Understanding Elliptic Curves in the Cryptographic World
|Cryptography is a topic that can get really complex very fast. Every cryptographic system is based on some type of problem. Elliptic curve cryptography is based on the discrete log problem using elliptic curves. Elliptic curves have their own group and field properties. The algorithms that come from elliptic curve cryptography are simple to follow, but hard to crack. Elliptic curve cryptography also has several successes and challenges in the corporate world. |
On the Algebra of Rotations in ℝ3: An Exploration of Representations by Quaternions and SU(2)
The need to represent rotations of objects in 3-D Euclidean space arises daily in many fields: animation, computer vision, and physics, to name a few. Ever since Euler first described his eponymous angles, without giving a tractable method for constructing them, mathematicians have longed for a better system to describe rotations. In 1843, William Rowan Hamilton had an epiphany whilst walking across Brougham Bridge in Dublin with his wife. Therein he inscribed the laws defining the quaternions, forever changing the face of rotations. The quaternions, when limited to having unit norm, form a group under multiplication which is isomorphic to SU(2). This presentation will discuss the interplay between these two groups and will clarify the use of quaternions to represent rotations. We will delve into the relationship between SU(2) and SO(3).