### 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$e$, m$m$) and private keys (d$d$, m$m$).

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

C(x)=xemodm

To decrypt a ciphertext C(x)${C}(x)$ 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

### 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> …