The Amalgamated product of free groups and residual solvability

In this paper we study residual solvability of the amalgamated product of two finitely generated free groups, in the case of doubles. We find conditions where this kind of structure is residually solvable, and show that in general this is not the case. However this kind of structure is always meta-residually-solvable.Comment: submitte

A new proof of a theorem of Ivanov and Schupp

The object of this note is to give a very short proof of the following theorem of Ivanov and Schupp. Let H be a finitely generated subgroup of a free group F and the index [F:H] infinite. Then there exists a nontrivial normal subgroup N of F such that N cap H = {1}

A Polynomial Time Algorithm For The Conjugacy Decision and Search Problems in Free Abelian-by-Infinite Cyclic Groups

In this paper we introduce a polynomial time algorithm that solves both the conjugacy decision and search problems in free abelian-by-infinite cyclic groups where the input is elements in normal form. We do this by adapting the work of Bogopolski, Martino, Maslakova, and Ventura in \cite{bogopolski2006conjugacy} and Bogopolski, Martino, and Ventura in \cite{bogopolski2010orbit}, to free abelian-by-infinite cyclic groups, and in certain cases apply a polynomial time algorithm for the orbit problem over $\Z^n$ by Kannan and Lipton

On the Dimension of Matrix Representations of Finitely Generated Torsion Free Nilpotent Groups

It is well known that any polycyclic group, and hence any finitely generated nilpotent group, can be embedded into $GL_{n}(\mathbb{Z})$ for an appropriate $n\in \mathbb{N}$; that is, each element in the group has a unique matrix representation. An algorithm to determine this embedding was presented in Nickel. 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 by Nickel

Non-commutative Digital Signatures

The objective of this work is to survey several digital signatures proposed in the last decade using non-commutative groups and rings and propose a digital signature using non-commutative groups and analyze its security

Decision and Search in Non-abelian Cramer Shoup Public Key Cryptosystem

A method for non-abelian Cramer-Shoup cryptosystem is presented. The role of decision and search is explored, and the platform of solvable/polycyclic group is suggested. In the process we review recent progress in non-abelian cryptography and post some open problems that naturally arise from this path of research

Polycyclic groups: A new platform for cryptology?

We propose a new cryptosystem based on polycyclic groups. The cryptosystem is based on the fact that the word problem can be solved effectively in polycyclic groups, while the known solutions to the conjugacy problem are far less efficient.Comment: 7 pages. submitte

A family of polycyclic groups over which the uniform conjugacy problem is NP-complete

In this paper we study the conjugacy problem in polycyclic groups. Our main result is that we construct polycyclic groups $G_n$ whose conjugacy problem is at least as hard as the subset sum problem with $n$ indeterminates. As such, the conjugacy problem over the groups $G_n$ is NP-complete where the parameters of the problem are taken in terms of $n$ and the length of the elements given on input

Publicly Verifiable Secret Sharing Using Non-Abelian Groups

In his paper Stadler develops techniques for improving the security of existing secret sharing protocols by allowing to check whether the secret shares given out by the dealer are valid. In particular, the secret sharing is executed over abelian groups. In this paper we develop similar methods over non-abelian groups

On the residual solvability of generalized free products of solvable groups

In this paper, we study the residual solvability of the generalized free product of solvable groups.Comment: Discrete Mathematics & Theoretical Computer Science, Vol 13, 201