Homomorphic Encryption (HE)

Supports unlimited operations of a single type—either addition or multiplication, but not both. RSA encryption is a classic example, exhibiting multiplicative homomorphism. Paillier cryptosystem provides additive homomorphism and is widely used in privacy-preserving federated learning for secure gradient aggregation. PHE schemes are computationally efficient and practical for production systems requiring only one type of arithmetic operation on encrypted data.

Supports both addition and multiplication but only for circuits of limited multiplicative depth. Each ciphertext contains a noise component that grows with every operation, especially multiplication. Once noise exceeds a threshold, decryption fails. SHE is the conceptual bridge between PHE and FHE, often used as a building block in leveled FHE constructions like BGV and BFV schemes.

Supports arbitrary computation on encrypted data with unlimited additions and multiplications. Achieved through bootstrapping, a technique introduced by Craig Gentry in 2009 that homomorphically evaluates the decryption circuit to refresh ciphertext noise. Modern FHE schemes include:

• CKKS: Optimized for approximate arithmetic on real numbers, ideal for neural network inference
• TFHE: Optimized for fast bootstrapping of individual gates, suitable for low-latency decision trees
• BGV/BFV: Integer arithmetic schemes for exact computation

The critical innovation enabling Fully Homomorphic Encryption. Bootstrapping homomorphically evaluates the decryption circuit on a ciphertext, producing a new ciphertext encrypting the same plaintext but with reset noise levels. This process breaks the limitation of multiplicative depth, allowing unbounded computation. The computational cost is significant—bootstrapping can dominate runtime—making it the primary target for hardware acceleration using FPGAs and ASICs.

Enables private inference as a service, where a client sends encrypted input to a cloud-hosted model and receives an encrypted prediction without the server ever seeing the raw data or the result. Key applications include:

• Encrypted neural network inference using CKKS for approximate activations
• Private information retrieval from vector databases
• Secure multi-party model evaluation where neither the data owner nor the model owner exposes their assets Frameworks like OpenFHE, Microsoft SEAL, and Concrete ML provide abstractions for compiling ML models into HE-compatible circuits.

Modern HE schemes derive their security from hard problems on mathematical lattices, specifically the Ring Learning With Errors (RLWE) problem. RLWE is believed to be resistant to attacks by both classical and quantum computers, making HE a post-quantum cryptographic primitive. The security parameter determines the polynomial modulus degree and ciphertext modulus, directly impacting performance and noise growth. NIST's post-quantum standardization process has validated lattice-based cryptography as a foundation for future-proof encryption.

Enables organizations to send encrypted queries to a proprietary AI model hosted in the cloud without exposing the raw input data. The cloud provider processes the encrypted data and returns an encrypted result, which only the client can decrypt.

• Privacy-preserving SaaS: A healthcare startup uses a diagnostic model without the cloud provider ever seeing patient symptoms.
• Secure code analysis: Developers submit encrypted proprietary source code to an AI vulnerability scanner.
• Financial fraud detection: Banks query a shared fraud model with encrypted transaction details, preventing leakage to the model host.

Combines Federated Learning with HE to allow a central server to compute aggregate statistics or model updates from multiple data owners without decrypting individual contributions. This protects against gradient leakage attacks.

• Cross-institution medical research: Hospitals jointly compute the average effectiveness of a treatment across their encrypted patient records without sharing raw data.
• Anti-money laundering consortium: Banks compute encrypted risk scores across shared transaction graphs without revealing customer details to competitors.
• Secure voting systems: Tally encrypted ballots to produce a final count while preserving individual vote secrecy.

Allows a client to execute private information retrieval and complex SQL-like queries on a remote database without the database server learning the query parameters or the results.

• Genomic sequence matching: A researcher queries a proprietary genomic database for a specific marker without revealing the marker to the database owner.
• Private location-based services: A user queries a map service for nearby restaurants without disclosing their exact location.
• Regulatory compliance checks: An auditor scans encrypted financial records for anomalies without exposing the underlying transactions.

Multiple parties collaboratively train a machine learning model on their combined, encrypted datasets without any party revealing its raw data to the others. HE ensures that intermediate computations during training remain confidential.

• Supply chain optimization: Manufacturers and logistics providers jointly train a demand forecasting model on encrypted inventory and shipment data.
• Competitive intelligence pooling: Retailers train a shared recommendation model on encrypted customer preference data without exposing proprietary sales figures.
• Pharmaceutical R&D consortia: Drug companies combine encrypted assay results to build more robust toxicity prediction models.

Addresses the inherent transparency of public blockchains by executing smart contracts on encrypted state. Transaction logic and values remain hidden from validators while still being verifiably correct.

• Sealed-bid auctions: Bids are encrypted on-chain, and the auction contract computes the winner homomorphically without revealing losing bids.
• Private decentralized finance (DeFi): Lending protocols process encrypted collateral ratios, preventing front-running and liquidation sniping.
• Confidential identity verification: On-chain credentials are verified against encrypted attributes without exposing the underlying personal data.

Enables financial institutions to comply with Know Your Customer (KYC) and Anti-Money Laundering (AML) regulations by performing checks on encrypted customer data across institutional boundaries.

• Cross-border KYC: Banks in different jurisdictions check a client's encrypted identity attributes against a shared sanctions list without exposing the client's full profile.
• Transaction monitoring: An encrypted transaction stream is analyzed for suspicious patterns by a third-party compliance service that never sees the raw transactions.
• Capital adequacy reporting: A holding company computes its consolidated risk exposure across encrypted subsidiary balance sheets.

The homomorphic encryption landscape is defined by the class of supported circuits:

• Partially Homomorphic Encryption (PHE): Supports only one operation type (e.g., RSA for multiplication, Paillier for addition). Simple and fast.
• Somewhat Homomorphic Encryption (SHE): Supports both addition and multiplication but only for circuits of limited depth.
• Fully Homomorphic Encryption (FHE): Supports arbitrary computation on ciphertexts through bootstrapping, which refreshes the noise accumulated during operations. FHE is the holy grail but remains computationally intensive.