The main properties: Double-spending is prevented with a peer-to-peer network. No mint or other trusted parties. Participants can be anonymous. New coins are made from Hashcash style proof-of-work. The proof-of-work for new coin generation also proof-of-workers the network to prevent double-spending.
Once the latest transaction in a coin is buried under enough blocks, the spent transactions before it can be discarded to save disk space. To facilitate this without breaking the block's hash, transactions are hashed in a Merkle Tree, with only the root included in the block's hash. Old blocks can then be compacted by stubbing off branches of the tree. The interior hashes do not need to be stored.
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.
We should have a gentleman's agreement to postpone the GPU arms race as long as we can for the good of the network. It's much easer to get new users up to speed if they don't have to worry about GPU drivers and compatibility. It's nice how anyone with just a CPU can compete fairly equally right now.
By convention, the first transaction in a block is a special transaction that starts a new coin owned by the creator of the block. This adds an incentive for nodes to support the network, and provides a way to initially distribute coins into circulation, since there is no central authority to issue them.
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. If the real transaction reaches 90% and the double-spent tx reaches 10%, the double-spender only gets a 10% chance of not paying, and 90% chance his money gets spent. For almost any type of goods, that's not going to be worth it for the scammer.
The design supports a tremendous variety of possible transaction types that I designed years ago. Escrow transactions, bonded contracts, third party arbitration, multi-party signature, etc. If Bitcoin catches on in a big way, these are things we'll want to explore in the future, but they all had to be designed at the beginning to make sure they would be possible later.
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