Eric Landquist

[Contact Information] [Employer Information] [Teaching] [Research]

Contact Information

I am currently holding a position as a postdoc working with the Centre for Information Security and Cryptography at the University of Calgary. Contact information may be found here.

Email: elandqui@ucalgary.ca or eric.landquist@gmail.com

Employer Information

Teaching

Past Courses:

Fall, 2001: Math 118: Numeracy, Section A2
Fall, 2002: Math 242.
Fall, 2003: Math 130
Spring, 2004: Math 242.
Fall, 2004: Math 242
Spring, 2006: Math 242 C&M
Fall, 2006: Math 241
Spring, 2007: Math 241 C&M
Fall, 2007: Math 241 C&M
Spring, 2008: Math 241 C8 C&M

Research

Ph.D. Thesis

E. Landquist, Infrastructure, Arithmetic, and Class Number Computations in Purely Cubic Function Fields of Characteristic at Least 5, University of Illinois at Urbana-Champaign, 2009. [PDF 1280 KB]

Publications and Preprints

M. Bauer, E. Landquist, and R. Scheidler, Arithmetic in the Jacobian of a purely cubic function field. In preparation.
E. Landquist, R. Scheidler, and A. Stein, Class number and regulator computation in cubic function fields. In preparation. (To be submitted to Math. Comp. shortly.)
E. Landquist, P. Rozenhart, R. Scheidler, J. Webster, and Q. Wu, An explicit treatment of cubic function fields, with applications. Accepted, Canadian Journal of Mathematics, 2008. Preprint in PDF format. (Revised May 28 2008)

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:

y³ = x⁶ + 5x⁵ + 6x⁴ + 5x² over F₇.

Click on the image to see a larger version.
Image created using Mathematica.

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 looks into optimizing algorithms to compute divisor class numbers of purely cubic function fields of large characteristic. At present, 28-digit class numbers have been computed in function fields of genus 3 and 25-digit class numbers have been computed in function fields of genus 4. In both cases, we found examples in function fields of unit ranks 0 and 1, extracting the regulator and computing in the infrastructure of the function field in the latter case.

My next project will be to extend the infrastructure methods used above to compute the regulator, R, of a function field of unit rank 2 infrastructure. While current methods require O(R) infrastructure operations, I believe that we can reduce this to O(R^0.5) operations and possibly as fast as O(R^0.4) operations.

Factoring Papers
It was easy to find a factor!

Paper I wrote for MATH 488, Cryptographic Algorithms on The Quadratic Sieve Factoring Algorithm (pdf). Disclaimer: There are some mistakes in this paper, so please refer to the published literature for accurate results. I plan to fix these mistakes when I have time. I do hope that you will find this as a gentle introduction to the algorithm written from the perspective of someone who (at the time) was just learning about the algorithm himself.
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)

Slides from Seminar and Conference Talks

Class Number Computation in Cubic Function Fields on December 19, 2007 at the West Coast Number Theory Conference in Pacific Grove, CA.
Class Number Computation in Cubic Function Fields on November 3, 2007 at the Midwest Number Theory Conference for Graduate Students in Madison, WI.
The Splitting of Primes in K(x)[Y]/(Y^3 -AY +B) on October 28, 2006 at the Midwest Number Theory Conference for Graduate Students here in Urbana, IL.
Donuts and Cubic Function Fields on Sep. 18, 2006, how real cubic function fields and donuts are related.
So you want to create a factoring algorithm? on Feb. 19, 2003, tying factoring in with the Curse of the Bambino.
Sieving Techniques, Football, and Taboo (pdf), Oct. 2, 2002. If you missed the talk, the slides may not make much sense.

PGP Public Key for landquis@uiuc.edu.

Last updated April 7, 2009.