FHE

Fully homomorphic encryption (FHE) is a form of public-key cryptography that permits arbitrary computations on ciphertexts, producing an encrypted result that, when decrypted, matches the result of the same computations on the plaintext. In effect, data can be processed while it remains encrypted, enabling secure outsourcing of computation to untrusted environments.

The concept originated with Craig Gentry's 2009 construction, which introduced bootstrapping to refresh noise and realize general computation on ciphertexts. Since then researchers developed leveled and practical variants, reducing impractical bootstrapping overhead and enabling deeper circuits with carefully managed noise growth. Most modern schemes rely on lattice-based hardness assumptions such as RLWE/LWE, providing post-quantum security foundations.

Homomorphic operations are implemented through ciphertext algebra: addition corresponds to the sum of ciphertexts, and multiplication corresponds to combining ciphertexts, with noise increasing with each operation. To perform many operations, schemes use techniques like noise management, modulus switching, and bootstrapping. Relatively recent schemes support batched or packed ciphertexts, enabling SIMD-style parallelism.

Prominent FHE schemes include BFV (Brakerski–Fan–Vaikuntanathan), BGV (Brakerski–Gentry–Vaikuntanathan), and CKKS (Cheon–Kim–Kim–Song). BFV/BGV use exact arithmetic over integers, while CKKS performs approximate arithmetic suitable for real-valued data. Implementations and libraries include Microsoft SEAL, HElib, PALISADE, and Lattigo.

Applications span privacy-preserving cloud computing, secure outsourced data analysis, and machine learning on encrypted data. Real-world use remains limited by computational overhead and memory requirements, though ongoing optimization and hardware acceleration are advancing practicality. FHE is a central area of post-quantum cryptography research with active standardization efforts.