The solution exists. It's called fully homomorphic encryption.

See:

https://researcher.ibm.com/researcher/view_page.php?id=2661


If you combine a) fully homomorphic encryption with b) mutation, genetic algorithms, and mitosis you could end up with truly autonomous software agents, that collect bitcoins entirely for their own benefit.

They would be the bacteria to the computer virus.

The interesting thing is about these "computer bacteria" is that, just like real bacteria, they don't need complex AI to be successful. They could do something relatively simple like the example of inducing people to solve captchas, mentioned above.  So this would be almost possible using today's technology.

As long as they keep finding people (and other software agents) who help them spread, they and their offspring could survive for a very long time and collect a large quantity of bitcoins that "belong" to them only and not a human being, not even the original author!

I'm not sure to what extent fully homomorphic encryption is resistant to brute force attacks though.

...and someone, posing himself as hosting provider, can collect them and scrape for their private keys
(or modify public keys)