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 F2 × F2 with finite number of generators are given, where the Mihailova subgroup of F2 × F2 enjoys the unsolvable subgroup membership problem. Particularly, a braid group Bn 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.