Homomorphic

Homomorphic Encryption

What is Homomorphic Encryption?
Encryption that allows computation on encrypted data without decrypting it, enabling privacy-preserving ML inference and training.

Types of Homomorphic Encryption
| Type | Operations | Performance |
|------|------------|-------------|
| Partial HE | One operation (add OR multiply) | Fast |
| Somewhat HE | Limited adds and multiplies | Medium |
| Fully HE (FHE) | Unlimited operations | Slow |

How it Works

[Plaintext Data] --> [Encrypt] --> [Ciphertext]
                                        |
                                        v
                              [Compute on Ciphertext]
                                        |
                                        v
                               [Encrypted Result]
                                        |
                                        v
                     [Decrypt] --> [Plaintext Result]

Key property: Decrypt(Compute(Encrypt(x))) = Compute(x)

Operations

# Conceptual example
from tenseal import BFVContext, BFVVector

# Setup
context = BFVContext.create(poly_modulus_degree=4096)

# Encrypt
encrypted_x = BFVVector.encrypt(context, [1, 2, 3])
encrypted_y = BFVVector.encrypt(context, [4, 5, 6])

# Compute on encrypted data
encrypted_sum = encrypted_x + encrypted_y
encrypted_product = encrypted_x * encrypted_y

# Decrypt
result = encrypted_sum.decrypt()  # [5, 7, 9]

HE for ML Inference

def encrypted_inference(encrypted_input, encrypted_weights):
    # Linear layer: y = Wx + b
    # Works because addition and multiplication are supported
    encrypted_output = encrypted_weights @ encrypted_input
    encrypted_output += encrypted_bias

    # Activation: approximate with polynomial
    # ReLU approximated as polynomial for HE compatibility
    encrypted_activated = polynomial_approx_relu(encrypted_output)

    return encrypted_activated

Limitations
| Limitation | Description |
|------------|-------------|
| Performance | 10,000-1,000,000x slower than plaintext |
| Noise growth | Operations accumulate noise |
| Bootstrapping | Refresh ciphertext (expensive) |
| Operations | Non-polynomial ops difficult |

Libraries
| Library | Features |
|---------|----------|
| TenSEAL | Python, tensor operations |
| Microsoft SEAL | C++, industry standard |
| PALISADE | Open source, many schemes |
| Concrete | Compiler for FHE |

Use Cases
| Use Case | Application |
|----------|-------------|
| Healthcare | Analyze encrypted patient data |
| Finance | Private credit scoring |
| Cloud ML | Inference on private data |
| Auction | Private bidding |

Practical Considerations
- Very computationally expensive
- Often combined with other techniques (MPC)
- Best for specific, high-value privacy scenarios
- Approximate operations needed for non-linear functions