BitCoin is a recently introduced currency, based on public-key cryptography combined with a peer-to-peer network for verifying transactions. I've been thinking a lot about BitCoin recently, and particularly about BitCoin's main weakness: if your computer is compromised, an attacker could copy your BitCoin wallet and use it to steal coins. That's bad. But I've come up with a possible improvement that would greatly mitigate this risk, and was hoping for some help confirming its viability and filling in the details.
The basic idea is to make it so that rather than having a single computer which can steal your coins if it's compromised, you have two computers (or a computer and a phone), such that your coins can only be spent if both devices cooperate. It is much harder to break into two computers belonging to the same person than just one, so this makes coins harder to steal. You could also have one of the computers involved be a third party that you trust to keep its files secure, and while that third party would be able to freeze your funds, it wouldn't be able to steal them. Using a third party this way, you could also add withdrawal rate limits and time delays, further improving security.
I believe that this can be done in a fully backwards-compatible way, without any changes to the BitCoin protocol, using homomorphic encryption. BitCoin is based on elliptic curve cryptography; a receiving address is a public key, and a wallet file is a collection of private keys. The goal is to create a protocol where two cooperating computers produce a split key, such that they can use it cooperatively to sign transactions later, but neither one can sign transactions or determine the whole key on its own. My understanding is that homomorphic encryption can be used to implement a simulated computer that does arbitrary trusted computation, so this should be possible. However, I'm a bit fuzzy on the details, and I don't have the time or comparative advantage to implement this myself.
(To deal with the risk of one one computer being lost or damaged, there could also be an override key; both computers would have the public half of the override key, and the private half would be kept offline in a bank deposit box or something similar. Then both computers use the override key to encrypt their halves of the split key, and send the encrypted keys to a cloud backup provider.)