Cracking DHCE (Diffie-Hellman color exchange)

I recently saw a video that explains Diffie–Hellman key exchange in terms of mixing colors of paint. It's a wonderfully simple and informative analogy, that Wikipedia actually uses as well. If you don't know about Diffie-Hellman, definitely watch the video and/or read the Wikipedia page to get a handle on it—it's not that complicated once you get the "trick." The color analogy intrigued me because I know just enough about both cryptography and color theory to be dangerous. So in this post, I'm going to attack the security of the color exchange protocol. ("Real" Diffie-Hellman remains secure, as far as I know.) Continue reading Cracking DHCE (Diffie-Hellman color exchange)

Efficient Integer Exponentiation in C

It's surprisingly difficult to find a good code snippet for this on Google, so here's an efficient computation of integer powers in C, using binary exponentiation:

Exact Bounding Boxes for Spheres/Ellipsoids

Finding the tightest axis-aligned bounding box for a sphere is trivial: the box extends from the center by the radius in all dimensions. But once the sphere is transformed, finding the minimal bounding box becomes trickier. Rotating a sphere, for example, shouldn't change its bounding box, but naïvely rotating the bounding box will expand it unnecessarily. Luckily there's a trick to computing minimal bounding boxes by representing the transformed sphere as a quadric surface.

A Beautiful Ray/Mesh Intersection Algorithm

In my last post, I talked about a beautiful method for computing ray/triangle intersections. In this post, I will extend it to computing intersections with triangle fans. Since meshes are often stored in a corner table, which is simply an array of triangle fans, this gives an efficient algorithm for ray tracing triangle meshes.

A Beautiful Ray/Triangle Intersection Method

3D ray/triangle intersections are obviously an important part of much of computer graphics. The Möller–Trumbore algorithm, for example, computes these intersections very quickly. But there is another method that I believe is more elegant, and in some cases allows you to compute the intersection for “free.”

Solving Cubic Polynomials

Although a closed form solution exists for the roots of polynomials of degree ≤ 4, the general formulae for cubics (and quartics) is ugly. Various simplifications can be made; commonly, the cubic $$a_3\,x^3+a_2\,x^2+a_1\,x+a_0$$ is transformed by substituting $$x = t-a_2/3a_3$$, giving Continue reading Solving Cubic Polynomials

Solving Polynomials

A well known (if not by name) theorem is the Abel–Ruffini theorem, which states that there is no algebraic expression for the roots of polynomials with degree higher than 4.

A not-so-well-known fact is that for any polynomial $$P(x)$$, it is possible to find (with exact arithmetic) a set of ranges each containing exactly one root of $$P(x)$$. One such algorithm is due to James Victor Uspensky in 1948. Continue reading Solving Polynomials