  The cycling operation is a special kind of conjugation that can be applied to
elements in Artin's braid groups, in order to reduce their length. It is a key
ingredient of the usual solutions to the conjugacy problem in braid groups. In
their seminal paper on braid-cryptography, Ko, Lee et al. proposed the {\it
cycling problem} as a hard problem in braid groups that could be interesting
for cryptography. In this paper we give a polynomial solution to that problem,
mainly by showing that cycling is surjective, and using a result by Maffre
which shows that pre-images under cycling can be computed fast. This result
also holds in every Artin-Tits group of spherical type.
  On the other hand, the conjugacy search problem in braid groups is usually
solved by computing some finite sets called (left) ultra summit sets
(left-USS), using left normal forms of braids. But one can equally use right
normal forms and compute right-USS's. Hard instances of the conjugacy search
problem correspond to elements having big (left and right) USS's. One may think
that even if some element has a big left-USS, it could possibly have a small
right-USS. We show that this is not the case in the important particular case
of rigid braids. More precisely, we show that the left-USS and the right-USS of
a given rigid braid determine isomorphic graphs, with the arrows reversed, the
isomorphism being defined using iterated cycling. We conjecture that the same
is true for every element, not necessarily rigid, in braid groups and
Artin-Tits groups of spherical type.
