Aims
This course aims to:
• introduce you to mathematics through the topic of codes and codebreaking
• introduce the mathematics of simple codes and codebreaking
• introduce modern cryptography from Enigma to internet encryption
Content
Codes and ciphers have been used for approximately 4,000 years to disguise our secret messages. This mathematics course will show you how to make and break ciphers, from simple historical ciphers to modern encryption.
Along the way, we will learn about different kinds of ciphers, including monoalphabetic and polyalphabetic ciphers. We will learn what these terms mean, why they were so difficult to break, and learn the mathematics behind making and breaking such ciphers.
By the Second World War, code making had become mechanised. The Enigma machine was a cipher machine used by the German military that many believed to be unbreakable. We will see exactly how this machine worked and how it was eventually broken by wartime code breakers.
After WWII mechanical cipher were replaced with electronic ciphers. These ciphers send messages using 1s and 0s and are the basis of how all messages are sent today. We will look at how the messages we send today are sent secretly and efficiently.
Finally, we will end the course with one of the most secure ciphers that we have today. Known as RSA, it is a method of encryption used on the internet and is an example of Public Key Encryption.
Cryptography touches on a broad range of topics in mathematics, and is one of the most exotic real life applications of maths. By the end of this course we will have introduced several fundamental ideas in code making and code breaking, and discover whether there is such a thing as an unbreakable code.
Presentation of the course
There are no books to read or essays to write. The course is intended to be self-contained. You will need to be comfortable with basic arithmetic and algebra, such as rearranging equations.
At the start of each session, you will be given notes with gaps, we will then fill the gaps together. This allows us to spend less time writing and more time absorbing the ideas. By the end of the course you will have a complete set of notes.
Course sessions
1. Monoalphabetic ciphers 1
An introduction to the simplest ciphers, and the mathematical ideas we will need in the course, including the Caesar shift and modular arithmetic.
2. Monoalphabetic ciphers 2
We will practice ideas learnt so far and apply them to other ciphers.
3. Polyalphabetic ciphers
We complete our look at monoalphabetic ciphers by looking at how they were eventually broken. A new type of cipher took their place, which were harder to break.
4. Polyalphabetic ciphers 2
We will look at how polyalphabetic ciphers were eventually broken. We look at the last of the pen and paper ciphers, taking us into the early 20th century.
5. Secret Sharing
We look at how to protect secrets so that they can only be read when several people cooperate.
6. Enigma
The infamous cipher machine used by Nazi Germany in World War II that was broken by the Allies. We will look in detail at how the Enigma machine worked and how it was broken.
7. Lorenz
We will look at the cipher machine used by Adolf Hitler himself, used for the most secret messages of WWII, and how this machine was broken.
8. Digital ciphers
Messages sent today are encoded as 1s and 0s. We will look at binary, and how it can be used to send messages in secret.
9. Digital coding
We look at how modern digital messages are made to be as short as possible, and how they fix any mistakes made when transmitted.
10. RSA and Internet encryption
In the second half of the 20th century, new mathematical methods were devised to encrypt messages. These are the ideas we currently use on the internet today.
Learning outcomes
You are expected to gain from this series of classroom sessions a greater understanding of the subject and of the core issues and arguments central to the course.
The learning outcomes for this course are:
• an understanding of topics of classical cryptography, including substitution ciphers, transposition ciphers and frequency analysis
• an understanding of cryptography and cryptanalysis in World War II and internet encryption
• a better understanding of the applications of mathematics and proof
