Homomorphic Encryption

The foundational property allowing the sum of two encrypted values to equal the encryption of their sum. Additive homomorphism is the basis for schemes like Paillier.

• Mechanism: Enc(a) ⊕ Enc(b) = Enc(a + b)
• Use Case: Encrypted vote tallying and private financial aggregation
• Limitation: Cannot perform multiplication, restricting it to linear functions

The property enabling the product of two ciphertexts to decrypt to the product of their plaintexts. Multiplicative homomorphism is a core feature of schemes like ElGamal and RSA.

• Mechanism: Enc(a) ⊗ Enc(b) = Enc(a * b)
• Use Case: Private key agreement and blind signatures
• Limitation: Cannot perform addition, limiting circuits to monomials

A scheme supporting both addition and multiplication but only for circuits of limited multiplicative depth. Each operation introduces noise that grows until decryption fails.

• Constraint: Can evaluate low-degree polynomials before noise ceiling is hit
• Efficiency: Significantly faster than fully homomorphic variants
• Example: Early BGV and BFV schemes operating without bootstrapping

The holy grail of encrypted computation, supporting arbitrary computation on ciphertexts. Achieved through bootstrapping, a technique that recursively refreshes ciphertext noise.

• Bootstrapping: Homomorphically evaluating the decryption circuit to reset noise
• Performance: Historically 1,000,000x slower than plaintext; modern schemes approach practical latency
• Scheme: Gentry's 2009 breakthrough using ideal lattices proved theoretical feasibility

A practical variant of FHE that avoids the cost of bootstrapping by pre-defining a fixed circuit depth L. Parameters are chosen to support exactly L sequential multiplications.

• Advantage: Drastically more efficient than pure FHE for known-depth circuits
• Constraint: Cannot evaluate circuits deeper than the pre-set level
• Application: Privacy-preserving neural network inference where layer depth is known at compile time

A property ensuring the evaluated ciphertext reveals no information about the function that was computed, only the output. Standard homomorphic evaluation leaks the circuit topology.

• Threat: An adversary can infer the algorithm from the structure of the evaluated ciphertext
• Solution: Rerandomization techniques that sanitize the output distribution
• Relevance: Critical when the function itself is proprietary intellectual property

Organizations can outsource sensitive data processing to third-party cloud providers without exposing plaintext. A financial institution encrypts transaction records client-side, uploads ciphertext to a cloud data warehouse, and executes aggregation queries, risk models, and regulatory reporting directly on encrypted data. The cloud provider never sees the underlying records, satisfying data residency requirements and shared responsibility models.

HE enables encrypted inference where a model owner and data owner remain mutually blind. A healthcare provider encrypts a patient's MRI scan and sends it to a diagnostic AI service. The service runs the model homomorphically, returning an encrypted diagnosis that only the hospital can decrypt. Neither the model weights nor the patient data are revealed, enabling HIPAA-compliant AI-as-a-service.

Competing entities can jointly compute insights without revealing proprietary datasets. Multiple banks encrypt their fraud indicators and contribute them to a shared encrypted fraud detection model. The homomorphic computation identifies cross-institutional patterns without any bank seeing another's transaction records. This unlocks anti-money laundering consortia and supply chain risk pooling previously blocked by antitrust concerns.

HE enables private information retrieval where a client queries a database without revealing the query itself. A law enforcement agency searches a third-party criminal database for a suspect's biometric hash. The database processes the encrypted query and returns matching records without learning who was searched or which records matched. This preserves operational security while leveraging external data sources.

HE satisfies data minimization principles by design. Under GDPR, a controller can process personal data for analytics without accessing plaintext, reducing the scope of processing activities. An HR platform computes salary equity audits across encrypted employee records, generating statistical reports without exposing individual compensation. This transforms compliance from a legal constraint into a cryptographically enforced property.

HE resolves the transparency-confidentiality tension in enterprise blockchain. Smart contracts execute business logic on encrypted state variables, preserving the auditability of distributed ledgers while hiding sensitive terms. A supply chain consortium records encrypted invoice amounts on-chain; the smart contract verifies payment conditions homomorphically without revealing pricing to competitors. This enables confidential DeFi and private institutional settlement.