Hoboken, NJ – Stevens professors Alexei Miasnikov and Alexander Ushakov from the Department of Mathematical Sciences in the Schaefer School of Engineering & Science are coauthors of a new book entitled, "Non-Commutative Cryptography and Complexity of Group-Theoretic Problems" (American Mathematical Society, December 2011).
The book explores “non-commutative ideas” in cryptography that have appeared in literature over the last decade or so. Since the authors have backgrounds in combinatorial and computational group theory, they pay particular attention to group-based cryptography – i.e., cryptography that uses non-commutative group theory. They also discuss ideas from other areas of mathematics, including the problem of authentication, one of the most important areas of applications of modern cryptography.
In the book, the coauthors note that there is remarkable feedback from cryptography to combinatorial and computational group theory because some of the problems motivated by cryptography appear to be new to group theory, and they open many interesting research avenues within group theory. Then they employ complexity theory, notably generic-case complexity of algorithms, for cryptanalysis of various cryptographic protocols based on infinite groups. They also use the ideas and machinery from the theory of generic-case complexity to study asymptotically dominant properties of some infinite groups that have been used in public-key cryptography so far. It turns out that for a relevant cryptographic scheme to be secure, it is essential that keys are selected from a very small subset rather than from the whole group. Detecting these subsets (“black holes”) for a particular cryptographic scheme is usually very challenging, but it holds the key to creating secure cryptographic primitives based on non-commutative groups.
Finally, a substantial part of the book deals with new directions in computational group theory itself, notably with search problems motivated by cryptography. The authors also study complexity of more traditional decision problems (like the word and conjugacy problems) in some groups that have been suggested as platforms for cryptographic protocols.
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