Skip to main content

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:


To decrypt a ciphertext C(x) one applies the private key:


The homomorphic property is that multiplication is preserved.


Due to the Distributive nature of the modulus operator this is rearranged to:


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)
>>> hex(c_101)

We can check that the decryption works as well:

>>> pow(c_101, d, m)

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

>>> cipher_multiply = 0x15c713c3db45595b17a5598471c36db * \
>>> cipher_multiply
>>> hex(cipher_multiply)

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

>>> pow(cipher_multiply, d, m)

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


  1. Thanks for sharing an informative blog keep rocking bring more details.I like the helpful info you provide in your articles. I’ll bookmark your weblog and check again here regularly. I am quite sure I will learn much new stuff right here! Good luck for the next!
    Web Design Training in Chennai | Web Designing Training and Placement
    Web Designing Classes in Chennai | Web Designing Institute in Chennai
    mobile application development training course in chennai
    data science course in chennai


Post a Comment

Popular posts from this blog

Matplotlib in Django

The official django tutorial is very good, it stops short of displaying
data with matplotlib - which could be very handy for dsp or automated
testing. This is an extension to the tutorial. So first you must do the
official tutorial!
Complete the tutorial (as of writing this up to part 4).

Adding an image to a view

To start with we will take a static image from the hard drive and
display it on the polls index page.
Usually if it really is a static image this would be managed by the
webserver eg apache. For introduction purposes we will get django to
serve the static image. To do this we first need to change the

Change the template
At the moment poll_list.html probably looks something like this:

<h1>Django test app - Polls</h1> {% if object_list %} <ul> {% for object in object_list %} <li><a href="/polls/{{}}">{{ object.question }}</a></li> {% endfor %} </ul> {% else %} <p>No polls are available.</p> …

Bluetooth with Python 3.3

Since about version 3.3 Python supports Bluetooth sockets natively. To put this to the test I got hold of an iRacer from sparkfun. To send to New Zealand the cost was $60. The toy has an on-board Bluetooth radio that supports the RFCOMM transport protocol.

The drive protocol is dead easy, you send single byte instructions when a direction or speed change is required. The bytes are broken into two nibbles: 0xXY where X is the direction and Y is the speed. For example the byte 0x16 means forwards at mid-speed. I was surprised to note the car continues carrying out the last given demand!

I let pairing get dealt with by the operating system. The code to create a Car object that is drivable over Bluetooth is very straight forward in pure Python: