Dr James HirschfeldASMAA MUSAED AYAD ALBALAWI2022-06-042022-04-222022-06-04110796https://drepo.sdl.edu.sa/handle/20.500.14154/66316The purpose of this dissertation is to provide an overview of affine and projective planes, as well as the use of projective plane in public key cryptography. We start with a review of affine and projective space and plane, then move on to all of the con- cepts necessary to explain Bezout’s theorem, which introduces the number of places at which two curves meet. The essential features of elliptic curves are then discussed in detail. We continue our dissertation by discussing public key cryptography and how elliptic curves relate to it. Finally, we describe a projective plane group law and its application for public key cryptography.69enThesis