Research
Index
| Research |
In the course of my studies, I solve (often partially) a lot of standard exercises. I provide these with no claims to correctness. However, if these are any help to you, please let me know!
Partial solutions to exercises in Gathmann's Algebraic Geometry notes:
Partial solutions to exercises in Silverman's Arithmetic of Elliptic Curves: most of Chapter 2 in PDF and HTML
Non-Vanishing of Uppuluri-Carpenter Numbers, Nick Alexander. Submitted.
We prove that the Uppuluri-Carpenter numbers vanish at most twice, answering a long-standing combinatorial question attributed to H. Wilf.
My Master's thesis, written under the supervision of Alfred Menezes and Edlyn Teske while I was at the University of Waterloo, considers constructive applications of algebraic tori in cryptography. The abstract reads:
``Communicating bits over a network is expensive. Therefore, cryptosystems that transmit as little data as possible are valuable. This thesis studies several cryptosystems that require significantly less bandwidth than conventional analogues. The systems we study, called torus-based cryptosystems, were analyzed by Karl Rubin and Alice Silverberg in 2003 [RS03]. They interpreted the XTR [LV00] and LUC [SL93] cryptosystems in terms of quotients of algebraic tori and birational parameterizations, and they also presented CEILIDH, a new torus-based cryptosystem. This thesis introduces the geometry of algebraic tori, uses it to explain the XTR, LUC, and CEILIDH cryptosystems, and presents torus-based extensions of van Dijk, Woodruff, et al. [vDW04, vDGP+05] that require even less bandwidth. In addition, a new algorithm of Granger and Vercauteren [GV05] that attacks the security of torus-based cryptosystems is presented. Finally, we list some open research problems.''
Janice Tytaneck and I compiled a list of errata detailing errors in Rene Schoof's paper, Elliptic Curves Over Finite Fields and the Computation of Roots mod $p$. From the introduction:
``This document lists errata for Rene Schoof's 1985 paper, \textit{Elliptic Curves Over Finite Fields and the Computation of Roots mod $p$}. As published, Schoof's work contains several typographical errors. To ease understanding, we have provided a corrected version of Schoof's text on pages 488 and 489, as well as listed individual errors. Changes are noted in boldface. We start at equation 17, at the bottom of page 488.''
I am happy to maintain this errata sheet; please email me (ncalexan [at] math [dot] uci [dot] edu) corrections.