Fully Homomorphic Encryption: Theory, Applications, and the BGV Scheme

Fully Homomorphic Encryption (FHE) enables computation on encrypted data without requiring decryption, offering powerful new possibilities for secure and privacy-preserving applications. This thesis provides a comprehensive overview of homomorphic encryption, with a focus on fully homomorphic schemes. We introduce the fundamental mathematical structures underlying FHE, such as rings, lattices, and Learning With Errors (LWE) assumptions, and discuss the challenges posed by noise growth and efficiency constraints. Applications of FHE in secure cloud computation, privacy-preserving machine learning, and encrypted search are examined, highlighting both their potential and current limitations. A detailed analysis of the BGV scheme is presented, covering its encryption and decryption processes, homomorphic operations, noise management techniques, and parameter selection strategies. Through this exploration, we aim to bridge theoretical concepts with practical implementation considerations, emphasizing the significance of FHE as an emerging technology in modern cryptography.