The RSA Challenge and Clock Arithmetic
Many years ago now I attended an inspirational presentation at Plymouth University which touched on a couple of fascinating (for me) mathematical ideas. One of these was the Mandelbrot Set; the other was Internet Encryption and the RSA Challenges. Whilst I have used The Mandelbrot Set in lessons on many occasions since then, the RSA Challenges and how they relate to Internet Encryption had remained dormant in my thinking ever since (although my interest deepened on reading Marcus du Sautoy’s book, The Music of the Primes). Basically, I needed an excuse to get my head around the mathematics underpinning the encryption process and that’s where the Researchers in Residence project comes in.
All the PhD projects outlined by the students sounded complex and bewildering when they were presented to us at our initial meeting and I would have been very happy to explore any of the three areas on offer. However, in retrospect I am very pleased with my choice as it not only related to a real-life problem that all the students involved were already aware of but also necessitated immersing myself in a branch of mathematics in which I was, at the time, completely unfamiliar – modular arithmetic.
It soon became clear to me that this was not a trivial undertaking. The more I explored the process of encryption, the deeper I needed to go in my understanding of modular arithmetic. I had forgotten how rewarding studying new ideas and concepts can be, but also how exhausting, frustrating and daunting it frequently is. The process was sobering and was useful as a wakeup call as this is what we expect from our students day in, day out!
It also became clear that presenting such a substantial collection of new concepts would require a considerable amount of lesson time and numerous activities, so I have outlined each session below and provided a link to a more detailed explanation and SMART board activities on this overview.
Time Taken (Overview of Sessions)
Session 1: Introduction to Modular Arithmetic
This series of lessons will require in the region of three to four hours of teaching time and is broken down into the following sessions:
Explore the basic workings and rules of modular arithmetic and provide an introduction to proof. Tests of divisibility.
Session 2: Basic Principles of Coding and Code Breaking
Look at using Caesar Ciphers as a basic coding system and the role of public and private keys. Understand the role of modular arithmetic in deciphering codes once the private key is known.
Session 3: Euler’s Totient Function
Explore Euler’s Totient function and find the conditions for multiplicative inverses to exist. Use the Totient Function to determine last digits of large powers of numbers.
Session 4: Fermat’s Little Theorem and RSA
Explore conditions for Fermat’s Little Theorem and find the conditions for when it works. Gain an awareness of RSA numbers and how they apply to internet encryption.
These sessions were designed to be used by very able mathematicians in Key Stage 5 although many of the ideas could be adapted to use with top end mathematicians in Key Stage 4 or even Key Stage 3. In the event, roughly 40 students were involved in the day – year 13 and year 12 Further Mathematicians, some of the most able Y11 students and a group of IB Higher students.
Students were put into vertical groups so the older and more experienced students were able to support and coach the younger students. Some of the Year 11s found it a little daunting but I think it was good for some of them to realise that they still have a long way to go with the subject. As the day progressed many of the Year 13 students took over the sessions and there were some exceptional examples of students explaining to other students from the board.
I was very pleased with the response the day had. To have so much interest for an event that would not directly relate to any of the students’ examinations (apart from some of the IB Higher Discrete Module), especially in this era of examination cramming was extremely encouraging. Feedback received for the day was generally very positive, although students did comment that they found it challenging and draining. In a way, I would have liked to break up the sessions a little but that would have had the effect of losing momentum between sessions and losing the ‘event’ aspect of the day.