CryptoSoc – The Mathematics Behind Encryption

The first few weeks of Cryptography Society (CryptoSoc) have been extremely successful – we were overwhelmed by the number of students wanting to take part and students have been very keen to decipher any secret messages sent their way.

What is CryptoSoc?
In the CryptoSoc we learn about the topic of Cryptography – the art of making and breaking secret messages.
We began with the basics – simple substitution. This involves each character of the message being substituted for another.
As an example, my original (plaintext) message of “meet me at eight thirty” could be encrypted using a ‘Caesar shift’ with key 2 – each letter is substituted with the letter 2 places after it in the alphabet.
The cipher alphabet would therefore look like this:

This was my plaintext message: “meet me at eight thirty”
The resulting ciphertext: “OGGV OG CV GKIJV VJKTVA”

All my recipient would have to do is reverse the process to decrypt the message sent to them.
This type of cipher is extremely easy to decrypt – even if you are not the intended recipient and don’t know the key – a shift of 2 in this case – and is therefore very unsafe.

After we have spent time familiarising ourselves with basic substitution ciphers and using computer software to help with laborious tasks, we’ll move on to more complicated ciphers such as the Vigenère cipher and how to break it.
The Vigenère cipher involves using different substitution alphabets depending on where the character is in the text. As an example, I might use the following 3 alphabets in repeated succession to encrypt the plaintext message of “better encryption” as shown below. All I would need to pass to my recipient would be the key “FHQ” – the starting letter of each alphabet- and they are able to easily reverse the process. The key could also be made longer to incorporate more alphabets, or perhaps a memorable word like “MATHS”.


My plaintext message: better encryption
The ciphertext message: GLJYLH JUSWFFYPES
There are a few things to note from this example:
The double ‘t’ in ‘better’ has become ‘JY’ – two different letters.
The ‘e’ in the plaintext has been enciphered twice as ‘L’ but another time as ‘J’.
The double ‘F’ in the ciphertext stands for ‘yp’ in “encryption” – two different letters.

This ciphering technique is extremely powerful as, without the key of “FHQ”, there seems little hope of deciphering the message. In fact, the Vigenère cipher went unbroken for 3 centuries and became known as “le chiffre indéchiffrable” – the indecipherable cipher. Over the next term we’ll learn how to break this cipher ourselves using some interesting mathematical techniques.

Throughout history all cultures have developed various means to disguise their messages from unauthorised eyes, some with more success than others, and the breaking of these messages has had monumental consequences. Examples range from Mary, Queen of Scots, executed due to the failed ‘Babington Plot’ where her orders to assassinate Queen Elizabeth were deciphered and used as evidence in her trial. A more recent example would be the work at Bletchley Park, building on work by the Polish, to break the Enigma ciphering system, allowing access to military intelligence that turned out to be decisive in bringing an end to World War 2.

Encryption has moved on from these basic techniques and is being used more frequently today than at any other time in history. My hope is that CryptoSoc may spark an interest in our students to take their mathematical ability beyond the scope of the curriculum and apply themselves to this growing field of mathematical research.

Daniel Knowles, Teacher of Mathematics