Solution of the Generalised Word Problem for Various Subgroups of the Braid Group and a Solutions of the Conjugacy Problem in the Braid Group

We present a near complete solution of the GWP in braid groups for cyclic subgroups and arbitrary subgroups. We generalise the GWP and give solutions of this generalisation. This open problem on the discrete logarithm was proposed in [2]. In this paper we point out that the discrete logarithm problem is nearly always solvable in the braid group using a fast polynomial time algorithm. By ‘nearly always solvable' we mean we can solve the discrete logarithm problem except when len_min0(v)=0 and v is not a fixed braid. We solve the conjugacy problem when the problem involves fixed braids. We propose a pseudo-random generator based on the discrete logarithm problem in the braid group