Band 7, Heft 1

Given a group G and a G -module A , we show how to determine up to isomorphism the extensions E of A by G so that A embeds as smallest non-trivial term of the derived series or of the lower central series into E .

A finite symmetric graph Γ is a pair (Γ,f)$(\Gamma ,f)$, where Γ is a finite graph and f:Γ→Γ$f:\Gamma \rightarrow \Gamma $ is a graph self equivalence or automorphism. We develop several tools for studying such symmetries. In particular, we describe in detail all symmetries with a single edge orbit, we prove that each symmetric graph has a maximal forest that meets each edge orbit in a sequential set of edges – a sequential maximal forest – and we calculate the characteristic polynomial χf(t)$\chi _f(t)$ and the minimal polynomial μf(t)$\mu _f(t)$ of the linear map H1(f):H1(Γ,ℤ)→H1(Γ,ℤ)$H_1(f):H_1(\Gamma ,\mathbb {Z})\rightarrow H_1(\Gamma ,\mathbb {Z})$. The calculation is in terms of the quotient graph Γ¯$\overline{\Gamma }$.

We present a new fast search algorithm for 〈 m , m , m 〉 Triple Product Property (TPP) triples as defined by Cohn and Umans in 2003. The new algorithm achieves a speed-up factor of 40 up to 194 in comparison to the best known search algorithm. With a parallelized version of the new algorithm we are able to search for TPP triples in groups up to order 55. As an application we identify lists “C1” and “C2” of groups that, if they contain a 〈5,5,5〉 TPP triple, could realize 5×5 matrix multiplication with under 100, respectively under 125, scalar multiplications, i.e., the best known upper bound by Makarov (1987), respectively the trivial upper bound. With our new algorithm we show that no group in this list can realize 5×5 matrix multiplication better than Makarov's algorithm. We also show a direction towards a modified group-theoretic search, not covered by the C1 list.

We present a multi-signature scheme based on bilinear pairings. The scheme is key escrow-free and does not require any secure channel for private key issuance to users. We use a binding-blinding technique to avoid the key escrow problem and to eliminate a secure channel requirement for the key issuance stage. The basic scheme is extended to sequential and parallel multi-signature schemes. We show that the basic scheme and multi-signature schemes are secure against adaptive chosen message attacks under standard assumptions.

Let G be a free metabelian group of rank r = 2. We introduce a faithful 2×2 real matrix representation of G and extend this to a group Gℤ[θ]$G^{\mathbb {Z}[\theta ]}$ of 2×2 matrices admitting exponents from the integral polynomial ring ℤ[θ]$\mathbb {Z}[\theta ]$. Identifying G with its matrix representation, we show that given γ(θ)∈Gℤ[θ]$\gamma (\theta )\in G^{\mathbb {Z}[\theta ]}$ and n∈ℤ$n\in \mathbb {Z}$, one has that limθ→nγ(θ)$\lim _{\theta \rightarrow n}\gamma (\theta )$ exists and lies in G . Furthermore, the maps γ(θ)↦limθ→nγ(θ)$\gamma (\theta )\mapsto \lim _{\theta \rightarrow n}\gamma (\theta )$ form a discriminating family of group retractions Gℤ[θ]→G$G^{\mathbb {Z}[\theta ]}\rightarrow G$ as n varies over ℤ. Although not explicitly carried out in this manuscript, it is clear that similar results hold for any countable rank r .

In this paper, we explore a method for forming the convex hull of a subset in a uniquely geodesic metric space due to Brunn and use it to show that with respect to the usual action of F m ×ℤ n on Tree ×ℝn${\mathrm {Tree}\times \mathbb {R}^n}$, every quasiconvex subgroup of F m ×ℤ n is convex. Further, we show that the Cartan–Hadamard theorem can be used to show that locally convex subsets of complete and connected CAT(0) spaces are convex. Finally, we show that the quasiconvex subgroups of F m ×ℤ n are precisely those of the form A × B , where A≤Fm${A\le F_m}$ is finitely generated, and B≤ℤn${B\le \mathbb {Z}^n}$.

We discuss a new attack, termed a dimension or linear decomposition attack, on several known group-based cryptosystems. This attack gives a polynomial time deterministic algorithm that recovers the secret shared key from the public data in all the schemes under consideration. Furthermore, we show that in this case, contrary to the common opinion, the typical computational security assumptions are not very relevant to the security of the schemes, i.e., one can break the schemes without solving the algorithmic problems on which the assumptions are based.