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The role of mathematics in a complex system such as the Internet has yet to be deeply explored. This paper summarizes some of the important and pressing problems in cyber security from the viewpoint of open science environments. The author starts by posing the question "What fundamental problems exist within cyber security research that can be helped by advanced mathematics and statistics?" The first and most important assumption is that access to real-world data is necessary to understand large and complex systems like the Internet. The second assumption is that many proposed cyber security solutions could critically damage both the openness and the productivity of scientific research.
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