Thursday, June 28, 2012

RSA encryption and the other Sun Tzu


Before we get started on how exactly RSA works and how it's related to Chinese mathematicians, it's important to have a basic comprehension of modular arithmetic, totients, and the "modular multiplicative inverse". None of these are very difficult, but they aren't commonly talked about, so here's a quick refresher:


In simplest terms, what you're looking for in a modular problem is the remainder of a division. Dividing 7 by 3 gives 2 with a remainder of 1. Expressed as a modular equation, 3 is the modulus, and 7 and 1 are congruent. The equation can be expressed like this:

7 ≡ 1 (mod 3)

Wednesday, June 13, 2012

Four ways to construct a pentagon

There are animations of the 4 compass and straightedge construction techniques at the bottom of this entry, but they require an HTML5 capable browser. (I suggest Chrome, but the animations should work with a modern Firefox, Safari, or Chromium... but let me know if they don't.)

Also included are animations of the first 6 "problems" in book 1 of Euclid's Elements, as a preview of a larger project I'm working on, and two variations on creating a heptadecagon (a seventeen-sided polygon).

Over the last few weeks I've been spending what free time I have obsessing over the odd mathematics of Argand diagrams, complex roots, and how they relate to constructing regular polygons with a compass and straightedge. I found, among other things, that I had forgotten quite a bit from my trig and geometry classes from 25 years ago.