Ievgen Bondarenko

On a family of Schreier graphs of intermediate growth associated with a self-similar group

For every infinite sequence ω = x 1 x 2 ? , with x i ? { 0 , 1 } , we construct an infinite 4-regular graph X ω . These graphs are precisely the Schreier graphs of the action of a certain self-similar group on the space { 0 , 1 } ∞ . We solve the isomorphism and local isomorphism problems for these graphs, and determine their automorphism groups. Finally, we prove that all graphs X ω have intermediate growth.

On the conjugacy problem for finite-state automorphisms of regular rooted trees (with an appendix by Raphaël M. Jungers)

We study the conjugacy problem in the automorphism group Aut(T) of a regular rooted tree T and in its subgroup FAut(T) of finite-state automorphisms. We show that under the contracting condition and the finiteness of what we call the orbit-signalizer, two finite-state automorphisms are conjugate in Aut(T) if and only if they are conjugate in FAut(T), and that this problem is decidable. We prove that both these conditions are satisfied by bounded automorphisms and establish that the (simultaneous) conjugacy problem in the group of bounded automata is decidable. Mathematics Subject Classification 2000: 20E08, 20F10

Encrypted Program Execution

Outsourcing computation becomes more and more popular in the new era of Cloud computing. Programs can be transmitted in encrypted form, but currently for execution they must be decrypted and program code may be disclosed. The main goal of this paper is to provide a reasonable method for executing arbitrary programs in encrypted space without decryption based on homomorphic cryptography. We revise the method of Brenner et al. [1], [2] for secure execution of arbitrary programs in encrypted space and show how to improve its performance in several aspects: machine architecture, memory access and solve the termination problem for encrypted execution. We implement the virtual machine capable of executing encrypted programs without decryption and the C compiler for it.

The Lamplighter Group ℤ3≀ℤ Generated by a Bireversible Automaton

We construct a bireversible self-dual automaton with three states over an alphabet with three letters which generates the lamplighter group ℤ3≀ℤ. In particular, this fact shows that not all groups defined by birevirsible automata are finitely presented.