This LaTeX paper is the culmination of many months of research, planning, and work under the guidance of my Mathematics Approaches and Analysis Professor. Attached is a shorter copy, which was modified and made more consice from submission for the IB Extended Essay project. This copy focuses more on an expository lens to these mathematical and computer-science related concepts, since I couldn’t assume the IB examiner would be familar with this area of math. I realized early on that my research and exploration would surpass the limits of the Extended Essay for this reason, so this project is beyond the scope of just my Extended Essay alone. I’m proud with the answers I reach in this paper, since I first considered this question more than a year ago when I stumbled upon the rich mathematics of elliptic curves. In essence, I wanted satisfactory reasoning as to why the group structure of an elliptic curve group was, on average, more resistant to cryptographic breaks than similarly sized classical groups used in crypotrgaphy, like the integers mod n under multiplication. The research process began this past summer, where I took the time to read up on implementations of elliptic curves—using my previous self-study of group theory as a starting point. I think this paper suceeded in my goal of making these concepts more intuitive, accessible and understandable to a wider audience. I hope you enjoy reading it as much as I enjoyed writing it!