All sound like things you might find on an episode of Star Trek, but are actually topics that can be found in Protecting Information, a new book by Susan Loepp and William Wootters, that covers these topics and many others in the fields of cryptography and coding theory. While several of the book's chapters deal with the classical (ie non-quantum) examples of cryptosystems and error correcting codes which can be found in any number of textbooks available, the bulk of the book is dedicated to explaining how the promise (or threat, depending on your point of view) of quantum computers has started to turn this whole field upside down.
The above description probably sounds quite daunting, and in the wrong hands it is easy to imagine a book covering those topics to be unreadable to anyone but an expert in the field. However, Loepp and Wootters have written a book that assumes very few prerequisites — some knowledge of linear algebra seems essential and some group or field theory would be useful — and in particular no prior knowledge of physics is assumed, which came as a great relief to this reviewer. With writing that is clear and to the point yet littered with copious examples, the authors explain many details of quantum mechanics and in particular how quantum information can become 'entangled' and how this idea of entanglement leads to ways to crack current methods of public-key cryptography, to design new alternatives to these protocols, and to design error-correcting codes.
Protecting Information began life as a textbook for a course the authors taught together at Williams College, where Loepp teaches in the mathematics department and Wootters teaches in the physics department, and it shows. One can easily imagine using this in a class for advanced undergraduate or beginning graduate students as, in addition to being extremely readable, the book has a large number of exercises. Most importantly, the topics will be appealing to a wide range of students from the mathematically inclined to computer science majors to phycisists, and the book has a very modern flavor — in fact, many of the references throughout the book are dated since our undergraduates were born. How many branches of mathematics can claim that?
Darren Glass (email@example.com) is an assistant professor of mathematics at Gettysburg College. His mathematical interests include number theory, algebraic geometry, and cryptography.
1. Cryptography: an overview; 2. Quantum mechanics; 3. Quantum cryptography; 4. An introduction to error-correcting codes; 5. Quantum cryptography revisited; 6. Generalized Reed-Solomon codes; 7. Quantum computing; Appendix.