Cracking the Code of Mathematics Research on Cryptography
Stevens students’ senior design project explores the strengths and weaknesses of cybersecurity systems
The idea of public key cryptosystems — with their complex algorithms for securing electronic data — has been around since at least 1976. One of the first such approaches was the Merkle-Hellman Knapsack Cryptosystem, which encrypted information using a one-way “trapdoor” mathematical function that appeared impossible to decrypt without knowing the secret solution to a complex mathematical problem. Less than a decade later, though, a successful cyberattack rendered this cryptosystem unreliable for security purposes.
However, as the saying goes, those who fail to learn from history are doomed to repeat it. That’s why Lauren Sheehan and Brendon Weiskott, two pure and applied pure and applied mathematics majors in the Stevens Institute of Technology Class of ’23, devoted their senior design project to creating a survey of cryptography and the trapdoor knapsack method. Specifically, their project details how codes can be used to protect information, then dives into the example of the Merkle-Hellman method — what it is, how to use it and how to break it.
“The goal was to give someone who is familiar with math an entry-level method to cryptography,” explained Weiskott, who will also graduate with a minor in computer science.
“At the start of the project, I was the target audience,” Sheehan recalled with a laugh. “I’m a finance minor, and I knew nothing about cryptography. My first project idea was related to cryptocurrency, and then Brendon suggested cryptography based on a class he was taking. It was something that we're both interested in, and it seemed a good way to get into research.”
As they got deeper and deeper into the dense research materials, they turned to simple tools to help keep all the complexity straight.
“The biggest challenge was that reading math is really hard,” Weiskott said. “You're keeping track of what each variable means, on top of trying to learn these abstract concepts. We often found ourselves sitting down in front of a whiteboard, drawing it out, step by step, repeatedly, so we could take it off the page and stop reading the same sentences over and over again. A lot of stick figures were drawn.”
All that advanced articulation — and elementary artwork — paid off.
“The goal of our mathematics department at Stevens is to prepare our students to do real research if they choose — to be able to read mathematical papers and understand what's going on,” said their advisor, Denis Serbin, associate professor of mathematics. “The papers they read were pretty heavy mathematically, and the consistent, clear way they explained everything in their report shows that they understood and are ready for the next stage of their education.”
Both students will be continuing their studies at Stevens in the fall through the master’s degree program in mathematics. In addition to providing insights for others interested in cryptography, their senior design project gave them insights into their own next steps.
“Learning how to read the papers and going into mathematical research was helpful, because it gave us a glimpse of what we will need to do next year as master's students,” Sheehan said. “It also confirmed that I’m more a fan of applied math, so that may be the focus of my master’s degree.”
“It helped me figure out what I want to do,” Weiskott noted. “I was deciding between going into data science or cybersecurity, and I decided on data science with a focus on data security. This project was a good way to learn the mathematical foundations behind everything that I'm going to be studying. I knew that fighting malware is a constant, uphill battle, but I never understood how that battle worked, or how even supposedly secure protections are breakable. Being aware of how vulnerable these systems can be, and that there are ways to be more careful, helps make our security systems stronger.”