Groups With Two Generators Having Unsolvable Word Problem and Presentations of Mihailova Subgroups of Braid Groups

Abstract

A presentation of a group with two generators having unsolvable word problem and an explicit countable presentation of Mihailova subgroup of F 2 × F 2 with finite number of generators are given, where the Mihailova subgroup of F 2 × F 2 enjoys the unsolvable subgroup membership problem. Particularly, a braid group B n with n ≥ 6 then contains Mihailova subgroups, and therefore, one possibly can apply the generators of these subgroups to create entities' private keys in a public key cryptsystem by taking a braid group as the correspondent platform.

Post-Quantum Signature Scheme Based on the Root Extraction Problem over Mihailova Subgroups of Braid Groups

“…In their paper [43], Bogopolski and Venura proved a theorem that gives an explicit representation of Mihailova subgroup M(H) in F k × F k under the assumption that the group H satisfies certain conditions. In [39], Wang, Li, and Lin gave a finite presentation of a group H, which is generated by just two elements and experiences an unsolvable word problem. Additionally, they proved that the presentation of H satisfies the conditions required in Bogopolski and Venura's theorem in [43].…”

“…Through this isomorphism we obtained the Mihailova subgroups M G i (H) of B n which have an unsolvable membership problem. Since the defining relations of Presentation C in [39] are of the following form…”

“…In [39], Wang, Li, and Lin have given an explicit presentation of the Mihailova subgroups of F 2 × F 2 , which experience unsolvable subgroup membership problems. Then by using an isomorphism, we show that for an index n ≥ 6, a braid group B n contains such Mihailova subgroups.…”

Public key cryptography based on some extensions of group

“…a finitely presented group on two generators with unsolvable word problem (for example the group presented in [WXLL14]). We then construct a group…”

Generalized small cancellation conditions, non-positive curvature and diagrammatic reducibility

“…For each i the diagram has an edge v i ei − → v which reads p i and an edge v i+1 αi − → v i Example 5.4. The presentations A and B from [36] have unsolvable word problem. A GAP computation using SmallCancellation shows that these presentations do not satisfy τ .…”