AMMCS-2011 Plenary Talk:
Group structures of elliptic curves:
Statistics, heuristics, algorithms
by Igor Shparlinki
We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection.
Some of these results are rigorous and based on recent advances in analytic number theory, some are conditional under certain widely believed conjectures, and others are purely heuristic in nature and exhibit several interesting and unexplained phenomena in the distribution of group structures.
Finally, we discuss some algorithms to compute group structures of elliptic curves over finite fields.
Igor Shparlinski is a Professor of Macquarie University, in 2010 he was awarded the title of Distinguished Professor. He is a fellow of the Australian Academy of Science and of the Australian Mathematical Society. He is a recipient of Australian Professorial Fellowship (2005-2010) and the Medal of the Australian Mathematical Society.
His research areas are number theory and its applications to computer science, cryptography and discrete mathematics. He servers on editorial boards of several journals specialising in these areas.