Explore the foundations and applications of lattice-based cryptography in this comprehensive lecture. Delve into lattice problems, hardness, and the significance of polynomial rings in cryptographic systems. Examine popular polynomial rings and their algebraic structures, with a focus on fully homomorphic encryption. Gain insights into zero-knowledge protocols, including Schnorr protocols, extractor protocols, and noise drowning techniques. Learn about rejection sampling, extraction methods, and the concept of Proof of Knowledge. Discover how these theoretical concepts translate into practical applications in modern cryptography.