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Groups Complex. Cryptol. 2 (2010), 67–81 DOI 10.1515/GCC.2010.005 © de Gruyter 2010 Challenge response password security using combinatorial group theory Gilbert Baumslag, Yegor Bryukhov, Benjamin Fine and Douglas Troeger Abstract. Challenge response methods are increasingly used to enhance password se- curity. In this paper we present a very secure method for challenge response password verification using combinatorial group theory. This method, which relies on the group randomizer system, a subset of the MAGNUS computer algebra system, handles most of the

Proceedings of the Singapore Group Theory Conference held at the National University of Singapore, June 8–19, 1987
Birdtracks, Lie's, and Exceptional Groups

Groups Complex. Cryptol. 2 (2010), 231–246 DOI 10.1515/GCC.2010.015 © de Gruyter 2010 Search and witness problems in group theory Vladimir Shpilrain Abstract. Decision problems are problems of the following nature: given a property P and an object O, find out whether or not the object O has the property P . On the other hand, witness problems are: given a property P and an object O with the property P , find a proof of the fact that O indeed has the property P . On the third hand(?!), search problems are of the following nature: given a property P and an object O

Proceedings of a Special Research Quarter at The Ohio State University, Spring 1992

3 Geometric group theory 3.1 Geometric group theory It is a classical idea to study a geometric object by looking at its group of isometries. Gromov [122] somewhat reversed this notion by making a group itself a geometric object. This was done by considering properties of groups whose Cayley graph (see Section 3) satisfies certain geometric properties. In particlar he introduced hyperbolic groupswhose Cayley graph satisfies a geometric property of hyperbolic geometry. This ushered in a whole new branch of combinatorial group theory called geometric group theory

2 Group Theory § 2.1 Representations Group theory is a branch of pure mathematics dealing with concepts that often seem too abstract for a typical physicist. A common reaction is to feel that it is difficult to comprehend because it is unclear where it is going, an impression which is not helped by the extensive literature on the subject, most of which is not used in theoretical physics. Nevertheless, some of its ideas are extremely useful. This account, which does not aim to be a detailed account of group theory, should therefore be regarded as an

2 Combinatorial group theory 2.1 Combinatorial group theory Most of the techniques, aswell as themotivation for the study of the elementary theory of groups arise, in combinatorial group theory, especially the theory of free groups. In this chapter we review the basic concepts from this discipline. Combinatorial group theory is roughly that branch of group theory which studies groups via their presentations, that is by generators and relations. Recall that a free group can be considered as a group with a free generating system for which there are no nontrivial