[geofft]

A good hash function doesn't, and cannot, ensure that you have no information about the input. It can't take information away from you.

A good hash function ensures that, whatever information you have about the input, you have no more information upon seeing the hash. If you believe x is equiprobable over some range, seeing H(x) should cause you to continue to believe it's equiprobable over that range. If you believe x is the square of some random variable y which is equiprobable in some range, seeing H(x) won't convince you that x is equiprobable!

If your prior for x is equiprobable over the space of Microsoft Word documents confessing to high treason and 0 probability elsewhere, seeing H(x) won't tell you which Word document it is, but it certainly won't cause you to believe that it might be an MP3, either.