Elliptic Curve Cryptography

Thompson, Dave

Publication

Elliptic Curve Cryptography

Thompson, Dave

Date

2016

Additional date(s)

2016-05-19

Abstract

While it is relatively well known that larger field orders in elliptic curves allow for increased security in a cryptographic setting, it was the goal of our research to discover if patterns would emerge when observing average point orders within the first five fields. Using SageMath, we computed the average point orders across every elliptic curve with constraints subject to a given finite field that passed through each point. From this, we also calculated an average of averages that we used to represent the average point order for the entire field. Our findings point towards the fact that if patterns do exist, they are complex and ultimately require more data sets than those which have been analyzed in this report. While our data suggests an almost linear relationship between the increase in field size with the increase in average point order, in truth this relationship may deviate significantly from one that is linear after larger data sets are taken into account.