Since blockstream released their fascinating sidechains paper I’ve been discussing related ideas with Gregory Maxwell. Sidechains are a method of automating the 1:1 bitcoin pegging (which pettycoin uses); it will require a soft-fork of the bitcoin protocol, which is why I never persued such a direction. The interest in side-chains suggests it’s a medium-term possibility, however.
More interesting to pettycoin directly is this writeup on how to change bitcoin to allow partial knowledge. Much of this (particularly the idea of transmitting proofs of incorrect behavior) pettycoin already does; the proof of spending a non-existing input, however, uses a completely different approach which is worth spelling out here.
In bitcoin, every node knows everything, so it’s easy to spot a
transaction which says “this transfers 100 bitcoins from
The miner inserts an index for each input into the block alongside the transaction: this says where the input transaction is. This “input_refs” array is hashed into the merkle hash tree (it’s every second leaf node).
You can prove a miner made up an input transaction by presenting the transaction, the input refs (both with proof that they’re in the block), and the transaction where the input ref said it would be (with proof). You still need to detect double-spends in the traditional way, though.
This is a bit more complicated, but it does more. The scheme has several parts, the first of which is UTXO Committments; a seemingly standard part of bitcoin-wizard lore which has several variants (Andrew Miller provided a wealth of links: Andrew Miller’s proposal and implementation, Alan Reiner’s proposal, Mark Friedenbach worked on one and Peter Todd has an implementation).
The important part of bitcoin is the Unspent Transaction Outputs (UTXO): stuff which can still be used. As you go through the blocks, you naturally have to track these, so you know whether a new transaction is spending only valid, unspent outputs.
The twist is that you formalize this structure into a UTXO tree; the specific species of tree doesn’t really matter for the idea. This tree maps “transaction id, output number” to “version, height, coinbase flag, scriptpubkey, value” (ie. everything you need to know about a transaction output to use it).
This tree then gets merkled together so you can describe the state of the tree with a single hash: this hash gets published in the block (and if it’s wrong, the block is invalid and rejected by the other miners). You can also compactly prove any part of the tree using a merkle proof, thus proving that a particular transaction is valid (hey, see, here are the unspent outputs in the last block, with proofs!).
The original motivation for this was to make light-weight clients more secure; without this, someone can prove to me that their transaction’s inputs are valid, but they can’t prove they’re not already spent.
In this model, every transaction in the block is accompanied by a proof that the inputs are in the UTXO tree; depending on how the UTXO tree is implemented, this may be sufficient to determine the new UTXO tree hash after the input is consumed (if not, proof of extra UTXO tree nodes is required). Similarly, you can provide the proof of the locations in the UTXO tree sufficient to determine the UTXO tree hash after the outputs are inserted.
This way, you can verify the inputs are correct and unspent for any random transaction. You can also verify that the UTXO tree is correct: if it’s not, some transaction must have updated it incorrectly and that can be proven.
The UTXO commitment scheme opens the door to “assisted transactions” where you supply a transaction along with UTXO proofs that all its inputs are unspent as of the last block. A recipient which knows only the block headers1 can confidently accept this as an unconfirmed transaction.
In fact, a miner can actually mine this without any extra checks: opening the door to partial-knowledge mining, which is a bar I decided was too high for pettycoin.
The UTXO solution is more complete: it provides both proof that the inputs exist, and that they’re unspent. This definitely wins.
Corollaries of this include (1) Gregory Maxwell is smarter than I am, and (2) if you can get the attention of the right bitcoin wizards, your own sidechain project will be greatly improved :)
 Bitcoin is unlikely to add the UTXO hash to the 80-byte header. Instead, it would become a compulsory part of the coinbase, is the order of 100 bytes long, and you need log2(num-transactions) 32 byte hashes to attach it to the header. But that’s still well under 1k per 10 minutes for a light-weight node.