  The role of mathematics in a complex system such as the Internet has yet to
be deeply explored. In this paper, we summarize some of the important and
pressing problems in cyber security from the viewpoint of open science
environments. We start by posing the question "What fundamental problems exist
within cyber security research that can be helped by advanced mathematics and
statistics?" Our first and most important assumption is that access to
real-world data is necessary to understand large and complex systems like the
Internet. Our second assumption is that many proposed cyber security solutions
could critically damage both the openness and the productivity of scientific
research. After examining a range of cyber security problems, we come to the
conclusion that the field of cyber security poses a rich set of new and
exciting research opportunities for the mathematical and statistical sciences.
