We consider several algorithmic problems concerning geodesics in finitely generated groups. We show that the three geodesic problems considered by Miasnikov et al [arXiv:0807.1032] are polynomial-time reducible to each other. We study two new geodesic problems which arise in a previous paper of the authors and Fusy [arXiv:0902.0202] .

We construct a short presentation of the ring of n × n matrices over Z with only 2 generators and 3 relations.

Algebraic attacks lead to the task of solving polynomial systems over F2. We study recent suggestions of using SAT-solvers for this task. In particular, we develop several strategies for converting the polynomial system to a set of CNF clauses. This generalizes the approach in [4]. Moreover, we provide a novel way of transforming a system over F2e to a (larger) system over F2. Finally, the efficiency of these methods is examined using standard examples such as CTC, DES, and Small Scale AES.

This is a short proof of the existence of finite sets of edges in graphs with more than one end, such that after removing them we obtain two components which are nested with all their isomorphic images. This was first done in “Cutting up graphs” [Dunwoody, Combinatorica 2: 15–23, 1982]. Together with a certain tree construction and some elementary Bass–Serre theory this yields a combinatorial proof of Stallings' theorem on the structure of finitely generated groups with more than one end.

We prove that for any finite Thurston-type ordering $<_{T}$ on the braid group\ $B_{n}$, the restriction to the positive braid monoid $(B_{n}^{+},<_{T})$ is a\ well-ordered set of order type $\omega^{\omega^{n-2}}$. The proof uses a combi\ natorial description of the ordering $<_{T}$. Our combinatorial description is \ based on a new normal form for positive braids which we call the $\C$-normal fo\ rm. It can be seen as a generalization of Burckel's normal form and Dehornoy's \ $\Phi$-normal form (alternating normal form).

Consider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators. For fixed k, we prove that the diameter of said graph is asymptotically (in q) of order k √ q. The same also holds when the generating set is taken to be a symmetric set of size 2k.

A Hurwitz group is any non-trivial finite quotient of the (2, 3, 7) triangle group, that is, any non-trivial finite group generated by elements x and y satisfying x2 = y3 = (xy)7 = 1. Every such group G is the conformal automorphism group of some compact Riemann surface of genus g > 1, with the property that |G| = 84(g – 1), which is the maximum possible order for given genus g. This paper provides an update on what is known about Hurwitz groups and related matters, following up the author's brief survey in Bull. Amer. Math. Soc.23 (1990).

In this paper we classify the coordinate ℕ-monoids of algebraic sets over the additive monoid of natural numbers.

Challenge response methods are increasingly used to enhance password security. 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 present problems with challenge response systems. Theoretical security is based on several results in asymptotic group theory.