Homomorphic Encryption

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

• RSA encryption: Multiplicatively homomorphic; ciphertext multiplication yields encrypted product of plaintexts
• Paillier cryptosystem: Additively homomorphic; commonly used in electronic voting and private aggregation
• ElGamal encryption: Multiplicatively homomorphic; foundational for threshold decryption schemes

PHE schemes are computationally efficient and widely deployed in production systems where only one operation type is required.

Supports both addition and multiplication but only for circuits of limited depth. Each operation introduces noise that accumulates, eventually making decryption impossible.

• BGN cryptosystem: Supports one multiplication followed by unlimited additions
• YASHE scheme: Optimized for polynomial evaluation with bounded multiplicative depth
• Ideal lattice-based constructions: Trade circuit depth for smaller ciphertext sizes

SHE is useful for evaluating low-degree polynomials and simple statistical functions where the computation depth is known in advance.

Supports arbitrary computation on encrypted data with unlimited additions and multiplications. Achieved through a breakthrough construction by Craig Gentry in 2009 using lattice-based cryptography.

• Bootstrapping: A noise-reduction technique that evaluates the decryption circuit homomorphically, resetting noise levels to enable unbounded computation
• Learning With Errors (LWE): The hardness assumption underlying most modern FHE schemes
• Ring-LWE variants: Improve efficiency by operating on polynomial rings rather than matrices

FHE enables general-purpose encrypted computation but incurs significant performance overhead compared to plaintext operations.

A practical variant of FHE that supports circuits of a pre-determined depth without bootstrapping. Parameters are chosen to accommodate the exact multiplicative depth required.

• BGV scheme (Brakerski-Gentry-Vaikuntanathan): Supports packed ciphertexts using Single Instruction Multiple Data (SIMD) operations
• BFV scheme (Brakerski-Fan-Vercauteren): Optimized for integer arithmetic with efficient batching
• CKKS scheme (Cheon-Kim-Kim-Song): Designed for approximate fixed-point arithmetic, ideal for machine learning inference

Leveled FHE is the dominant approach for production applications, avoiding the prohibitive cost of bootstrapping.

Combines homomorphic encryption with distributed key management, requiring multiple parties to collaborate for decryption. No single entity holds the complete private key.

• Threshold decryption: The secret key is split into shares; a minimum quorum must combine partial decryptions
• Multi-party computation (MPC) integration: Enables joint computation where inputs remain private from all participants
• Use case: Secure federated learning where model updates are aggregated homomorphically and only decrypted when sufficient participants contribute

This prevents any single party from unilaterally accessing the plaintext results.

Enables privacy-preserving inference where a model owner can classify encrypted user data without ever seeing the input, and the user learns only the result.

• CryptoNets: Early demonstration of neural network inference on encrypted data using leveled FHE
• nGraph-HE: Intel's compiler framework for homomorphic inference on deep learning models
• Concrete-ML: Open-source toolkit by Zama for converting trained models into FHE-compatible circuits

Key challenge: Non-linear activation functions like ReLU must be approximated as polynomials, introducing accuracy trade-offs.

The most powerful variant of homomorphic encryption, supporting arbitrary computation (both addition and multiplication) on ciphertexts without any theoretical limit on circuit depth. FHE schemes, such as CKKS and TFHE, rely on lattice-based cryptography and remain computationally intensive. Key characteristics:

• Unlimited sequential operations on encrypted data
• Enables general-purpose programs without decryption
• Bootstrapping refreshes ciphertext noise to enable deep circuits
• Latency measured in seconds to minutes per operation

A restricted form of homomorphic encryption that supports only one type of operation—either addition or multiplication—but not both. RSA (multiplicative) and Paillier (additive) are classic examples. PHE is significantly faster than FHE and is widely deployed in production:

• Paillier enables encrypted vote tallying and private aggregation
• ElGamal supports multiplicative operations on ciphertexts
• Much lower computational overhead than FHE
• Suitable for single-purpose privacy applications

An intermediate category that supports both addition and multiplication but only for circuits of limited depth. SHE schemes can evaluate low-degree polynomials before noise accumulation renders decryption impossible. They serve as the foundation for many FHE constructions:

• Supports a bounded number of operations
• More efficient than full FHE for shallow computations
• Often used as a building block before bootstrapping is applied
• Ideal for fixed-complexity tasks like private set intersection

A cryptographic protocol where multiple parties jointly compute a function over their private inputs without revealing those inputs to each other. Unlike homomorphic encryption, MPC requires interactive communication between participants. Common techniques include garbled circuits and secret sharing:

• Distributes trust across multiple non-colluding parties
• Often combined with HE for hybrid privacy architectures
• Used in privacy-preserving machine learning and auctions
• Communication overhead scales with circuit complexity

A hardware-based secure enclave within a CPU that isolates code and data from the host operating system. Intel SGX and AMD SEV provide TEE implementations. Data is decrypted only inside the enclave, offering a complementary approach to cryptographic privacy:

• Hardware root of trust rather than mathematical guarantees
• Significantly faster than FHE for computation
• Vulnerable to side-channel attacks and hardware exploits
• Requires trust in the silicon vendor's implementation

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 quantum computer attacks. Key properties:

• Post-quantum security guarantees
• Enables noise-based encryption essential for HE
• NIST has standardized lattice-based algorithms (CRYSTALS-Kyber)
• Forms the security basis for CKKS, BGV, and TFHE schemes