The field of cryptography revolves around constructing provably secure communication schemes, which are used in all areas of life from online messaging to blockchain and internet protocols. Encryption schemes often rely on an underlying "hard" mathematical problem for computers to break. Unfortunately, the most common mathematics problem used in cryptography for the last 50 years (the Diffie-Hellman problem) is susceptible to a devastating attack by quantum computers.
In post-quantum cryptography, we construct classical encryption and signature schemes that are resistant against quantum attacks. Many of these schemes rely on mathematical primitives 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. In essence, if (Gen, Enc, Dec) is a fully homomorphic encryption scheme and m1, m2 are plaintexts, then
Enc(m1+m2)=Enc(m1)+Enc(m2)
Enc(m1*m2)=Enc(m1)*Enc(m2)
These systems allow untrusted users of data and third parties to perform mathematical computations on encrypted data without uncovering its contents. Current systems are not efficient enough for widespread implementation and require an expensive bootstrapping step following a set number of operations to maintain the security of data. However, ongoing research seeks to produce both hardware and algorithmic improvements.
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: