. The definition of a commutative group is presented, detailing the necessary axioms: closure, associativity, identity element, inverse element, and commutativity. The lecture includes exercises to identify commutative groups and explores the properties of groups formed under modular arithmetic. The instructor explains Euler's totient function, which counts the integers co-prime to a given integer, and its significance in determining the number of elements in a group. The concept of isomorphism is also introduced, illustrating how different groups can exhibit the same structure. The lecture concludes with a discussion on exponentiation within groups, highlighting its relevance in cryptographic applications. Overall, the lecture provides a comprehensive overview of the mathematical foundations necessary for advanced studies in cryptography.