Thursday, June 28, 2012

RSA encryption and the other Sun Tzu

Background

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:

Modulus

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.