GROUPS WITH TWO GENERATORS HAVING UNSOLVABLE WORD PROBLEM AND PRESENTATIONS OF MIHAILOVA SUBGROUPS OF BRAID GROUPS

A presentation of a group with two generators having unsolvable word problem and an explicit countable presentation of Mihailova subgroup of F-2 x F-2 with finite number of generators are given, where the Mihailova subgroup of F-2 x 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.