| In this expository thesis we study elliptic curves and their role in cryptography. In doing so we examine an intersection of linear algebra, abstract algebra, number theory, and algebraic geometry, all of which combined provide the necessary background. First we present background information on rings, fields, groups, group actions, and linear algebra. Then we delve into the structure and classification of finite fields as well as construction of finite fields and computation in finite fields. We next explore logarithms in finite fields and introduce the Diffie-Hellman key exchange system. Subsequently, we take a look at the projective and affine planes and we examine the action of the general linear group of degree 3 (over K) on the points of the projective plane P2(K ). We then explore the geometry of the projective plane with Desargues Theorem. Next, we study conics, quadratic forms, and methods of counting intersection of curves. Finally, we study forms of degree 3 and we are able to explore cubics and the group law on an elliptic curve which leads us to our ultimate goal of examining the role of elliptic curves in cryptography. |
