Cryptography, literally defined as ‘secret writing,’ requires a solid background in algebra and number theory and is used to ensure confidentiality and authentication of information when transferring over an insecure channel. In this paper, he explored the applications of finite field arithmetic in cryptography by explaining the Advanced Encryption Standard (private-key) and Elliptic Curve Cryptography (public-key).
“It was great to see how abstract mathematical structures like finite fields have real-world application to information security through cryptography,” said Sharma.
He explains that his thesis offers a complete and quick overview of finite fields and how they are used in both private and public key cryptosystems.
“My thesis not only includes these essential mathematical and computational prerequisites to understand the materials covered but also provides ample references to literature for motivated readers,” said Sharma. “Though the thesis does not invent “new” math, it presents the applications in an organized and succinct manner.”
Sharma, a native of Nepal, will take his talents to Chicago this summer, working as an Associate Software Engineer for a Quantitative Research Team at Morningstar Inc., a financial-services company headquartered in Chicago.
“While the thesis directly doesn’t have a connection with my job, the skills I have learned through it are transferable,” said Sharma. “In a few years, I might end up going to a graduate school and I wouldn’t be surprised if I continue my research in cryptography.”
Because his was a joint thesis in both computer science and mathematics, he received extensive help from both of his advisors, Professor Liz Milićević from the math department and Professor Steve Lindell from the computer science department. “I was interested in looking at application of finite fields to cryptography and though this topic was not the focus of my advisors’ research, they graciously agreed to move forward and provided guidance along the way. For instance, in the beginning, Liz particularly helped me develop necessary background in field theory by covering essential definitions and proofs. Another example includes Steve suggesting new directions to the thesis, specifically the idea of exploring the elliptic-curve cryptography for public key cryptosystems. In our weekly meetings, all three of us would gather in a room, discuss new things I had learned over the week, and ponder our curiosities!”
As for his biggest takeaway from the project, he says the process itself was a valuable learning experience. “Of course, I learnt the algorithms and the math behind them but the biggest takeaway would be the process itself. In the beginning, I had no clear clue where my thesis was headed. However, the more I moved along, the clearer my path became. I learned that it is okay to not have the entire plan in place before getting started. Things are unpredictable and that is the beauty of something complicated like research!”
“What They Learned” is a blog series exploring the thesis work of recent graduates.