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:

C(x)=xemodm

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

m=C(x)dmodm

The homomorphic property is that multiplication is preserved.

C(x1)â‹…C(x2)=(xe1modm)â‹…(xe2modm)

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

xe1xe2modm=(x1x2)emodm=E(x1â‹…x2)

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

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
template.



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.id}}">{{ object.question }}</a></li> {% endfor %} </ul> {% else %} <p>No polls are available.</p> …

My setup for downloading & streaming movies and tv

I recently signed up for Netflix and am retiring my headless home media pc. This blog will have to serve as its obituary. The box spent about half of its life running FreeNAS, and half running Archlinux. I’ll briefly talk about my experience with FreeNAS, the migration, and then I’ll get to the robust setup I ended up with.

The machine itself cost around $1000 in 2014. Powered by an AMD A4-7300 3.8GHz cpu with 8GB of memory. A SilverStone DS380 case is both functional, quiet and looks great. The hard drives have been updated over the last two years until it had a full compliment of 6 WD Green 4TiB drives - all spinning bits of metal though.

Initially I had the BSD based FreeNAS operating system installed. I had a single hard drive in its own ZFS pool for TV and Movies, and a second ZFS pool comprised of 5 hard drives for documents and photos.

FreeNAS is straight forward to use and setup, provided you only want to do things supported out of the box or by plugins. Each plugin is install…

Python and Gmail with IMAP

Today I had to automatically access my Gmail inbox from Python. I needed the ability to get an unread email count, the subjects of those unread emails and then download them. I found a Gmail.py library on sourceforge, but it actually opened the normal gmail webpage and site scraped the info. I wanted something much faster, luckily gmail can now be accessed with both pop and imap.

After a tiny amount of research I decided imap was the better albiet slightly more difficult protocol. Enabling imap in gmail is straight forward, it was under labs.

The address for gmail's imap server is:

imap.gmail.com:993

Python has a library module called imaplib, we will make heavy use of that to access our emails. I'm going to assume that we have already defined two globals - username and password. To connect and login to the gmail server and select the inbox we can do:

importimaplibimap_server=imaplib.IMAP4_SSL("imap.gmail.com",993)imap_server.login(username,password)imap_server.select(…