Explore the groundbreaking concepts of homomorphic encryption based on learning with errors in this 23-minute talk from Crypto 2013. Delve into the work of Craig Gentry, Amit Sahai, and Brent Waters as they present a leveled fully homomorphic encryption (FHE) scheme. Learn about the toy scheme, its vulnerabilities, and the innovative methods used to overcome them. Discover techniques for addition and multiplication while maintaining small errors, flattening ciphertexts, and implementing identity-based homomorphic encryption. Gain insights into this cutting-edge cryptographic technology that enables computations on encrypted data without compromising privacy or security.