Generative Adversarial Networks as a Method to Produce Elliptic Curves by Rank Open Access

Abstract

Elliptic curves are studied in many areas of mathematics, yet there is still much to learn about their properties. One key property—rank—is particularly tricky. Conjecture states that nearly all elliptic curves are of rank zero or one, with relatively few exceeding rank one.(1) However, it is believed that there is no bound on how high the rank of elliptic curves can be, and the current discovered record is a curve of rank 28. Because of their varied uses in mathematics, it is common for researchers to search for elliptic curves with specific properties such as a rank value. One approach to searching for elliptic curves by rank is to randomly check curves for their rank, a time-consuming and inefficient method. Generative Adversarial Networks (GANs) are an emerging machine learning technique that allows for the creation of fake data that imitates real data. Using GANs, it is possible to create a model that generates elliptic curves by specified rank, given a sufficient training set of elliptic curves to feed the model. This project proposes GANs as a new method for generating elliptic curves and creates a GAN model that generates elliptic curves of rank one at a more effective rate than a guess and check approach. 

Table of Contents

Introduction and Background Information.................................................................... 1

Elliptic Curves............................................................................................................ 1

Elliptic Curves Over the Rational Numbers.................................................................. 2

Group Law................................................................................................................ 3

Rank of Elliptic Curves................................................................................................ 4

Generative Adversarial Networks (GANs).................................................................... 6

The Generator .......................................................................................................... 6

The Discriminator ..................................................................................................... 7

The Adversarial Relationship Between the Generator and Discriminator ...................... 8

Data................................................................................................................................ 8

Source...................................................................................................................... 8

Filtering.................................................................................................................... 9

Methods........................................................................................................................ 10

Loss Functions, Backpropagation, and Optimization................................................... 11

Parameters.............................................................................................................. 12

GAN Architecture..................................................................................................... 13

Issues During Training.............................................................................................. 14

Final Model............................................................................................................. 16

Results........................................................................................................................... 17

Discussion..................................................................................................................... 18

References ................................................................................................................... 20

About this Honors Thesis

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School	Emory College
Department	Quantitative Science
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Statistics Mathematics
Keyword	Machine Learning Rank Elliptic Curves Generative Adversarial Networks Math Statistics
Committee Chair / Thesis Advisor	Jeremy Jacobson, Emory University
Committee Members	Christoph Breunig, Emory University David Zureick-Brown, Emory University

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