Cryptography: unlocking the future

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 the meaning of our most secret messages, from the most elementary ciphers of ancient times to the modern encryption methods used today. In this short 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 for 2024 is an expansion of the course run in previous years, including more codes and ciphers from history and the modern day.

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.
   
   Monoalphabetic ciphers 3
   Completing our look at the fundamental ideas in codes and codebreaking, such as frequency analysis.
    
3. Polyalphabetic ciphers
   These ciphers are harder to break. We will look at how polyalphabetic ciphers work and how they were eventually broken. 
    
4. Polyalphabetic ciphers 2
   Completing our look at classic codes and ciphers and looking forward to how things began to change in the 20th century.
    
5. 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.
    
6. 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.
    
7. A digital world 1
   Messages sent today are encoded as 1s and 0s. We will look at binary, and how it is used to transmit messages, photos and sound, and how we can do so secretly.
    
8. A digital world 2 
   Continuing our look at how messages are transmitted today, and how mistakes can be reduced or fixed.
    
9. 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

Typical week: Monday to Friday 

Courses run from Monday to Friday. For each week of study, you select a morning (Am) course and an afternoon (Pm) course. The maximum class size is 25 students.   

Courses are complemented by a series of daily plenary lectures, exploring new ideas in a wide range of disciplines. To add to your learning experience, we are also planning additional evening talks and events. 

Evaluation and Academic Credit  

If you are seeking to enhance your own study experience, or earn academic credit from your Cambridge Summer Programme studies at your home institution, you can submit written work for assessment for one or more of your courses.  

Essay questions are set and assessed against the University of Cambridge standard by your Course Director, a list of essay questions can be found in the Course Materials. Essays are submitted two weeks after the end of each course, so those studying for multiple weeks need to plan their time accordingly. There is an evaluation fee of £75 per essay. 

For more information about writing essays see Evaluation and Academic Credit. 

Certificate of attendance 

A certificate of attendance will be sent to you electronically after the programme.