MSRI-UP 2010: Elliptic Curves and Applications

• Jose Ayala (California State University at Pomona)
• Alex Barrios (Brown University)
• Renee Brady (Florida A&M University)
• Juan Cervantes (Lewis and Clark)
• Naleceia Davis (Spelman College)
• Alexander Diaz (University of Puerto Rico at Mayaguez)
• Zachary Flores (Michigan State University)
• Erin Jones (Carlton College)
• Kelsy Kinderknecht (University of Kansas)
• Megan Ly (Loyola Marymount University)
• Keatra Nesbitt (University of Northern Colorado)
• Toya Skeete (Spelman College)
• Caleb Tillman (Reed College)
• Anna Tracy (Sewanee: the University of the South)
• Shawn Tsosie (University of Massachusetts at Amherst)
• Pam Urresta (Union College)
• Markus Vasquez (Oklahoma State University)
• Charles Watts (Morehouse College)

• Alexander Diaz (University of Puerto Rico at Mayaguez), Zachary Flores (Michigan State University), and Markus Vasquez (Oklahoma State University); co-mentored by Ebony Harvey (Boston College).

• ABSTRACT: In 1640, Pierre de Fermat sent a letter to Bernard Frénicle de Bessy claiming that that there are no four rational squares in a nontrivial arithmetic progression; this statement was shown in a posthumous work by Leonhard Euler in 1780. A modern proof reduces this statement to showing that a certain elliptic curve has no rational points other than torsion. A 2009 paper on the ArXiv by Enrique González-Jiménez and Jörn Steuding discussed a slight generalization by looking at four squares in an arithmetic progression over quadratic extensions of the rational numbers. In this project, we give explicit examples of four squares in arithmetic progressions, and recast many ideas by performing a complete 2-descent of quadratic twists of certain elliptic curves.

• Preliminary Results in the MSRI Journal (2010): [Download Article]
  Final Presentation at MSRI on July 23, 2010: [Download Slides] | [Watch Video] | [Download Video]
  Poster Presentation at the SACNAS National Conference on October 2, 2010: [Download Poster]

• Renee Brady (Florida A&M University), Naleceia Davis (Spelman College), and Anna Tracy (Sewanee: the University of the South).

• ABSTRACT: Many of us communicate via text messages, and some even "microblog" via Twitter. Both services, following a suggestion in 1985 from Friedhelm Hillebrand, use just 160 characters, thus giving a nice application to cryptography over finite fields generated by primes with at least 2560 bits. We discuss how to use elliptic curves to send such messages securely. In the process, we will explain how to use Unicode and an idea from 1987 by Neal Koblitz to encode a message as a point on a fixed elliptic curve; combine ideas of Diffie-Hellman and Massey-Omura to send an encrypted message via a key exchange; then use Lagrange's Theorem to reconstruct the original message as plaintext.

• Preliminary Results in the MSRI Journal (2010): [Download Article]
  Final Presentation at MSRI on July 23, 2010: [Watch Video] | [Download Video]
  Poster Presentation at the SACNAS National Conference on October 2, 2010: [Download Poster]

• Alex Barrios (Brown University), Caleb Tillman (Reed College), and Charles Watts (Morehouse College); co-mentored by Luis Lomelí (University of Iowa).

• ABSTRACT: Given three positive, relative prime integers such that the first two sum to the third, it is rare to have the product of the primes dividing them to be smaller than each of the three. In 1985, David Masser and Joseph Osterlé made this precise by defining a "quality" for such a triple of integers; their celebrated "ABC Conjecture" asserts that it is rare for this quality to be greater than 1 -- even through there are infinitely many examples where this happens. In 1987, Gerhard Frey offered an approach to understanding this conjecture by introducing elliptic curves. In this project, we introduce families of triples so that the Frey curve has nontrivial torsion subgroup, and explain how certain triples with large quality appear in these families. We also discuss how these families contain infinitely many examples where the quality is greater than 1.

• Preliminary Results in the MSRI Journal (2010): [Download Article]
  Final Presentation at MSRI on July 23, 2010: [Download Slides] | [Watch Video] | [Download Video]
  Poster Presentation at the SACNAS National Conference on October 2, 2010: [Download Poster]

• Juan Cervantes (Lewis and Clark), Keatra Nesbitt (University of Northern Colorado), and Kelsy Kinderknecht (University of Kansas); co-mentored by Katie Ansaldi (University of Notre Dame).

• ABSTRACT: There is only one abelian group of order 8 which is noncyclic yet contains a cyclic subgroup of order 4; in 1973, Andrew Ogg showed there exist infinitely many elliptic curves defined over the rationals with this group as its torsion subgroup. It is natural to ask what are the properties of the elliptic curves in this family. In 2005, K. Ansaldi, A. Ford, J. George, K. Mugo, and C. Phifer generalized ideas of N. Rogers from 2000 to find curves with rank 0 through 6. That same year, Noam Elkies did better: he found a curve in this family with rank 8. This project seeks to extend ideas of Ansaldi et al. to find an elliptic curve in this family with rank 9. We explain how to sort through such curves by focusing on a subfamily with aforementioned torsion subgroup and positive rank, and discuss the implementation of our ideas on a high performance computing cluster at Miami University of Ohio.

• Preliminary Results in the MSRI Journal (2010): [Download Article]
  Final Presentation at MSRI on July 23, 2010: [Download Slides] | [Watch Video] | [Download Video]
  Poster Presentation at the SACNAS National Conference on October 2, 2010: [Download Poster]

• Jose Ayala (California State University at Pomona), Erin Jones (Carlton College), and Toya Skeete (Spelman College).

• ABSTRACT: In 50 B.C., Julius Caesar wrote to Marcus Cicero using a "secret code" which had shifted the Latin alphabet three places. Today, we may generalize the Caesar cipher to any affine cipher by translating characters via a linear transformation, where we canonically identify characters as nonnegative integers via Unicode. Modern cryptosystems generalize this even further by translating characters via exponential maps. In this project, we list various attacks to these cryptosystems. We explain how to decrypt affine ciphers through frequency analysis, as well as how to decrypt public key systems through factorization of large integers. We also explain the importance of understanding Discrete Logarithms, John Pollard's p-1 Method as introduced in 1974, and Hendrik Lenstra's Elliptic Curve Factorization Method as introduced in 1987.

• Preliminary Results in the MSRI Journal (2010): [Download Article]
  Final Presentation at MSRI on July 23, 2010: [Watch Video] | [Download Video]
  Poster Presentation at the SACNAS National Conference on October 2, 2010: [Download Poster]

• Megan Ly (Loyola Marymount University), Shawn Tsosie (University of Massachusetts at Amherst), and Pam Urresta (Union College); co-mentored by Luis Lomelí (University of Iowa).

• ABSTRACT: Leonhard Euler noted that there exists an infinite set of rational points on the unit circle such that the pairwise distance of any two is also rational; the same statement is nearly always true for lines and other circles. In 2004, Garikai Campbell considered the question of a rational distance set consisting of four points on a parabola. We introduce new ideas to discuss a rational distance set of four points on a hyperbola. We will also discuss the issues with generalizing to a rational distance set of five points on an arbitrary conic section.

• Preliminary Results in the MSRI Journal (2010): [Download Article]
  Final Presentation at MSRI on July 23, 2010: [Watch Video] | [Download Video]
  Poster Presentation at the SACNAS National Conference on October 2, 2010: [Download Poster]