• University of York

The Mathematics of Cryptography: From Ancient Rome to a Quantum Future

Explore the history of code breaking and cryptography to prepare for the future of communications and quantum computing.

621 enrolled on this course

A lock in the shape of a castle

The Mathematics of Cryptography: From Ancient Rome to a Quantum Future

621 enrolled on this course

  • 3 weeks

  • 4 hours per week

  • Digital certificate when eligible

  • Open level

Find out more about how to join this course

Crack the code of cryptography with the University of York

The ability to use codes to send secret messages has been of critical importance to civilisations for thousands of years. Nowadays, the need for messages to remain hidden from anyone intercepting them underpins services we use every day: internet banking, online shopping, instant messaging, and more.

On this course from the University of York, you’ll travel back in time to discover the earliest codes and ciphers. With an emphasis on mathematical developments that have helped make and break some of the strongest codes in history, you’ll learn what makes cryptography vital to humans and society.

Trace the history of cryptography, from Roman times to the Second World War

You’ll start by tracing two millennia of codes and code cracking, from the first ciphers used by Roman emperors to Nazi communications decoded during the Second World War. In Week 2 of the course, you’ll explore in detail the maths which was used to break the legendary Enigma machine.

As you move through the chronology of cryptography, you’ll investigate key mathematical concepts, including modular arithmetic, permutations, and prime numbers.

Explore current and future developments in secure communication and quantum computing

In the last week of the course, you’ll flash forward to the present day of cryptography. You’ll learn about cutting-edge cryptographic techniques, and the mathematics of modern-day secure communications.

Finally, you’ll look to the future of encryption and cryptography. You’ll discuss how quantum computers could transform the way we communicate and share information.

By the end of the course, you’ll have a solid foundation in the mathematical theory underlying a variety of cryptographic techniques.

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Skip to 0 minutes and 6 seconds The ability to use codes to send secret messages has been of critical importance to civilisations for thousands of years. In modern everyday life, our ability to share confidential messages with one another underpins the security of internet banking, online shopping and instant messaging services. Mathematics has famously been used to break codes such as that used by the Enigma Machine in World War II. But maths also forms the foundation of all cutting edge cryptographic techniques. In this three-week course you will explore the history of cryptography, with an emphasis on the mathematical developments used to create – and break – some of the strongest codes known.

Skip to 0 minutes and 53 seconds You will investigate permutations, modular arithmetic, prime numbers and more, as we examine the codes used by Roman emperors, Nazi Germany, and modern electronic communications. You’ll also learn about the exciting development of quantum computers, and what they might mean for the future of cryptography. You will join some of York’s world-renowned mathematicians – specialists in number theory, statistics and quantum information – as they lead you through this fascinating and far-reaching topic. So join us to explore the mathematics of cryptography, and learn to unlock the secrets behind secure communication.


  • Week 1

    From the Romans to the 1800s

    • Welcome and introduction

      An introduction to the course objectives and a chance to think about how you will learn. There is a welcome video from the lead educator and also an opportunity to introduce yourself and meet fellow learners for the first time.

    • The Caesar Cipher

      How did the Romans send encrypted messages? And how can we describe this using the language of mathematics?

    • Substitution Ciphers

      Shift ciphers are pretty simple to break: once you've worked out one letter, you've worked out them all! Our next type of cipher -- at least on the face of it -- is much more complicated.

    • The Vigenère Cipher

      The Vigenère cipher used a codeword, making it much more secure than a simple substitution cipher: it went unbroken for over 300 years!

    • Looking back, and ahead

      A short quiz to end the week, as well as another chance for you to discuss with classmates what we've been learning about.

  • Week 2

    The Engima machine

    • Introduction

      An introduction to the Enigma machine, which will form the focus of this week's activities.

    • How Enigma was used

      Here we explain how the Enigma machine was set up each day, and how it was used to send messages.

    • Permutations

      In order to understand how Enigma was broken, we need to learn more about permutations. In particular, we need to investigate their cycle structure.

    • Breaking Enigma

      How the mathematics of permutations led to the breaking of Enigma.

    • Looking back, and ahead

      A short quiz to end the week, as well as another chance for you to discuss with classmates what we've been learning about.

  • Week 3

    21st century cryptography

    • Public key cryptography

      An introduction to the idea of public key cryptography

    • Diffie–Hellman key exchange

      The first public key algorithm we will look at was invented by Whit Diffie and Martin Hellman. Let's look at the mathematics behind how it worked.

    • The RSA algorithm

      The RSA algorithm is another public key method, but one which allows messages (i.e. secrets that the sender can control) to be encoded. It forms the basis for all modern day secure communications.

    • Quantum cryptography

      A look into the future of cryptography offered by quantum computers.

    • Looking back, and ahead

      The conclusion to our course. A short quiz to end the week, as well as another chance for you to discuss with classmates what we've been learning about.

When would you like to start?

Start straight away and join a global classroom of learners. If the course hasn’t started yet you’ll see the future date listed below.

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Learning on this course

On every step of the course you can meet other learners, share your ideas and join in with active discussions in the comments.

What will you achieve?

By the end of the course, you‘ll be able to...

  • Summarise the history of cryptography, and the role played by mathematics in cracking some famous ciphers.
  • Assess how the concepts of frequency analysis and modular arithmetic can be applied to code-breaking.
  • Calculate with permutations and analyse their cycle structure.
  • Explain the importance of prime numbers and factorisation to the security of modern day communications

Who is the course for?

This course is designed for anyone interested in learning more about the history and future of code breaking, with an emphasis on mathematical concepts that have shaped cryptography.

It will particularly interest A-Level or undergraduate students who want to go beyond their syllabus and explore a cutting-edge field of mathematics and technology.

Who will you learn with?

I'm a mathematician working at the University of York, where I do a mix of teaching and research.

Senior lecturer in probability and statistics in the Department of Mathematics at the University of York, UK.

Who developed the course?

University of York

The University of York combines the pursuit of academic excellence with a culture of inclusion, which encourages everyone – from a variety of backgrounds – to achieve their best.

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Develop skills to further your career

  • Access to this course
  • Access to 1,000+ courses
  • Learn at your own pace
  • Discuss your learning in comments
  • Tests to boost your learning
  • Digital certificate when you're eligible

Cancel for free anytime

Buy this course

$109/one-off payment

Fulfill your current learning need

  • Access to this course
  • Learn at your own pace
  • Discuss your learning in comments
  • Tests to boost your learning
  • Printed and digital certificate when you’re eligible

Limited access


Sample the course materials

  • Access expires 4 May 2024

Find out more about certificates, Unlimited or buying a course (Upgrades)

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