Homomorphic Encryption

Homomorphic Encryption enables computations on encrypted data without decryption. A system is homomorphic if an operation (e.g., addition) on ciphertexts, when decrypted, yields the same result as performing that operation on the original plaintexts. This is expressed as Decrypt(Encrypt(a) ⊕ Encrypt(b)) = a ⊗ b, where ⊕ and ⊗ are operations in ciphertext and plaintext spaces, respectively.

In Web3, HE allows smart contracts to process sensitive user data (e.g., financial records, medical info) without exposing the raw data to the public blockchain or the contract logic itself. This enables confidential DeFi transactions, private voting mechanisms, and secure identity verification, moving beyond the transparency-efficiency trade-off of zero-knowledge proofs.

HE is foundational for trustless data marketplaces. Data providers can sell access to encrypted datasets, and buyers can perform queries and analytics on the encrypted data via smart contracts. Decentralized Oracles (like Chainlink) could use HE to fetch and compute on private off-chain data, delivering verifiable results to the chain without leaking the source data.

HE facilitates privacy across blockchain interoperability protocols. It allows a verifiable computation to be performed on encrypted state data from multiple chains, enabling use cases like:

• Private cross-chain asset swaps where order sizes and prices remain hidden.
• Federated learning across decentralized networks without exposing individual node data.

HE schemes are categorized by supported operations:

• Partially Homomorphic Encryption (PHE): Supports one operation (e.g., addition or multiplication). RSA is multiplicatively homomorphic.
• Somewhat Homomorphic Encryption (SHE): Supports limited operations.
• Fully Homomorphic Encryption (FHE): Supports arbitrary computations. FHE is the goal for Web3 but is currently computationally intensive, leading to significant overhead in gas costs and latency.

Blockchain projects are integrating HE to enable privacy-preserving AI models on decentralized data.

• Federated Learning: Models can be trained on encrypted data from multiple sources without exposing the raw data, crucial for healthcare or financial datasets.
• Inference on Encrypted Data: Users can submit encrypted data to an on-chain AI model and receive an encrypted result, ensuring complete data confidentiality. This is a research focus for networks aiming to become decentralized AI hubs.

While powerful, homomorphic encryption introduces significant computational overhead and engineering complexity.

• Performance: FHE operations are orders of magnitude slower than plaintext computation, requiring specialized hardware or optimistic proving systems for feasibility.
• Circuit Limitations: Operations are often limited to arithmetic (addition, multiplication), making complex contract logic challenging.
• Key Management: Requires secure systems for generating, distributing, and rotating encryption keys, often solved via Threshold FHE or trusted execution environments.

A core limitation of Somewhat Homomorphic Encryption (SHE) and Fully Homomorphic Encryption (FHE) schemes is noise. Each homomorphic operation increases the noise embedded in the ciphertext. If noise exceeds a threshold, decryption fails. This necessitates:

• Bootstrapping: A computationally intensive operation to "refresh" the ciphertext and reduce noise, enabling further computations.
• Parameter Sizing: Choosing larger security parameters to accommodate deeper computational circuits, which further impacts performance.

Not all schemes support unlimited operations. Key limitations include:

• Somewhat Homomorphic Encryption (SHE): Supports only a limited number of multiplications or a specific circuit depth before noise becomes unmanageable.
• Leveled Homomorphic Encryption: Supports circuits up to a pre-set depth L without bootstrapping.
• Function-specific Schemes: Some are optimized only for specific operations (e.g., addition in Paillier, polynomial evaluation). Truly unbounded FHE requires bootstrapping.

Secure implementation is non-trivial and introduces new attack vectors:

• Side-channel attacks: Timing, power analysis, or electromagnetic leaks during the intensive computation phase can potentially reveal secret keys.
• Parameter selection: Incorrect choice of security parameters (e.g., polynomial degree, ciphertext modulus) can catastrophically reduce actual security below theoretical guarantees.
• Complexity: The cryptographic complexity increases the attack surface for implementation bugs.

Security often depends on initial setup conditions, which can be a point of failure:

• Trusted Setup: Some FHE schemes require a one-time trusted setup to generate public parameters. If compromised, all subsequent encryptions are vulnerable.
• Key Management: The security of the entire system relies on the secure generation, distribution, and storage of the secret key. Loss of the secret key means permanent loss of data access.

A user receiving an encrypted result faces a trust problem: how to verify the computation was performed correctly on the intended data? This requires additional cryptographic machinery:

• Verifiable Computation: Combining FHE with zero-knowledge proofs or other verifiable computing techniques to prove execution integrity.
• Malicious Servers: Without verification, a malicious cloud server could return an encrypted result that decrypts to garbage, undetectably.

A cryptographic method allowing one party (the prover) to prove to another (the verifier) that a statement is true without revealing any information beyond the validity of the statement itself. Unlike homomorphic encryption, which processes encrypted data, ZKPs verify computations.

• Key Use: Privacy-preserving authentication and transaction validation (e.g., ZK-Rollups).
• Contrast: Proves a result is correct; does not inherently allow computation on the encrypted data.

A protocol that enables multiple parties to jointly compute a function over their private inputs while keeping those inputs confidentially segregated. No single party sees the others' data.

• Key Use: Distributed key generation, private auctions, and collaborative data analysis.
• Contrast with HE: MPC distributes trust among participants, whereas homomorphic encryption often involves a single data holder and a computational service.

The most powerful form of homomorphic encryption, allowing unlimited computations (both addition and multiplication) on ciphertext. The result, when decrypted, matches the result of operations performed on the plaintext.

• Key Use: Enabling truly private cloud computing and confidential smart contracts.
• Challenge: Historically computationally intensive, though modern schemes (e.g., CKKS, BFV) have improved performance.

Limited forms of homomorphic encryption that support only one type of operation (additive or multiplicative) or a limited number of operations.

• Partially HE: Supports either addition or multiplication indefinitely (e.g., Paillier for addition, RSA for multiplication).
• Somewhat HE: Supports both addition and multiplication but only for a limited number of operations before noise corrupts the ciphertext.
• Foundation: These were crucial stepping stones to achieving Fully Homomorphic Encryption (FHE).

A secure, isolated area within a main processor (e.g., Intel SGX, ARM TrustZone) that guarantees code integrity and data confidentiality. Computations run on plaintext data inside this 'secure enclave'.

• Key Use: Private smart contracts, secure key management.
• Contrast with HE: Relies on hardware-based security and trusted manufacturers rather than pure cryptographic guarantees.

An advanced encryption scheme where decrypting a ciphertext yields the result of a specific function of the original data, rather than the data itself. A holder of a secret key (for a function 'f') can learn f(x) from an encryption of x.

• Key Use: Fine-grained access control to encrypted data analytics.
• Relation to HE: Can be seen as a more targeted approach, revealing only a specific output rather than allowing arbitrary computation.