  We study the centralizer of a braid from the point of view of Garside theory,
showing that generically a minimal set of generators can be computed very
efficiently, as the ultra summit set of a generic braid has a very particular
structure. We present an algorithm to compute the centralizer of a braid whose
generic-case complexity is quadratic on the length of the input, and which
outputs a minimal set of generators in the generic case.
