Vladimir Shpilrain

Generic-case complexity, decision problems in group theory, and random walks

Abstract We give a precise definition of “generic-case complexity” and show that for a very large class of finitely generated groups the classical decision problems of group theory—the word, conjugacy, and membership problems—all have linear-time generic-case complexity. We prove such theorems by using the theory of random walks on regular graphs.

Thompson's group and public key cryptography

Recently, several public key exchange protocols based on symbolic computation in non-commutative (semi)groups were proposed as a more efficient alternative to well established protocols based on numeric computation. Notably, the protocols due to Anshel-Anshel-Goldfeld and Ko-Lee et al. exploited the conjugacy search problem in groups, which is a ramification of the discrete logarithm problem. However, it is a prevalent opinion now that the conjugacy search problem alone is unlikely to provide sufficient level of security no matter what particular group is chosen as a platform. In this paper we employ another problem (we call it the decomposition problem), which is more general than the conjugacy search problem, and we suggest to use R. Thompsons group as a platform. This group is well known in many areas of mathematics, including algebra, geometry, and analysis. It also has several properties that make it fit for cryptographic purposes. In particular, we show here that the word problem in Thompsons group is solvable in almost linear time.

Non-Commutative Cryptography and Complexity of Group-Theoretic Problems

This book is about relations between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory. It explores how non-commutative (infinite) groups, which are typically studied in combinatorial group theory, can be used in public-key cryptography. It also shows that there is remarkable feedback from cryptography to combinatorial 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. In particular, a lot of emphasis in the book is put on studying search problems, as compared to decision problems traditionally studied in combinatorial group theory. Then, complexity theory, notably generic-case complexity of algorithms, is employed for cryptanalysis of various cryptographic protocols based on infinite groups, and the ideas and machinery from the theory of generic-case complexity are used to study asymptotically dominant properties of some infinite groups that have been applied in public-key cryptography so far. This book also describes new interesting developments in the algorithmic theory of solvable groups and another spectacular new development related to complexity of group-theoretic problems, which is based on the ideas of compressed words and straight-line programs coming from computer science.

Combinatorial methods : free groups, polynomials, and free algebras

Preface.- Introduction.- I. Groups: Introduction. Classical Techniques. Test Elements. Other Special Elements. Automorphic Orbits.- II. Polynomial Algebras: Introduction. The Jacobian Conjecture. The Cancellation Conjecture. Nagatas Problem. The Embedding Problem. Coordinate Polynomials. Test Polynomials.- III. Free Nielsen-Schreier Algebras: Introduction. Schreier Varieties of Algebras. Rank Theorems and Primitive Elements. Generalized Primitive Elements. Free Leibniz Algebras.- References.- Notations.- Author Index.- Subject Index.

Combinatorial Group Theory and Public Key Cryptography

After some excitement generated by recently suggested public key exchange protocols due to Anshel–Anshel–Goldfeld and Ko–Lee et al., it is a prevalent opinion now that the conjugacy search problem is unlikely to provide sufficient level of security if a braid group is used as the platform. In this paper we address the following questions: (1) whether choosing a different group, or a class of groups, can remedy the situation; (2) whether some other “hard” problem from combinatorial group theory can be used, instead of the conjugacy search problem, in a public key exchange protocol. Another question that we address here, although somewhat vague, is likely to become a focus of the future research in public key cryptography based on symbolic computation: (3) whether one can efficiently disguise an element of a given group (or a semigroup) by using defining relations.

The Conjugacy Search Problem in Public Key Cryptography: Unnecessary and Insufficient

The conjugacy search problem in a group G is the problem of recovering an

Public key exchange using matrices over group rings

x \in G

A practical attack on a braid group based cryptographic protocol

from given