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.

In this paper, by introducing an isomorphism from the Mihailova subgroup of F2×F2 to the Mihailova subgroups of a braid group, we give an explicit presentation of Mihailova subgroups of a braid group. Hence, in a braid group, there are some Mihailova subgroups experiencing unsolvable subgroup membership problem. Based on this, we propose a post-quantum signature scheme of the Wang–Hu scheme, and we show that the signature scheme is free of quantum computational attack.

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