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**
```python
# 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**
```python
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