Master's Theses

Fall 2011

Degree Type

Thesis - Campus Access Only

Degree Name

Master of Science (MS)

Mathematics

Brian Peterson

Keywords

continued, elliptic, factorization, integer, lenstra, sieve

Mathematics

Abstract

In the past, factoring integers was thought to be of little benefit to the mathematical world. The subject of integer factorization methods is one that has only relatively recently become of interest to mathematicians, as cryptography and other fields have made its thorough study a necessity.

Here, we study the continued fraction method of factorization, which is an early method that relies on useful formulas derived from the theory of continued fractions. We also consider the more efficient successor of the continued fraction method, called the quadratic sieve. Sieving greatly reduces the time it takes to find values that have a particular characteristic that is of use when trying to factor a given integer. Both the continued fraction and quadratic sieve methods are predecessors of the number field sieve, which we do not outline here. The number field sieve is used today to factor large integers.

We also discuss Lenstra's elliptic curve factorization method, in which a group of elliptic curves over a particular ring is used, incredulously, to factor integers. We touch on some of the theory of elliptic curves for the unfamiliar reader. Variations of the elliptic curve method are used today when attempting to find smaller factors of a given integer.

In this thesis, a thorough description of these three integer factorization algorithms will be presented. The discussion of each method includes background information, a summary, and detailed examples of the factorization of large integers that help illustrate these complex factorization algorithms.

COinS