TITLE: Complexity of testing integer square-freeness
QUESTION [25 upvotes]: How fast can an algorithm tell if an integer is square-free?
I am interested in both deterministic and randomized algorithms.  I also care about both unconditional results and ones conditional on GRH (or other reasonable number-theoretic conjectures).
One reference I could find was on the Polymath4 wiki, where it states

No unconditional polynomial-time
  deterministic algorithm for
  square-freeness seems to be known. (It
  is listed as an open problem in this
  paper
  from 1994.)

I can't tell if that quote implies that both conditional and randomized polynomial-time algorithms exist, but it might (the exception that proves the rule?).
Thanks in advance.

REPLY [2 votes]: I was involved in the conversations about this topic on the Polymath4 blog (actually, looking back, it looks like I was the one who dug up that old paper...) and I came to believe that there was no such algorithm (randomized, conditional, whatever). Certainly I searched the literature as best I could and didn't find one. But I'm pessimistic about finding a reduction from factoring, for reasons I touched on in the linked post.
I was going to mention this beautiful argument, but actually I don't think it applies here -- you can only use squarefreeness to tell if a prime factor $p | N$ ramifies over some extension (Edit: I think this is true -- but something weird might happen if the extension isn't Galois? Maybe? I know so little algebraic number theory it's not even funny), but that's only possible if p divides the discriminant -- but you can do that already by the Euclidean algorithm. So squarefreeness would only let you maybe factor if for some reason you could do the algorithm quickly in number fields with huge discriminant, which admittedly might be possible. Edit: Although of course if the discriminant is big enough to make a difference, it's unclear how you'd extract information about p anyway. Which, modulo a whole bunch of holes and handwaving, would seem to rule out any naive attempt to adapt that "reduction."