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Arman Darbinyan

Publications and source records attributed to Arman Darbinyan.

10 recordsLinked to original sources
arXiv2025

Hard Instances of Discrete Logarithm Problem and Cryptographic Applications

Let f be an arbitrary positive integer valued function. The goal of this note is to show that one can construct a finitely generated group in which the discrete log problem is polynomially equivalent to computing the function f. In particular, we provide infinite, but finitely generated groups, in which the discrete logarithm problem is arbitrarily hard. As another application, we construct a family of two-generated groups that have polynomial time word problem and NP-complete discrete log problem. Additionally, using our framework, we propose a generic scheme of cryptographic protocols, which might be of independent interest.

math.GR↗
arXiv2025

A robust family of residually finite groups; spectra of residual finiteness growth, computability properties, and other applications (with an appendix by Arman Darbinyan and Emmanuel Rauzy)

In this paper, we introduce a family of residually finite groups that helps us to systematically study the residual finiteness growth function (RFG) from various perspectives. First, by strengthening results of Bou-Rabee and Seward and also of Bradford, we show that any non-decreasing function $f: \nn \rightarrow \nn$ that satisfies $f(n) > \exp{(\varepsilon n\log{n})}$ for some $\varepsilon>0$ can be realized (up to the standard equivalence) as RFG function of a two-generated residually finite group. Moreover, such a group can be found among solvable groups of derived length $3$; due to what, in a strengthened way, we extend a theorem of Kharlampovich, Miasnikov and Sapir. {Next, we consider computability aspects related to those growth functions. In particular, we characterize the decidability of the word problem in residually finite groups with respect to \emph{individual residual finiteness depth} functions. Then, we give a full description of sufficiently fast {growing} functions that are realizable as RFG for some group \emph{with decidable word problem} in terms of \emph{left-computable functions.} We also show that a Turing degree can be realized via RFG of a group with decidable word problem if and only if it is recursively enumerable. Finally, applying the introduced theoretical framework, we answer several open questions and extend known results. For example, answering a question of Minasyan, by providing a construction, we show the existence of conjugacy separable groups with decidable word problem and undecidable conjugacy problem. Two more applications, including an answer to a question by Nies, can be found in the appendix coauthored with Rauzy.}

math.GR↗
arXiv2025

On what finitely generated (left-orderable) simple groups can know about their subgroups

In this paper, we survey some of the recent advances on embeddings into finitely generated (left-orderable) simple group such that the overgroup preserves algorithmic, geometric, or algebraic information about the embedded group. We discuss some new consequences and also extend some of those embedding theorems to countable classes of finitely generated groups.

math.GR↗
arXiv2021

Finitely generated subgroups of free groups as formal languages and their cogrowth

For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the minimal deterministic finite automaton that recognizes $L_H$. Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and for such groups explicitly construct ergodic automaton that recognizes $L_H$. This construction gives us an efficient way to compute the cogrowth series $L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method and a comparison is made with the method of calculation of $L_H(z)$ based on the use of Nielsen system of generators of $H$.

math.GR↗
arXiv2020

Embeddings into left-orderable simple groups

We prove that every countable left-ordered group embeds into a finitely generated left-ordered simple group. Moreover, if the first group has a computable left-order, then the simple group also has a computable left-order. We also obtain a Boone-Higman-Thompson type theorem for left-orderable groups with recursively enumerable positive cones. These embeddings are Frattini embeddings, and isometric whenever the initial group is finitely generated. Finally, we reprove Thompson's theorem on word problem preserving embeddings into finitely generated simple groups and observe that the embedding is isometric.

math.GR↗
arXiv2019

Computability, orders, and solvable groups

The main objective of this paper is the following two results. (1) There exists a computable bi-orderable group that does not have a computable bi-ordering; (2) There exists a bi-orderable, two-generated recursively presented solvable group with undecidable word problem. Both of the groups can be found among two-generated solvable groups of derived length $3$. (1) answers a question posed by Downey and Kurtz; (2) answers a question posed by Bludov and Glass in Kourovka Notebook. One of the technical tools used to obtain the main results is a computational extension of an embedding theorem of B. Neumann that was studied by the author earlier. In this paper we also compliment that result and derive new corollaries that might be of independent interest.

math.GR↗
arXiv2017

The word and conjugacy problems in lacunary hyperbolic groups

We study the word and conjugacy problems in lacunary hyperbolic groups (briefly, LHG). In particular, we describe a necessary and sufficient condition for decidability of the word problem in LHG. Then, based on the graded small-cancellation theory of Olshanskii, we develop a general framework which allows us to construct lacunary hyperbolic groups with word and conjugacy problems highly controllable and flexible both in terms of computability and computational complexity. As an application, we show that for any recursively enumerable subset $\mathcal{L} \subseteq \mathcal{A}^*$, where $\mathcal{A}^*$ is the set of words over arbitrarily chosen non-empty finite alphabet $\mathcal{A}$, there exists a lacunary hyperbolic group $G_{\mathcal{L}}$ such that the membership problem for $ \mathcal{L}$ is `almost' linear time equivalent to the conjugacy problem in $G_{\mathcal{L}}$. Moreover, for the mentioned group the word and individual conjugacy problems are decidable in `almost' linear time. Another application is the construction of a lacunary hyperbolic group with `almost' linear time word problem and with all the individual conjugacy problems being undecidable except the word problem. As yet another application of the developed framework, we construct infinite verbally complete groups and torsion free Tarski monsters, i.e. infinite torsion-free groups all of whose proper subgroups are cyclic, with `almost' linear time word and polynomial time conjugacy problems. These groups are constructed as quotients of arbitrarily given non-elementary torsion-free hyperbolic groups and are lacunary hyperbolic. Finally, as a consequence of the main results, we answer a few open questions.

math.GR↗
arXiv2017

Group Embeddings with Algorithmic Properties

We show that every countable group H with solvable word problem (=computable group) can be subnormally embedded into a 2-generated group G which also has solvable word problem. Moreover, the membership problem for H < G is also solvable. We also give estimates of time and space complexity of the word problem in G and of the membership problem for H < G.

math.GR↗
arXiv2012

On Order-Preserving and Verbal Embeddings of the Group $\mathbb{Q}$

We show that there is an order-preserving embedding of the additive group of rational numbers $\mathbb{Q}$ into a 2-generator group $G$. The group $G$ can be chosen to be a solvable group $G$ of length 3, which is a minimal result in the sense that it cannot be chosen to be neither solvable of length 2, nor a nilpotent group. For any non-trivial word set $V \subseteq F_\infty$ there is an order-preserving verbal embedding of $\mathbb{Q}$ into a 2-generator group $G$. The embeddings constructed are subnormal.

math.GR↗