Primality Testing and Sub-Exponential Factorization

This paper discusses the problems of primality testing and large number factorization. The first section is dedicated to a discussion of primality testing algorithms and their importance in real world applications. Over the course of the discussion the structure of the primality algorithms are developed rigorously and demonstrated with examples. This section culminates in the presentation and proof of the modern deterministic polynomial-time Agrawal-Kayal-Saxena algorithm for deciding whether a given n is prime. The second section is dedicated to the process of factorization of large composite numbers. While primality and factorization are mathematically tied in principle they are very different computationally. This fact is explored and current high powered factorization methods and the mathematical structures on which they are built are examined.