Homomorphic Encryption: Compute on Encrypted Data

Homomorphic encryption lets you run computations on encrypted data without decrypting it first.

Let’s see this in action. Imagine you have sensitive customer data – purchase history, personal details – that you want to use to train a recommendation engine. You can’t just send this data to a cloud provider for training because it’s encrypted. With homomorphic encryption, you can.

Here’s a simplified workflow.

Encryption: Your data is encrypted using a homomorphic encryption scheme. This results in ciphertext that looks like random noise to anyone without the secret key.

from concrete import fhe
# Assuming a context is set up for a specific scheme (e.g., CKKS, BFV)
# For demonstration, let's use a simplified representation
secret_key, public_key = fhe.new_keys()
context = fhe.Context(secret_key) # Simplified context

data = [10, 20, 30, 40]
encrypted_data = [fhe.encrypt(d, public_key) for d in data]

Computation: You send the encrypted_data to a compute service. This service, which doesn’t have the secret key, performs operations on the ciphertext. For example, calculating the average:

# Assume 'compute_service' is a remote entity with the public_key
# and homomorphic operations defined.
# This is a conceptual representation of the computation.

# Encrypted sum
encrypted_sum = encrypted_data[0]
for item in encrypted_data[1:]:
    encrypted_sum = fhe.add(encrypted_sum, item) # Homomorphic addition

# Encrypted count (if not known beforehand, it can be encrypted too)
encrypted_count = fhe.encrypt(len(data), public_key)

# Encrypted average = encrypted_sum / encrypted_count
# Division is more complex in HE, often approximated or done differently.
# For simplicity, let's assume we have an encrypted reciprocal.
encrypted_reciprocal_of_count = fhe.encrypt(1.0 / len(data), public_key)
encrypted_average = fhe.multiply(encrypted_sum, encrypted_reciprocal_of_count) # Homomorphic multiplication

Decryption: The result, encrypted_average, is sent back to you. Only you, with the secret_key, can decrypt it to get the actual average.

decrypted_average = fhe.decrypt(encrypted_average, secret_key)
print(f"Decrypted Average: {decrypted_average}") # Output: Decrypted Average: 25.0

This entire process allows for privacy-preserving machine learning inference. A model can be trained on sensitive data, and then inferences can be run on new, encrypted data without ever exposing the raw data. The compute provider only ever sees ciphertext.

The core problem homomorphic encryption solves is the conflict between the utility of data and its privacy. Traditionally, to gain insights from data (utility), you need to see it. But seeing sensitive data is a privacy risk. Homomorphic encryption breaks this link: you can compute on data as if you could see it, without actually seeing it.

Internally, homomorphic encryption schemes rely on complex mathematical structures, often lattices. Operations like addition and multiplication on encrypted data correspond to specific operations on these mathematical objects. The "noise" in the ciphertext, which grows with each operation, is the primary challenge. Too much noise, and decryption becomes impossible. Different schemes (like BFV, BGV, CKKS, TFHE) have different trade-offs in terms of supported operations, performance, and noise management.

The exact levers you control are primarily the choice of scheme, parameter tuning (like polynomial degree, modulus sizes), and the algorithm design for the computation itself. For instance, CKKS is good for approximate arithmetic (like floating-point operations) and commonly used in ML, while BFV/BGV are exact for integers. Parameter tuning is critical; too small, and noise overwhelms the computation; too large, and operations become prohibitively slow.

The one thing most people don’t realize is that the "cost" of homomorphic operations isn’t just about raw CPU cycles; it’s also about the depth of the computation. Each homomorphic multiplication significantly increases the noise in the ciphertext. Therefore, algorithms that require many sequential multiplications (like deep neural networks) become exponentially harder to run homomorphically. This has led to research in techniques like "bootstrapping" (a way to refresh the ciphertext and reduce noise, but it’s computationally expensive) and developing ML models that are inherently "shallow" or optimized for HE.

The next hurdle you’ll face is optimizing the performance of complex ML models, as basic homomorphic operations can be orders of magnitude slower than their plaintext equivalents.