**Groups With Two Generators Having Unsolvable Word Problem And Presentations of Mihailova Subgroups**

*Xiaofeng Wang and Chen Xu and Guo Li and Hanling Lin*

**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 Mihailova subgroup of F_2×F_2 enjoys the unsolvable subgroup membership problem.One then can use the presentation to create entities' private key in a public key cryptsystem.

**Category / Keywords: **foundations / word problem, subgroup membership problem, unsolvability, Mihailova subgroup

**Date: **received 7 Jul 2014, last revised 7 Jul 2014

**Contact author: **wangxf at szu edu cn

**Available format(s): **PDF | BibTeX Citation

**Note: **In this paper, we give an explicit countable presentation of Mihailova subgroup of F_2×F_2 with finite number of generators which has unsolvable subgroup membership problem. Followed we know that in a braid group B_n with n \geq 6, there are some such subgroups. Thus one possibly can use this property to set up highly secured public key cryptsystems.

**Version: **20140708:063620 (All versions of this report)

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