Eric Landquist

This image represents an object called the infrastructure of principal ideal class and is a set of reduced principal ideals that almost forms a group under ideal composition, which is multiplication followed by reduction. The infrastructure exists in any number field or function field whose ring of integers has positive unit rank and is useful for computing regulators and fundamental units in these fields. In unit rank 1, by far the most understood case, the infrastructure is a cycle. In the unit rank 2 setting, however, the infrastructure is bicyclic like a torus, as the image shows. The spheres represent reduced principal ideals, the red and orange lines represent so-called baby steps in one direction, and the blue lines represent baby steps in the other direction in the infrastructure. Specifically, this is the infrastructure corresponding to the cubic function field defined by the curve:

My research interests fall under the general category of Computational Algebraic Number Theory and Cryptography. In the past I have studied integer factorization and the discrete logarithm problem, but my current research focuses on cubic function fields of large characteristic.

One current project is looking into optimizing algorithms to compute divisor class numbers in cubic function fields. Currently the arithmetic is developed for purely cubic fields of rank 0, but extensions to each unit rank and to general cubic function fields is planned. At present, class numbers as large as 10²⁴ have been computed in parallel in genus 3 and 4 fields. Computations have taken 4 and 72 machine days, in the respective cases.

Another, and more interesting, component of my research examines cubic function fields whose maximal order has unit rank 2. In this setting the ideal class group is small and usually trivial, but within the principal ideal class we find a very unusual object called the infrastructure. This has been studied extensively in quadratic and cubic number fields and function fields of unit rank 1, but very little work has been done in the unit rank 2 case. The goal of my research then is to develop a theory of unit rank 2 infrastructure. I am currently focusing on the particular case of singular genus 3 fields over F_q for q = 1 (mod 3), and looking into how to navigate the infrastructure more quickly using a baby step-giant step procedure. This will allow one to compute divisor and ideal class numbers, the regulator, and fundamental units of unit rank 2 cubic function fields much more quickly.

• Paper I wrote for MATH 488, Cryptographic Algorithms on The Quadratic Sieve Factoring Algorithm (pdf).
• Paper I wrote for MATH 420, Computer Algebra Systems on The Number Field Sieve Factoring Algorithm (pdf).
• Paper I wrote for MATH 488, Integer Factorization on An Implementation of Zhang's Special Quadratic Sieve and Possible Extension (pdf). Disclaimer: The possible extension as presented in this paper has some mistakes in it. It turns out that the extension one can do does factor integers of a special form, but it does not use a sieve. Instead, it finds integers of special forms that factor nicely, such as x^2 - y^2 = (x+y)(x-y).
• Possible Ways to Extend Zhang's Special Quadratic Sieve (pdf)