I believe that once could sit down with a computer and another person, and demonstrate to his satisfaction that I have a private key that holds one bitcoin, which nobody else has access to.
This would be by creating a new private key in his presence, in some way where he does not see it or all of it.
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But how do you prove that the key "generated" is "unique" and that no-one else has access to it?
I know it's a bit of a stupid argument really,
because proving the absence of something is always difficult... but just because they see you "generate" it... doesn't mean that someone else doesn't have access. Although given that this is a "thought experiment" and therefore we're free to dream up all sorts of theoretical scenarios... I suppose if you were both in a sealed room with dice and an offline device that could convert the dice rolls to private key/public key/address, you could be fairly certain the key you generated is "unique"... So as long as the other person had no access to a transmitting device, you could send 1 BTC to the address... show the confirmation and they could be fairly certain you had sole access to that 1 BTC at that point in time... and you wouldn't have to worry about them stealing it as they have no way of "using" the private key.
The problem then would be that to ensure they don't steal your 1 BTC, you either have to leave them in the sealed room forever, kill them... or move the 1 BTC to another address. At which point, they can no longer be certain you have sole access
Yes this is all correct. RE the bolded above, an example would be if the key I presented was part of a heuristic chain, for which another party had knowledge of.
This reminds me of playing cards, where in normal conditions one assumes that a party other than the dealer cuts the deck, and randomness is established. That's knowing that cutting the deck is insufficient if a talented mechanic is the dealer.
Suppose that I generated a private key, and then allowed the other party to see and to change ten digits of it. He can be certain its then secure from 3rd parties at that moment. I then generate a public key, and show that it contains zero. Then I put this private key in my wallet, move a bitcoin to it.
You mentioned, "The problem then would be that to ensure they don't steal your 1 BTC, you either have to leave them in the sealed room forever, kill them... "
That's kind of funny. "Okay now I've proved I have one bitcoin, but you have to die..." It's like the joke about the spy who says "If you find the secret I'll have to kill you."
But the other party, in the above scenario, does not know the key, only ten digits of it. He cannot find it from just ten digits.