Codes and ciphers have been used for approximately 4,000 years to disguise the meaning of our most secret messages, from the most elementary ciphers of ancient times to the modern encryption methods used today. In this course we will be studying the mathematics behind these systems.
The course will begin with some of the more elementary ciphers used in the ancient world. Known as monoalphabetic ciphers, these methods are very easy to break using some basic statistical techniques. Clearly needing a stronger form of encryption, these were soon replaced by a more secure cipher that resisted such attacks, known as polyalphabetic ciphers.
In our study of monoalphabetic and polyalphabetic ciphers, we will define precisely what these terms mean, and introduce some fundamental ideas of mathematics such as functions, statistics, combinatorics, elementary number theory and methods of proof. We will describe several ciphers used throughout history and show some of the techniques used to break them.
By the 20th century, code making had become mechanised, and we will look at one of the more advanced ciphers of the Second World War: that of the infamous Enigma machine, as used by the German military. This cipher machine was thought to be unbreakable, until Polish, British and American mathematicians, working in secret, broke this remarkable code – changing the course of the war. We will be looking at the mathematics of this cipher, and that of the wartime code breakers.
Finally, we will end the course with a brief look at one of the most secure ciphers 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 short course we will have introduced several fundamental ideas in mathematics, and maybe even answer the question: "is there such a thing as an unbreakable code?”
The course will be self-contained. You will need to be confident with basic arithmetic and algebra.
Learning outcomes
- An appreciation of the many different areas of mathematics used for cryptography;
- An understanding of the different types of cryptography, namely: transposition ciphers,
substitution ciphers, monoalphabetic ciphers, polyalphabetic ciphers;
- An understanding of cryptanalysis techniques to decode messages;
- An appreciation of the application of cryptography and cryptanalysis in Second World War and internet encryption;
- An introduction to mathematics and proof.
