  Several key agreement protocols are based on the following "Generalized
Conjugacy Search Problem": Find, given elements b_1,...,b_n and
xb_1x^{-1},...,xb_nx^{-1} in a nonabelian group G, the conjugator x. In the
case of subgroups of the braid group B_N, Hughes and Tannenbaum suggested a
length-based approach to finding x. Since the introduction of this approach,
its effectiveness and successfulness were debated.
  We introduce several effective realizations of this approach. In particular,
a new length function is defined on B_N which possesses significantly better
properties than the natural length associated to the Garside normal form. We
give experimental results concerning the success probability of this approach,
which suggest that very large computational power is required for this method
to successfully solve the Generalized Conjugacy Search Problem when its
parameters are as in existing protocols.
