Paul Dundon’s Weblog


A little cheese and a little whine

The Mathematics of RSA

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.

Filed under: Maths

Defying Gravity

A comment (from John Z) on an earlier post highlights some interesting research on anti-gravity. Unfortunately, I can find nothing relating directly to the original research after 2000, although New Scientist reports some similar work from 2006.

Filed under: Maths, Tech

Less spin, more swing

Not a post about the upcoming election, but another physics problem.

There’s an ad on TV at the moment where a couple hook their car up to a crane and accelerate underneath it so that, when the cables become taut, they are propelled into the air. Is this possible, we ask? The answer is: sort of – you just need a really high performance engine.

Full details (apologies for the PDF) are here. The solution rests on considering a situation in which the car is a frictionless pendulum (with a hole underneath the crane to allow it to swing) and then calculating the velocity of the car as it ascends past the level of the road. I assume that this is the velocity needed to achieve take-off in the actual scenario. I think this is valid but stand to be corrected.

Incidentally, before you go out and try this, I’m not convinced the cables would remain taut for the descent or what speed the car would hit the ground at.

Filed under: Maths

You spin me right round

The series “Defying Gravity” features a spaceship which generates “artificial gravity” by spinning parts of its body around a central axis. The question naturally occurs:- how fast would a cylinder have to spin to generate a centripetal force equivalent to gravity?

It turns out that the answer, in rpm, is very close to 30 times the reciprocal of the root of the radius (ie 30/sqrt(r)).

The derivation is here (PDF – only way I can get the equations to show reliably).

Filed under: Maths

My Bookshelf

The Golden Bough
The Value of Nothing
The Fire
A Wolf at the Table
Devil Bones

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