Cryptography has become essential as bank transactions, credit card infor-mation, contracts, and sensitive medical information are sent through inse-cure channels. This book is concerned with the mathematical, especially algebraic, aspects of cryptography. It grew out of many courses presented by the authors over the past twenty years at various universities and covers a wide range of topics in mathematical cryptography. It is primarily geared towards graduate students and advanced undergraduates in mathematics and computer science, but may also be of interest to researchers in the area. Besides the classical methods of symmetric and private key encryption, the book treats the mathematics of cryptographic protocols and several unique topics such as Group-Based Cryptography Groebner Basis Methods in Cryptography Lattice-Based Cryptography

From the Preface:"...Magnus has had such a profound influence on combinatorial group theory because many of his ideas, startingly and strikingly simple, have provided not only deep insights into a very difficult subject but also powerful methods for dealing with these difficulties...His ideas have also found application in topology, K-theory, the theory of Lie and associative algebras, computational complexity, and also in logic.The expert in group theory, however, will be astonished to find that this reprinting of Magnus' papers contains a very large amount of very important work on diffraction problems and related topics in analysis. Indeed Magnus is one of the very few mathematicians who has done significant work in two completely different fields. There is a large number of mathematicians who know Magnus for his work in analysis but are totally unaware of his work in group theory...His books, his teaching...his many doctoral students, his effect on the thinking of his colleagues both in private conversation and in seminars have also helped to establish him as a mathematician of the first rank and enriched the mathematical community."

In January 1989 a Workshop on Algorithms, Word Problems and Classi- fication in Combinatorial Group Theory was held at MSRl. This was part of a year-long program on Geometry and Combinatorial Group Theory or- ganised by Adyan, Brown, Gersten and Stallings. The organisers of the workshop were G. Baumslag, F.B. Cannonito and C.F. Miller III. The pa- pers in this volume are an outgrowth of lectures at this conference. The first three papers are concerned with decision problems and the next two with finitely presented simple groups. These are followed by two papers dealing with combinatorial geometry and homology. The remaining papers are about automatic groups and related topics. Some of these papers are, in essence, announcements of new results. The complexity of some of them are such that neither the Editors nor the Reviewers feel that they can take responsibility for vouching for the completeness of the proofs involved. We wish to thank the staff at MSRl for their help in organising the workshop and this volume.

Combinatorial group theory is a loosely defined subject, with close connections to topology and logic. With surprising frequency, problems in a wide variety of disciplines, including differential equations, automorphic functions and geometry, have been distilled into explicit questions about groups, typically of the following kind: Are the groups in a given class finite (e.g., the Burnside problem)? Finitely generated? Finitely presented? What are the conjugates of a given element in a given group? What are the subgroups of that group? Is there an algorithm for deciding for every pair of groups in a given class whether they are isomorphic or not? The objective of combinatorial group theory is the systematic development of algebraic techniques to settle such questions. In view of the scope of the subject and the extraordinary variety of groups involved, it is not surprising that no really general theory exists. These notes, bridging the very beginning of the theory to new results and developments, are devoted to a number of topics in combinatorial group theory and serve as an introduction to the subject on the graduate level.