Advancements in fully homomorphic encryption over the integers: a comprehensive survey and analysis

Fully Homomorphic Encryption (FHE) schemes represent a groundbreaking development that enables computations on encrypted data while ensuring data privacy. Since Gentry's pioneering work on FHE, various variants of FHE schemes have emerged. These variants, particularly (Ring) Learning with Errors-based schemes, NTRU-based schemes, and Approximate Greatest Common Divisor (AGCD)-based schemes, also known as FHE over the integers (FHE-OI), have garnered substantial interest and remain active areas of research. Notably, FHE-OI stands out for its conceptual simplicity and has undergone significant improvements over time. These improvements include reductions in public key sizes, batched processing capabilities, scale-invariant properties, faster bootstrapping, and parameter selection for enhanced security. In light of these noteworthy advancements and the growing importance of AGCD-based FHE, this survey paper aims to explore existing FHE-OI schemes, providing detailed insights into their protocols, strengths, and limitations. Moreover, we propose a technique for seamless transitions between different variants of FHE schemes without requiring decryption, enabling the utilization of their respective advantages. Additionally, we present several open problems in FHE-OI and expand the scope of this encryption paradigm.