A few years ago, I spent a few months with *way* too much time on my hands. One result of this was a paper on the mathematics underpinning the RSA encryption system written for people with an interest in maths but not much knowledge of it beyond high school. Yesterday, an old friend asked for a copy; so here it is (apologies for the PDF; let me know if you’d like it in another format). The first 11 pages review the basic mathematical concepts needed, so if you’re already familiar with prime numbers and modular arithmetic, you can skip them.

**A very quick summary:** Imagine a clock face with millions of numbers, instead of just 12. Suppose that all the numbers 1 to n appear on the clock face, in random order. Now suppose you have two numbers, p and q, so that p+q=n. Point the minute hand of the “clock” to the number that corresponds to the message you want to send. Now, advance it p places clockwise. The number it now points to is your encrypted message. You send the message. The recipient makes their own clock with the numbers in the same order, and points the minute hand at the number you’ve sent them. Then, they advance the minute hand by q places clockwise. The minute hand is now pointing at your message. So, p and q form an asymmetric key pair – one key is used for encryption, and one for decryption. The actual algorithm is more sophisticated (for example, p and q don’t add up to n, since this would make it easy to work out one given the other) but the basic principle is the same.

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