Rubber-hose-squad:

We will never be able to show that Alice has revealed the last of her keys. Further, even if Alice has co-operated fully and has revealed all of her keys, she will not be able to prove it. Therefore, we must assume that at every stage that Alice has kept secret information from us, and continue to beat her, even though she may have revealed the last of her keys. But the whole time we will feel uneasy about this because Alice may have co-operated fully. Alice will have realised this though, and so presumably it's going to be very hard to get keys out of her at all.

Alice:

I can never prove that I have revealed the last of my keys. In the end I'm bound for continued beating, even if I can buy brief respites by coughing up keys from time to time. Therefor, it would be foolish to divulge my most sensitive keys, because (a) I'll be that much closer to the stage where I have nothing left to divulge at all (it's interesting to note that this seemingly illogical, yet entirely valid argument of Alice's can protect the most sensitive of Alice's keys the "whole way though", like a form mathematical induction), and (b) the taste of truly secret information will only serve to make my aggressors come to the view that there is even higher quality information yet to come, re-doubling their beating efforts to get at it, even if I have revealed all. Therefor, my best strategy would be to (a) reveal no keys at all or (b) depending on the nature of the aggressors, and the psychology of the situation, very slowly reveal my "duress" and other low-sensitivity keys."