Homomorphic Encryption

Supports a single type of operation—either addition or multiplication—unlimited times on encrypted data.

• RSA Encryption: Multiplicatively homomorphic; encrypting two messages and multiplying the ciphertexts yields the encryption of the product.
• Paillier Cryptosystem: Additively homomorphic; the product of two ciphertexts decrypts to the sum of the plaintexts.
• Use Case: Secure e-voting systems where encrypted votes are tallied without individual decryption.

PHE schemes are computationally efficient and widely deployed in production privacy systems.

Permits both addition and multiplication but only for circuits of limited depth. Each operation introduces noise that grows until decryption becomes impossible.

• Noise Budget: A finite resource consumed by each homomorphic operation; once exhausted, the ciphertext is corrupted.
• Bootstrapping: A technique to refresh the noise budget, enabling deeper computation.
• Example: Evaluating low-degree polynomials on encrypted genomic data for privacy-preserving diagnostics.

SHE represents the intermediate step between PHE and fully homomorphic encryption, balancing flexibility with performance.

Supports arbitrary computation on encrypted data, enabling any function to be evaluated without ever decrypting the inputs.

• Gentry's Blueprint (2009): First construction proved FHE possible using ideal lattices and bootstrapping.
• Modern Schemes: CKKS for approximate arithmetic on real numbers; BGV/BFV for exact integer arithmetic; TFHE for fast boolean gates.
• Performance: FHE operations are typically 1,000x to 1,000,000x slower than plaintext equivalents, though hardware acceleration is closing this gap.
• Use Case: Outsourcing ML inference to an untrusted cloud where both the model weights and user query remain encrypted throughout.

A variant of FHE that supports circuits up to a pre-determined multiplicative depth without requiring bootstrapping.

• Parameter Selection: The scheme parameters are chosen based on the known depth of the target computation, avoiding the overhead of bootstrapping.
• Efficiency: Significantly faster than bootstrapped FHE for computations with bounded depth.
• Example: Evaluating a fixed-depth neural network where the number of layers is known at encryption time.

Leveled schemes are the practical workhorses in most real-world FHE deployments today.

Enables privacy-preserving machine learning by encrypting both the user's input query and the model's prediction.

• Linear Layers: Matrix multiplications and convolutions map naturally to homomorphic additions and multiplications.
• Non-Linear Activations: ReLU and sigmoid require polynomial approximations (e.g., Chebyshev polynomials) to be evaluated homomorphically.
• Frameworks: Concrete-ML, HElayers, and TenSEAL provide high-level APIs for compiling neural networks to FHE circuits.
• Latency: Encrypted inference on a ResNet-20 can run in under 1 second on modern hardware with optimized cryptographic parameters.

Distributes the decryption key across multiple parties, requiring a quorum to decrypt any result.

• Key Generation: A distributed protocol generates shares of the secret key without any single party ever holding the full key.
• Decryption Protocol: Each party computes a partial decryption; combining a threshold number of shares reconstructs the plaintext.
• Use Case: Multi-institutional medical research where patient data remains encrypted during analysis and only aggregate results are revealed when all hospitals consent.

This combines the confidentiality of FHE with the governance of multi-party computation.