Homomorphic Encryption

Supports unlimited operations of a single type—either addition or multiplication, but not both.

• RSA encryption: Unbounded modular multiplication of ciphertexts
• Paillier cryptosystem: Additively homomorphic, widely used in electronic voting and private information retrieval
• ElGamal: Multiplicatively homomorphic, foundational for threshold decryption schemes

PHE schemes are computationally efficient and deployed in production today, but limited to single-operation workflows like encrypted aggregation.

Permits both addition and multiplication, but only for circuits of limited depth. Each operation introduces noise that accumulates, and decryption fails once a threshold is exceeded.

• Built on ideal lattices and ring-learning-with-errors (RLWE) problems
• Suitable for low-degree polynomial evaluations, such as mean and variance computations
• Often used as a stepping stone to fully homomorphic encryption via bootstrapping

SHE bridges the gap between PHE and FHE, enabling more complex analytics on encrypted data without the full cost of bootstrapping.

Supports arbitrary computation on encrypted data through bootstrapping—a technique that recursively evaluates the decryption circuit homomorphically to refresh ciphertext noise.

• Gentry's blueprint (2009): First construction using ideal lattices and squashing
• Modern schemes: BGV, BFV (integer arithmetic), CKKS (approximate floating-point)
• Enables encrypted machine learning inference and private database queries

FHE remains computationally intensive, with ciphertext expansion factors of 100x–1000x, but hardware acceleration is rapidly closing the practicality gap.

A practical variant that supports circuits of a pre-determined multiplicative depth without bootstrapping. Parameters are chosen at key generation to accommodate the exact computation.

• Eliminates the performance penalty of bootstrapping for known-depth circuits
• Used in private neural network inference where layer count is fixed
• BGV and BFV schemes natively support leveled operation

Leveled HE is the dominant approach for production deployments where the computation graph is known in advance, such as fixed-architecture model inference.

All HE ciphertexts contain structured noise that grows with each operation. Bootstrapping is the recursive self-decryption procedure that resets this noise to a baseline level.

• Gentry's bootstrapping theorem: A scheme that can homomorphically evaluate its own decryption circuit plus one additional operation is fully homomorphic
• Programmable bootstrapping (TFHE): Simultaneously refreshes noise and evaluates a lookup table, enabling non-linear functions
• Noise growth is asymmetric: Multiplication increases noise exponentially faster than addition

Efficient bootstrapping remains the central research challenge, with recent advances achieving sub-second latency on consumer GPUs.

Modern HE schemes derive security from the hardness of lattice problems believed resistant to both classical and quantum attacks.

• Learning With Errors (LWE): Distinguishing noisy linear equations from random
• Ring-LWE (RLWE): LWE over polynomial rings, enabling compact keys and ciphertexts
• NTRU and GGH: Early lattice cryptosystems that influenced HE design

Lattice-based cryptography is the backbone of post-quantum standardization (NIST), giving HE schemes long-term security guarantees against future quantum adversaries.

The most powerful variant of homomorphic encryption that supports arbitrary computation (both addition and multiplication) on ciphertexts without any limit on circuit depth. FHE schemes, such as CKKS and TFHE, enable general-purpose programs to run on encrypted data, but incur substantial computational overhead—often 4-5 orders of magnitude slower than plaintext computation. Recent hardware acceleration and algorithmic improvements are closing this gap for specific workloads like private inference.

A restricted form of homomorphic encryption that supports only one operation type—either addition or multiplication, but not both. Classic examples include RSA (multiplicative) and Paillier (additive). PHE is significantly faster than FHE and is practical for specific use cases like encrypted vote tallying or privacy-preserving aggregation of medical statistics, where only one mathematical operation is required.

An intermediate scheme that supports both addition and multiplication but only for circuits of limited depth. SHE is often used as a building block for bootstrapped FHE. The BGV and BFV schemes are prominent examples, supporting leveled computation where the multiplicative depth is known in advance. SHE is well-suited for evaluating fixed-depth machine learning models like shallow neural networks on encrypted patient data.

A cryptographic protocol where multiple parties jointly compute a function over their private inputs without revealing those inputs to each other. Unlike homomorphic encryption, SMPC requires interactive communication between parties. Common techniques include garbled circuits and secret sharing. SMPC is often combined with HE in hybrid models—using HE for local computation and SMPC for collaborative aggregation across institutions.

A hardware-based secure enclave within a CPU that isolates code and data from the host operating system. Technologies like Intel SGX and AMD SEV provide confidentiality at processor speed, but require trust in the hardware vendor and are vulnerable to side-channel attacks. TEEs offer a complementary approach to HE—TEEs protect data during computation on specific hardware, while HE protects data algorithmically regardless of infrastructure.

The mathematical foundation underlying most modern homomorphic encryption schemes. Lattice problems like Learning With Errors (LWE) and Ring-LWE are believed to be resistant to attacks from both classical and quantum computers. The security of schemes like CKKS and BFV reduces to the hardness of finding short vectors in high-dimensional lattices. This post-quantum security property makes HE a strategic investment for long-term protection of genomic and longitudinal health data.