PhD in Combinatorics and Optimization - Cryptography in Waterloo Canada | University of Waterloo

Combinatorics examines discrete structures and their characteristics. Contemporary scientific progress frequently utilizes combinatorial models to represent physical phenomena, with computational advancements enabling practical exploration. Given that computers handle discrete data, combinatorics has become essential to computer science. Optimization, or mathematical programming, focuses on maximizing or minimizing functions within defined constraints. The rise of computing power spurred significant theoretical growth in optimization, benefiting both combinatorics and traditional mathematical analysis. These optimization functions find applications in engineering, physical sciences, management, and diverse mathematical fields. The PhD program typically spans four years, comprising two years of graduate coursework followed by research and dissertation work.

Traditional cryptography employed algorithms reliant on secret keys exchanged before communication to encrypt and decrypt messages. (In well-designed systems, security should depend solely on the key, with the assumption that adversaries know the algorithm.) A revolutionary cryptosystem emerged in 1976—asymmetric or public-key systems. These systems distinctly separate encryption and decryption roles, allowing public disclosure of one key while keeping the other secret, eliminating the need for pre-communication key exchange. Such systems rely on computationally difficult mathematical problems like integer factorization. Identifying suitable complex problems and developing associated public-key systems has created a vibrant and dynamic research field.