Project : Supersingular Curves and the Weil Pairing in Elliptic Curve Cryptography

Even first semester calculus students are aware of how calculus, hence analysis, is used to solve problems in engineering. In recent decades the engineering world is gaining more exposure to algebra through the powerful problem solutions it provides. One area that algebra has made significant contributions to is cryptography and, more specifically, public key cryptography. In this paper we aim to illustrate such enhancements by discussing how supersingular elliptic curves and the Weil pairing have impacted elliptic curve cryptography. The paper is structured as follows: In the first few sections we give an overview of public key cryptography and elliptic curves in this context; Then we discuss supersingular curves and the Weil pairing and see how the pairing can be used to break and construct supersingular elliptic curve cryptosystems; Finally, to provide contrast and a little food for thought, we discuss other categories of elliptic curves, an alternative to the Weil pairing (the Tate pairing) and some exciting new index calculus methods for elliptic curves.