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The problem of course is the payee can't verify that one of the owners did not double-spend the coin. A common solution is to introduce a trusted central authority, or mint, that checks every transaction for double spending. After each transaction, the coin must be returned to the mint to issue a new coin, and only coins issued directly from the mint are trusted not to be double-spent. The problem with this solution is that the fate of the entire money system depends on the company running the mint, with every transaction having to go through them, just like a bank.

We propose a solution to the double-spending problem using a peer-to-peer network. The network timestamps transactions by hashing them into an ongoing chain of hash-based proof-of-work, forming a record that cannot be changed without redoing the proof-of-work. The longest chain not only serves as proof of the sequence of events witnessed, but proof that it came from the largest pool of CPU power. As long as a majority of CPU power is controlled by nodes that are not cooperating to attack the network, they'll generate the longest chain and outpace attackers. The network itself requires minimal structure.

In this paper, we propose a solution to the double-spending problem using a peer-to-peer distributed timestamp server to generate computational proof of the chronological order of transactions. The system is secure as long as honest nodes collectively control more CPU power than any cooperating group of attacker nodes.

Long before the network gets anywhere near as large as that, it would be safe for users to use Simplified Payment Verification (section 8) to check for double spending, which only requires having the chain of block headers, or about 12KB per day.

The attacker isn't adding blocks to the end. He has to go back and redo the block his transaction is in and all the blocks after it, as well as any new blocks the network keeps adding to the end while he's doing that. He's rewriting history. Once his branch is longer, it becomes the new valid one.

The guy who received the double-spend that became invalid never thought he had it in the first place. His software would have shown the transaction go from "unconfirmed" to "invalid". If necessary, the UI can be made to hide transactions until they're sufficiently deep in the block chain.

The receiver of a payment must wait an hour or so before believing that it's valid. The network will resolve any possible double-spend races by then.

When there are multiple double-spent versions of the same transaction, one and only one will become valid.

There's no need for reporting of "proof of double spending" like that. If the same chain contains both spends, then the block is invalid and rejected.
Same if a block didn't have enough proof-of-work. That block is invalid and rejected. There's no need to circulate a report about it. Every node could see that and reject it before relaying it.

We're not "on the lookout" for double spends to sound the alarm and catch the cheater. We merely adjudicate which one of the spends is valid. Receivers of transactions must wait a few blocks to make sure that resolution has had time to complete. Would be cheaters can try and simultaneously double-spend all they want, and all they accomplish is that within a few blocks, one of the spends becomes valid and the others become invalid. Any later double-spends are immediately rejected once there's already a spend in the main chain.

The race is to spread your transaction on the network first. Think 6 degrees of freedom -- it spreads exponentially. It would only take something like 2 minutes for a transaction to spread widely enough that a competitor starting late would have little chance of grabbing very many nodes before the first one is overtaking the whole network. During those 2 minutes, the merchant's nodes can be watching for a double-spent transaction. The double-spender would not be able to blast his alternate transaction out to the world without the merchant getting it, so he has to wait before starting.

Any owner could try to re-spend an already spent coin by signing it again to another owner. The usual solution is for a trusted company with a central database to check for double-spending, but that just gets back to the trust model. In its central position, the company can override the users, and the fees needed to support the company make micropayments impractical.
Bitcoin's solution is to use a peer-to-peer network to check for double-spending. In a nutshell, the network works like a distributed timestamp server, stamping the first transaction to spend a coin. It takes advantage of the nature of information being easy to spread but hard to stifle.