Volume 5, Issue 2

Abstract. We argue that it is unnatural and undesirable to use the non-uniform model of complexity for practice-oriented security reductions in cryptography.

Abstract. We propose a slight weakening of the definitions of Artin monoids and Coxeter monoids. We study one `infinite series' in detail.

Abstract. We construct non-associative key establishment protocols for all left self-distributive (LD), multi-LD-, and other left distributive systems. Instantiations of these protocols using generalized shifted conjugacy in braid groups lead to instances of a natural and apparently new group-theoretic problem, which we call the (subgroup) conjugacy coset problem.

Abstract. It is well known that any polycyclic group, and hence any finitely generated nilpotent group, can be embedded into for an appropriate ; that is, each element in the group has a unique matrix representation. An algorithm to determine this embedding was presented in [J. Algebra 300 (2006), 376–383]. In this paper, we determine the complexity of the crux of the algorithm and the dimension of the matrices produced as well as provide a modification of the algorithm presented in [J. Algebra 300 (2006), 376–383].

Abstract. A subgroup H of a free group F is called inert in F if for every . In this paper we expand the known families of inert subgroups. We show that the inertia property holds for 1-generator endomorphisms. Equivalently, echelon subgroups in free groups are inert. An echelon subgroup is defined through a set of generators that are in echelon form with respect to some ordered basis of the free group, and may be seen as a generalization of a free factor. For example, the fixed subgroups of automorphisms of finitely generated free groups are echelon subgroups. The proofs follow mostly a graph-theoretic or combinatorial approach.

Abstract. We explain and perform the steps for an ( n , t ) secret sharing scheme based on the closest vector theorem. We then compare this scheme and its complexity to the secret sharing schemes of both Shamir and Panagopoulos. Finally we modify the ( n , t ) secret sharing scheme to a private key cryptosystem.