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# Our National Security Depends On "Algebra"

How Would Our National Security Depend on Algebra?
The National Security Agency (NSA) makes use of mathematicians in the field of cryptology, the science of encoding and decoding messages.

NSA Mathematicians use math, and specifically Abstract Algebra, to:

• Design and analyze complex algorithms
• Express difficult cryptographic problems in Mathematic terms
• Apply their work to find a solution or demonstrate that a solution cannot be found1

What is Abstract Algebra and How Is It Used To Break Codes?
Abstract Algebra, a course typically taken my Mathematics or Physics majors in college, actually is a generalized form of the algebra one learns in high school. For example, an Algebraic Group consists of an operation that must be associative, an identity must exist for each element of the group, and an inverse must exist for each element of the group.

In order to break a code, one must be able to reproduce the coding algorithm along with a matching decoding algorithm. In other words, one must be able to develop a formula or formulas that will both code messages and reverse the coding process in order to decode an encrypted message. We call these formulas functions.  We would call the formula that reverses the operation the inverse function.

An Example of Coding and Decoding
An encryption method used by Julius Caesar to communicate with his generals is called the Caesar Cipher and simply shifts each letter a specified number of places to make a new letter.  For example, if each letter were shifted 2 places, the word CAT is encoded as ECV since C shifts to E, A shifts to C, and T shifts to V. An online encoding program that I wrote up allowing you to encode your own messages is given here.

Some interesting examples of coding used in driver's licenses, ISBN numbers, and other areas is given at  Washington State Check Digit Scheme