It should be noted that fan-out, where a transaction depends on several transactions, and those transactions depend on many more, is not a problem here. There is never the need to extract a complete standalone copy of a transaction's history.
The traditional banking model achieves a level of privacy by limiting access to information to the parties involved and the trusted third party. The necessity to announce all transactions publicly precludes this method, but privacy can still be maintained by breaking the flow of information in another place: by keeping public keys anonymous. The public can see that someone is sending an amount to someone else, but without information linking the transaction to anyone. This is similar to the level of information released by stock exchanges, where the time and size of individual trades, the "tape", is made public, but without telling who the parties were.
New transaction broadcasts do not necessarily need to reach all nodes. As long as they reach many nodes, they will get into a block before long. Block broadcasts are also tolerant of dropped messages. If a node does not receive a block, it will request it when it receives the next block and realizes it missed one.
We define an electronic coin as a chain of digital signatures. Each owner transfers the coin to the next by digitally signing a hash of the previous transaction and the public key of the next owner and adding these to the end of the coin. A payee can verify the signatures to verify the chain of ownership.
We need a way for the payee to know that the previous owners did not sign any earlier transactions. For our purposes, the earliest transaction is the one that counts, so we don't care about later attempts to double-spend. The only way to confirm the absence of a transaction is to be aware of all transactions. In the mint based model, the mint was aware of all transactions and decided which arrived first. To accomplish this without a trusted party, transactions must be publicly announced, and we need a system for participants to agree on a single history of the order in which they were received. The payee needs proof that at the time of each transaction, the majority of nodes agreed it was the first received.
The solution we propose begins with a timestamp server. A timestamp server works by taking a hash of a block of items to be timestamped and widely publishing the hash, such as in a newspaper or Usenet post. The timestamp proves that the data must have existed at the time, obviously, in order to get into the hash. Each timestamp includes the previous timestamp in its hash, forming a chain, with each additional timestamp reinforcing the ones before it.
Although it would be possible to handle coins individually, it would be unwieldy to make a separate transaction for every cent in a transfer. To allow value to be split and combined, transactions contain multiple inputs and outputs. Normally there will be either a single input from a larger previous transaction or multiple inputs combining smaller amounts, and at most two outputs: one for the payment, and one returning the change, if any, back to the sender.
It's not a problem if transactions have to wait one or a few extra cycles to get into a block.
The recipient just needs to verify it back to a depth that is sufficiently far back in the block chain, which will often only require a depth of 2 transactions. All transactions before that can be discarded.
Receivers of transactions will normally need to hold transactions for perhaps an hour or more to allow time for this kind of possibility to be resolved. They can still re-spend the coins immediately, but they should wait before taking an action such as shipping goods.
Instantant non-repudiability is not a feature, but it's still much faster than existing systems. Paper cheques can bounce up to a week or two later. Credit card transactions can be contested up to 60 to 180 days later. Bitcoin transactions can be sufficiently irreversible in an hour or two.
A basic transaction is just what you see in the figure in section 2. A signature (of the buyer) satisfying the public key of the previous transaction, and a new public key (of the seller) that must be satisfied to spend it the next time.
If a merchant actually has a problem with theft, they can make the customer wait 2 minutes, or wait for something in e-mail, which many already do. If they really want to optimize, and it's a large download, they could cancel the download in the middle if the transaction comes back double-spent. If it's website access, typically it wouldn't be a big deal to let the customer have access for 5 minutes and then cut off access if it's rejected. Many such sites have a free trial anyway.
There are two ways to send money. If the recipient is online, you can enter their IP address and it will connect, get a new public key and send the transaction with comments. If the recipient is not online, it is possible to send to their Bitcoin address, which is a hash of their public key that they give you. They'll receive the transaction the next time they connect and get the block it's in. This method has the disadvantage that no comment information is sent, and a bit of privacy may be lost if the address is used multiple times, but it is a useful alternative if both users can't be online at the same time or the recipient can't receive incoming connections.
A digital coin contains the public key of its owner. To transfer it, the owner signs the coin together with the public key of the next owner. Anyone can check the signatures to verify the chain of ownership.
For greater privacy, it's best to use bitcoin addresses only once.
That would be nice at point-of-sale. The cash register displays a QR-code encoding a bitcoin address and amount on a screen and you photo it with your mobile.
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.
As you figured out, the root problem is we shouldn't be counting or spending transactions until they have at least 1 confirmation. 0/unconfirmed transactions are very much second class citizens. At most, they are advice that something has been received, but counting them as balance or spending them is premature.