Homomorphic Encryption

A scheme that supports an unlimited number of operations but only of a single type (either addition or multiplication) on ciphertexts.

• Examples: RSA (multiplicative), Paillier (additive), ElGamal (multiplicative).
• Use Case: Secure electronic voting (Paillier for tallying encrypted votes) or private set intersection.
• Property: The simplest form, offering high efficiency for its specific operation but limited functionality.

A scheme that supports both addition and multiplication on ciphertexts, but only for a limited number of operations (a bounded circuit depth).

• Limitation: Noise grows with each operation; after a certain threshold, decryption fails.
• Foundation: Serves as the building block for Fully Homomorphic Encryption (FHE). Early FHE constructions (e.g., Gentry's) first created an SHE scheme, then applied bootstrapping to make it fully homomorphic.

A scheme that supports an unlimited number of both addition and multiplication operations on ciphertexts, enabling the evaluation of arbitrary functions (any boolean or arithmetic circuit).

• Breakthrough: First realized by Craig Gentry in 2009 using lattice-based cryptography and the bootstrapping technique.
• Core Challenge: Computational overhead is significant, often 10,000x to 1,000,000x slower than plaintext computation, making it a major area of optimization.
• Modern Schemes: BGV, BFV, CKKS (for approximate arithmetic), and FHEW/TFHE (for fast bootstrapping).

A practical variant of FHE that supports circuits of a pre-determined, bounded depth without requiring bootstrapping.

• Advantage: Eliminates the massive computational cost of bootstrapping for applications where the maximum computational circuit depth is known in advance (e.g., a specific machine learning inference graph).
• Trade-off: Parameters (and thus ciphertext size) are set for the worst-case depth, which can be less efficient than bootstrapping for very deep or unpredictable circuits.

A scheme, most notably the CKKS scheme, designed to perform efficient homomorphic operations on approximate numbers (complex or real numbers), introducing a small, controlled error.

• Primary Use: Enables privacy-preserving machine learning on encrypted data, as ML models are inherently robust to small numerical noise.
• Mechanism: Encodes numbers into the plaintext polynomial with a scaling factor, allowing for approximate arithmetic. The error is managed through rescaling operations.
• Benefit: Offers a much better performance profile for real-world data science applications compared to exact arithmetic FHE schemes.

The key technique that transforms a Somewhat Homomorphic Encryption scheme into a Fully Homomorphic one. It is the process of homomorphically decrypting a ciphertext.

• Purpose: To refresh a ciphertext by reducing the accumulated noise from previous operations, effectively resetting the noise level and allowing for unlimited further computations.
• Analogy: Like cleaning the lenses of glasses to see clearly again after they've become foggy.
• Cost: Historically the most computationally expensive operation in FHE. Modern research focuses on optimizing bootstrapping to make FHE practical (e.g., the FHEW and TFHE schemes).

Facilitates cross-institutional research on encrypted patient data. Hospitals or research labs can encrypt genomic sequences or medical records and share them for collaborative studies. Researchers can perform analyses—such as genome-wide association studies (GWAS), calculating disease risk scores, or training predictive models—without ever decrypting the individual-level data. This directly supports healthcare federated learning initiatives while providing a stronger, cryptographic guarantee of patient privacy beyond statistical methods.

Allows users to submit encrypted data to a Machine Learning as a Service (MLaaS) platform and receive an encrypted prediction. The model owner never sees the user's raw input. For example, a user could encrypt a medical image, send it to a diagnostic AI service, and get back an encrypted classification (e.g., 'benign' or 'malignant') that only they can decrypt. This protects both the user's data privacy and the service provider's intellectual property (the model weights), as the model can also be kept encrypted.

Enables multi-party financial computations where inputs must remain secret. Use cases include:

• Private bidding and auctions: Bids are encrypted, and the auctioneer determines the winner without learning individual bid amounts.
• Secure credit scoring: A bank can compute a credit score using encrypted data from multiple sources (other banks, utilities) without accessing the raw data.
• Confidential blockchain smart contracts: Transaction logic executes on encrypted state data, enhancing privacy for decentralized finance (DeFi) applications beyond public ledgers.

Allows clients to run search and analytical queries on an encrypted database. A user with the decryption key can formulate a query (e.g., SELECT SUM(salary) WHERE department = 'Engineering') that the database server can execute on the encrypted salary and department fields. The server returns an encrypted result to the user. This provides a stronger guarantee than Transparent Data Encryption (TDE), which only protects data at rest, by ensuring data remains confidential even during processing.

Supports the generation of high-quality synthetic data while protecting source data. A trusted curator can train a generative model (like a GAN or VAE) using homomorphic encryption. The training data remains encrypted throughout, and the resulting model parameters are also encrypted. This model can then be used to generate synthetic data that statistically mirrors the original sensitive dataset (e.g., patient records, financial transactions) without any plaintext exposure, mitigating risks of membership inference or model inversion attacks.