Elliptic Curve Cryptography and Quantum Computing

In the year 2007, a slightly nerdy girl fell in love with all things math. Even though she only was exposed to a small part of the immense field of mathematics, she knew that math would always have a place in her heart. Ten years later, that passion for math is still burning inside. She never thought she would be interested in anything other than strictly mathematics. However, she discovered a love for computer science her sophomore year of college. Now, she is graduating college with a double major in both mathematics and computer science. This nerdy girl is me. It was my first semester, freshman year at Ouachita Baptist University when I discovered an interest in cryptography. Discrete Mathematics had a small portion of the class that discussed various types of cryptosystems. These were the simplified versions that were easier to comprehend so that we could truly grasp the underlying concepts of cryptography. However, I craved more. I wanted to delve further into the advanced world of cryptography. It was not until my Honors Directed Study that I was able to do so. During my directed study, I was able to research the history of cryptography and what it has transformed into today. While it was fascinating to look at all the various cryptosystems that have been created throughout time, one cryptosystem caught my eye in particular—Elliptic Curve Cryptography, commonly referred to as ECC. I had heard of a few cryptosystems but never the ECC. As such, my interest was piqued. Having the opportunity to research Elliptic Curve Cryptography, I was ready to try and understand a new mathematical concept. I was ready to further expand my love and passion for mathematics, and I was not disappointed. Learning even the basics of elliptical curves proved challenging. The math behind ECC is immense and quite frankly, intimidating. Nevertheless, I do not back down from a challenge…at least not a mathematical one. My research of elliptic curves, the basis of Elliptic Curve Cryptography, opened up my eyes to an entirely new field of mathematics. The challenges that arose during my studies, while frustrating at times, are well worth the wealth of knowledge I was able to acquire. The research in my study of elliptical curve cryptography only increased the ten-year passion for mathematics that is still inside the slightly nerdy girl. This paper is the culmination of all my research over elliptic curves. It reflects the knowledge that I was able to acquire while studying elliptic curve cryptography and quantum computers