#### Date of Award

6-2009

#### Document Type

Union College Only

#### Degree Name

Bachelor of Science

#### Department

Mathematics

#### First Advisor

Kathyrn Lesh

#### Language

English

#### Keywords

algorithm, intersection, equation, curve, theorem

#### Abstract

The security of many cryptosystems relies on the diﬃculty of factoring a number that is a product of two prime numbers (usually hundreds of digits long) so it is important to understand ways that one might attempt to ﬁnd divisors of these large composite numbers. The Elliptic Curve Factoring Algorithm, ﬁrst developed by H.W. Lenstra, is a method to factor numbers using elliptic curves over a ﬁnite ﬁeld. By taking an equation for an elliptic curve (of the form y2 = x3 + Ax + B) and performing addition on the points on the elliptic curve over a ﬁnite ﬁeld, we are sometimes able to ﬁnd a factor. Demonstrating when the algorithm is successful is dependent on Hasse’s Theorem, which we also prove.

#### Recommended Citation

Britton, Sarah G., "The elliptic curve factoring algorithm" (2009). *Honors Theses*. 1274.

https://digitalworks.union.edu/theses/1274