Introduction to Homomorphic Encryption

Homomorphic encryption allows computations to be performed directly on encrypted data without decrypting it first. The result, when decrypted, is the same as if the computation had been performed on the plaintext. This is the holy grail of encrypted computation.

What Is Homomorphic Encryption?

Homomorphic encryption (HE) is a form of encryption that permits computation on ciphertexts, producing an encrypted result that, when decrypted, matches the result of the same operations performed on the plaintext data.

Mathematically, if Enc() is the encryption function and f() is the computation:

# The homomorphic property:
Dec( f( Enc(x) ) ) = f(x)

# Example with addition:
Dec( Enc(3) + Enc(5) ) = 3 + 5 = 8

# Example with multiplication:
Dec( Enc(3) × Enc(5) ) = 3 × 5 = 15

# For ML inference:
Dec( model( Enc(input) ) ) = model(input)

Types of Homomorphic Encryption

Why HE for AI?

Homomorphic encryption enables several important use cases in AI:

• Privacy-preserving inference as a service: A cloud provider runs your ML model on clients' encrypted data. The provider never sees the inputs or outputs, and clients never see the model weights.
• Secure data analysis: An analyst can run queries on encrypted medical, financial, or personal data without accessing the raw data.
• Regulatory compliance: Process data while it remains encrypted, helping satisfy data protection requirements.
• Cross-organization collaboration: Multiple organizations can run joint analytics without sharing raw data.

HE vs Other Privacy Technologies

• HE vs MPC: HE does not require interaction between parties during computation - the data owner encrypts, the server computes, and the data owner decrypts. MPC requires ongoing communication between parties. HE is better for single-server outsourced computation; MPC is better for multi-party collaborative computation.
• HE vs Differential Privacy: HE protects the input data during computation; DP protects the output from revealing information about individuals. They address different threats and can be combined.
• HE vs TEEs: Trusted Execution Environments (SGX, TrustZone) rely on hardware trust; HE relies on mathematical hardness. HE is more expensive but does not require trust in hardware vendors.

The State of HE for ML

HE for ML has made remarkable progress:

• 2009: Gentry's first FHE construction - theoretically possible but impractically slow.
• 2017: CKKS scheme enables approximate arithmetic, making HE practical for ML.
• 2020s: Libraries like Concrete ML and TenSEAL make encrypted ML accessible to Python developers.
• Current: Encrypted inference for small-to-medium neural networks runs in seconds to minutes. Active research on hardware acceleration and compiler optimization.