Homomorphic encryption using RSA

I recently had cause to briefly look into Homomorphic Encryption, the process of carrying out computations on encrypted data. This technique allows for privacy preserving computation. Fully homomorphic encryption (FHE) allows both addition and multiplication, but is (currently) impractically slow.

Partially homomorphic encryption just has to meet one of these criteria and can be much more efficient.
An unintended, but well-known, malleability in the common RSA algorithm means that the multiplication of ciphertexts is equal to the multiplication of the original messages. So unpadded RSA is a partially homomorphic encryption system.

RSA is beautiful in how simple it is. See wikipedia to see how to generate the public ($e$, $m$) and private keys ($d$, $m$).

Given a message $x$ it is encrypted with the public keys it to get the ciphertext ${C}(x)$with:

$$\mathcal{C}(x) = x^e \;\bmod\; m$$
To decrypt a ciphertext ${C}(x)$ one applies the private key:

$$\mathcal m = {C}(x)^d \;\bmod\; m$$
The homomorphic property is that multiplication is preserved.

$$\mathcal{C}(x_1) \cdot \mathcal{C}(x_2) = (x_1^e \;\bmod\; m) \cdot (x_2^e \;\bmod\; m)$$
Due to the Distributive nature of the modulus operator this is rearranged to:

$$x_1^e x_2^e \;\bmod\; m = (x_1x_2)^e \;\bmod\; m = \mathcal{E}(x_1 \cdot x_2)$$

An example in python

Say these values in hexadecimal format are my public/private keys:

m = 0x1d7777c38863aec21ba2d91ee0faf51
e = 0x5abb
d = 0x1146bd07f0b74c086df00b37c602a0b

I will choose two numbers (273, 101) which I want an untrusted third party to multiply together. First I need to encrypt the two plaintext messages:

Encryption is one call to Python's builtin pow() function, giving a little known third parameter for the modulus:

>>> c_243 = pow(243, e, m)
>>> c_101 = pow(101, e, m)

>>> hex(c_243)
'0x15c713c3db45595b17a5598471c36db'
>>> hex(c_101)
'0x12314f0fe732e421017cf710dd1834c'

We can check that the decryption works as well:

>>> pow(c_101, d, m)
101

At this point we can now ask our untrusted party to carry out the multiplication on the ciphertext:

>>> cipher_multiply = 0x15c713c3db45595b17a5598471c36db * \
                      0x12314f0fe732e421017cf710dd1834c
>>> cipher_multiply
2734418524132665852913864980612094018180511394708197352750873115983960580
>>> hex(cipher_multiply)
'0x18c3138575668d2753d4acf635bb4d09b4a67df66ac9eb8891e15743d5a04'

Now we can decrypt this new ciphertext that has been created by multiplying two ciphertexts together.

>>> pow(cipher_multiply, d, m)
24543

Which luckily is equal to our two messages multiplied together (101 * 243).

This field of study will be an interesting one to watch over the next few years as several researchers are working on Fully Homomorphic Encryption. A C++ library called HElib comprises computing primitives for fully homomorphic encryption - assembly language for HE. A good introductory tutorial can be found on tommd.github.io