Multiplicative orders of elements in finite fields

For many applications in cryptography and coding theory it is important to find elements of high multiplicative order in a finite field. This motivates the search for upper and lower bounds on such multiplicative orders. In this talk, I will outline some bounds which arise from counting certain partitions (for lower bounds) and the analytic class number formula (for upper bounds). This talk should be accessible to advanced undergraduate students.

About the speaker

Florian Breuer grew up in Stellenbosch, South Africa, where he completed school as well as his undergraduate studies in Mathematics and Theoretical Physics. He then obtained a bursary from the French government to complete his graduate studies in Paris, where he completed his D.E.A. (Masters) at the Université Pierre et Marie Curie in 1999 and his PhD at the Université Denis Diderot in 2002, both under supervision of Marc Hindry. After two postdoctoral fellowships in Taiwan and Germany, he returned to Stellenbosch University in July 2004 as a senior lecturer. He was promoted to full professor in 2013, and in April 2018 he moved with his family to the University of Newcastle.

Florian's main research area is the arithmetic of function fields. This started with his PhD thesis, in which he proved an analogue of the André-Oort Conjecture for products of Drinfeld modular curves. Most recently he was involved in joint work with former PhD student Dirk Basson (Stellenbosch), and friend and mentor Richard Pink (ETH-Zurich), which laid the foundations for the analytic theory of Drinfeld modular forms in arbitrary rank.