Building A Fully Homomorphic Encryption Scheme In Python

Okay, so picture this: I'm at a party, trying to explain cryptography to my incredibly-not-techy aunt Mildred. She's giving me that "bless your heart" look, right? I'm babbling about encryption, decryption, and how we keep our online bank accounts safe. Then, she asks, "But can't they still see my data when they're processing it?" Boom. Aunt Mildred drops the truth bomb. That's when I started diving deeper into fully homomorphic encryption (FHE). Because, spoiler alert, she's right!

See, traditional encryption protects data at rest and in transit. But while it's being processed? Usually, it has to be decrypted first. FHE, however, allows you to perform computations on encrypted data without decrypting it. Mind. Blown. Basically, it’s like performing surgery while wearing oven mitts - you get the desired outcome, but you never had to directly touch the sensitive stuff. Pretty cool, right?

And guess what? We're going to try and build a rudimentary one... in Python! Now, before you run screaming, I'm not promising enterprise-grade security here. This is a simplified version, purely for educational purposes. Think of it as building a Lego version of the Death Star – impressive-ish, but probably not battle-ready.

The (Ridiculously Simple) Toy Scheme

First, let's acknowledge that real-world FHE schemes are insanely complex, often relying on lattice-based cryptography and concepts that would make your head spin faster than a fidget spinner. We're going to sidestep all that and use a very basic (and very insecure) additive homomorphic scheme. Don't use this for anything important!

The core idea is simple: we'll encrypt a number by adding a large, random multiple of a key to it. That's it. Seriously. I told you it was simple!

Here's the Python code:

import random

def generate_key(n_length):
  key = random.randint(2(n_length-1), 2n_length-1)
  return key

def encrypt(message, key):
  # Make sure to choose a much larger random number,
  # the larger, the more secure.
  # BUT: larger is more safe, but also slower.
  noise = random.randint(1, key//2)
  ciphertext = message + noise * key
  return ciphertext

def decrypt(ciphertext, key):
  message = ciphertext % key
  return message

See? It’s basically just addition and modulo operations. The generate_key function generates a random key. The encrypt function adds a random multiple of the key to the message. And the decrypt function uses the modulo operator to get back the original message. (Psst... notice how the security relies entirely on the key staying secret? Yeah, this is why it's not secure.)

Homomorphic Properties: Where the Magic Happens

Now, here's where the "homomorphic" part comes in. Let's say we have two encrypted messages, ciphertext1 and ciphertext2, both encrypted with the same key. If we add them together, the result is an encryption of the sum of the original messages!

encrypt(message1, key) + encrypt(message2, key) will decrypt to message1 + message2. This is because

(message1 + noise1 * key) + (message2 + noise2 * key) = (message1 + message2) + (noise1 + noise2) * key

The result is still an encryption of message1 + message2.

encrypted_sum = ciphertext1 + ciphertext2

decrypted_sum = decrypt(encrypted_sum, key)

This means we can perform addition on encrypted data without decrypting it! We didn't need to know what message1 or message2 were to know their sum.

This is very limited though. What about multiplication?

Well, as you might expect, this scheme isn't multiplicatively homomorphic. Trying to multiply encrypted values with this simple approach won’t magically give you the encrypted product of the original messages. Real FHE schemes use way more complicated math to allow for both addition and multiplication (and, therefore, any arbitrary computation) on encrypted data.

Why This Matters

Imagine being able to train a machine learning model on encrypted medical records without ever seeing the raw data. Or running financial calculations on encrypted data without exposing sensitive information. That's the potential of FHE. It's a game-changer for data privacy and security. Fully homomorphic encryption enables computation on encrypted data, meaning your data is still protected while being processed.

Of course, there are challenges. FHE is computationally expensive, which limits its practical applications right now. But research is progressing rapidly, and we're getting closer to a future where FHE is a widespread technology.

So, next time Aunt Mildred asks about data privacy, you can confidently tell her about the magic of FHE. Just maybe skip the Python code example… unless she's secretly a coding genius.