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.

We prove that Whitehead's algorithm for solving the automorphism problem in a fixed free group F k has strongly linear time generic-case complexity. This is done by showing that the "hard" part of the algorithm terminates in linear time on an exponentially generic set of input pairs. We then apply these results to one-relator groups. We obtain a Mostow-type isomorphism rigidity result for random one-relator groups: If two such groups are isomorphic then their Cayley graphs on the given generating sets are isometric. Although no nontrivial examples were previously known, we prove that one-relator groups are generically complete groups, that is, they have trivial center and trivial outer automorphism group. We also prove that the stabilizers of generic elements of F k in Aut(F k ) are cyclic groups generated by inner automorphisms and that Aut(F k )-orbits are uniformly small in the sense of their growth entropy. We further prove that the number I k (n) of isomorphism types of k-generator one-relator groups with defining relators of length n satisfies c 1 n (2k − 1)where c 1 , c 2 are positive constants depending on k but not on n. Thus I k (n) grows in essentially the same manner as the number of cyclic words of length n.

Abstract. 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. Thompson's 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 Thompson's group is solvable in almost linear time.

Basic definitions and notation 2.2. Presentations of groups by generators and relators 2.3. Algorithmic problems of group theory: decision, witness, search 2.3.1. The word problem 2.3.2. The conjugacy problem 2.3.3. The decomposition and factorization problems 2.3.4. The membership problem 2.3.5. The isomorphism problem 2.3.6. More on search/witness problems 2.4. Nielsen's and Schreier's methods 2.5. Tietze's method 2.6. Normal forms Chapter 3. Background on Computational Complexity 3.1. Algorithms 3.1.1. Deterministic Turing machines 25 3.1.2. Non-deterministic Turing machines 26 3.1.3.

We investigate the average-case complexity of decision problems for finitely generated groups, in particular, the word and membership problems. Using our recent results on ''generic-case complexity'', we show that if a finitely generated group G has word problem solvable in subexponential time and has a subgroup of finite index which possesses a nonelementary word-hyperbolic quotient group, then the average-case complexity of the word problem of G is linear time, uniformly with respect to the collection of all length-invariant measures on G: This results applies to many of the groups usually studied in geometric group theory: for example, all braid groups B n ; all groups of hyperbolic knots, many Coxeter groups and all Artin groups of extra-large type. r (I. Kapovich), alexeim@att.net (A. Myasnikov), schupp@math.uiuc.edu (P. Schupp), shpil@groups.sci.ccny.cuny.edu (V. Shpilrain).