The field of cryptography revolves around constructing provably secure communication schemes, which are used in all areas of the digital world, from online messaging to blockchain and internet protocols. Encryption schemes often rely on an underlying "hard" mathematical problem for computers to break. Unfortunately, many of the most common math problems used in cryptography for the last 50 years are susceptible to a devastating attack by quantum computers, called Shor's Algorithm.
In post-quantum cryptography, we construct cryptography that can be run on classical computers and is resistant to quantum attacks. Many of these schemes rely on mathematical objects called lattices. I am passionate about researching lattice-based post-quantum schemes for newly evolving cryptographic problems.
One problem of interest is Fully Homomorphic Encryption (FHE) schemes. The question FHE answers is: how can a user perform some program or computation on a server without revealing their private data?
In mathematical terms, a user has a private input x and would like to use a server's function f to compute the output value y = f (x), without revealing any information about x. To do this, the user can send the encryption of x to the server, and using an FHE scheme, the server can send the encryption of f(x) back to the user, which can decrypt and reveal f(x). The server can do all of this without ever being able to learn f(x). Interestingly, we only know how to create FHE from lattices!
This has important applications across all domains, including financial services (e.g., private credit checks), healthcare (e.g., testing health analytics without revealing private medical data), and artificial intelligence (e.g., sending and receiving encrypted messages with ChatGPT).
Another interesting problem is Private Information Retrieval (PIR). The question PIR answers is: how can we perform a Google search without ever leaking the text we're searching for to Google? While we can use FHE to implement PIR, these schemes are much too inefficient, and ongoing research seeks to create new algorithmic speedups so PIR schemes can be used in practical settings.
In fall of 2024, I completed my Honours Research Thesis under the supervision of Prof. Henri Darmon in the area of modular forms, Hecke operators, and Weierstrass Uniformization. I read the text "Course in Arithmetic" by J.P. Serre and excerpts of "Advanced Topics in the Arithmetic of Elliptic Curves" by Joseph Silverman.
Here is my research thesis:
I presented an introduction to these interests at Les séminaires universitaires en mathématiques à Montréal 2025 (SUMM) hosted by the Université de Montréal (UdeM). Here are my slides: