summary of group projects

History of RSA
Cryptography goes back to ancient Greece and Rome. Julius Caesar used a simple shift code to send and receive messages, this was called the Caesar Cipher. More recently RSA has taken over the encryption process. RSA stands for Rivest, Shamir, and Adleman. These were the people that first publicly described this particular process of cryptography, in 1978. This was the first algorithm suitable for signing and encryption. It was also one the first great advances in public key cryptography. RSA was released to the public in Sept. 2000 by the RSA security, but the National Security Agency wanted to keep it secretive. RSA involves three steps: key generation, encryption, and decryption. There is still the open question on whether or not RSA can be broken, i.e. intercepted and decoded by a math ninja without the decryption key. At the time our book was written the largest factored number is 135 digits long. Today the code is considered "secure" if the key is about 400 digits long and is the product of two 200 digit primes. It is not difficult to find two random primes, but to factor a number, 150 digits long, into the product of two large primes would take 100 million computers operating at 10 million instructions per second about 50 million years under the fastest algorithms currently known. (That's pretty epic!) Factoring a large number into two primes is very difficult because there is no efficient algorithm currently known.

Ceasar Cipher
The original Ceasar Cipher is a letter shift of three. So when you want to say the letter "A" you would encrypt it to letter "D"

Wallpaper groups

Definition of wallpaper group: (loosely) Simply repeating patterns in the plane $\bb{R}^2$. (P.162, book)
The study of wallpaper groups is the study of the symmetries when a pattern in the shape of a square, parallelogram, rhombus or hexagon is repeated by translations along two nonparallel vector directions to fill the entire plane.

To construct a wallpaper you must start with a base unit known as the fundamental domain. From this given information and using the different isometries you can construct the infinite wallpaper.

** Representation Theory**
Deals with linear transportations of vector space. Permutation groups under matrix multiplication.

Group Action
A group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set."Group actions generalize group multiplication. If G is a group and X is an arbitrary set, a group action of an element $g\in$G and $x\in$X is a product,$gx$, living in X. Many problems in algebra may best be attacked via group actions. For example, the proofs of the Sylow theorems and of Burnside's Counting Theorem are most easily understood when they are formulated in
terms of group actions."

** Classifications of finite simple groups**

$Simple Group$
A simple group in mathematics is a nontrivial group whose only normaly subgroups are the trivial group and the group itself. A normal subgroup exists if the left and right cosets of that subgroup are equal. An example of a simple group would be: $A_5$ or $Z_p$ where p is a prime number.

Monster Group
The Monster Group is a part of the classification of simple finite groups. It is the group of 20 groups out of 26 that are a part of the Sporadic sub-groups. It is one of the largest groups that is known.

page revision: 11, last edited: 18 May 2010 11:34