Bruce Richmond

Intersection: combinatorics and number theory

Combinatorics studies finite collections of objects (integers, polynomials, etc.) that satisfy specific criteria. Collections may be counted, analyzed against criteria, examined for largest or smallest instances or explored to find algebraic structures. The study involves analysis of partitions of numbers (the order of numbers in a set does not matter) or permutations of numbers (ordered selection and number of them does matter).

Bruce Richmond works in Department of Combinatorics & Optimization and finds many questions related to number theory. He explains, “I was always interested in number theory, but I was surprised to find that there were some objects that I didn’t consider part of number theory that behaved like integers.”

For instance, factoring polynomials is much like factoring integers. Primes are the smallest factors of integers and irreducible polynomials are the smallest factors of polynomials. These irreducible polynomials over finite fields create building blocks for cryptographers. Bruce clarifies: “Cryptographers use finite fields and come up with schemes based on the fact that it’s difficult to factor a polynomial. So in my work, we proposed a way to find irreducible polynomials and then analyzed algorithms to generate them.”

Bruce has applied his understanding of combinatorics and number theory to the analysis of Ben-Or versus Rabin algorithms (with D. Panario) and a comparison of Feynman’s Path Integral with Schrödinger’s Psi Function for quantum mechanics (with H. Carteret and N. Temme).

University of Waterloo Mathematics, Annual Report 2005