Previously published as a reply on Reddit.

Why aren’t we as a community talking about Sharding as a scaling solution? It seems to me that sharding would be the ideal solution for scaling bitcoin up indefinitely, but all I keep hearing about is lightning and sidechains, which are sort of bolt-on afterthoughts operating a level above bitcoin. Will existing wallets work smoothly with lightning? dunno, but I doubt it. And anyway, they do nothing to help bitcoin itself scale up, or to reduce disk/bandwidth requirements of individual nodes.

In a system where everyone mostly trusts each other, sharding works great! You just split up the blockchain the same way you’d shard a database, assigning miners/validators a subset of the txid space. Transaction validation would assume that if you don’t have the history for an input yourself, you assume that history is valid. In a banking-like environment where there’s a way to conduct audits and punish those who lie, this could certainly be made to work. (I myself have worked on and off on a scheme to do exactly that for a few different clients: Proofchains)

But in a decentralized environment sharding is far, far, harder to accomplish… There’s an old idea we’ve been calling “fraud proofs”, where you design a system where for every way validation can fail, you can create a short proof that part of the blockchain was invalid. Upon receiving that proof your node would reject the invalid part of the chain and roll back the chain. In fact, the original Satoshi whitepaper refers to fraud proofs, using the term “alerts”, and assumed SPV nodes would use them to get better guarantees they’re using a valid chain. (SPV as implemented by bitcoinj is sometimes referred to as “non-validating SPV”) The problem is, how do you guarantee that the fraud will get detected? And How do you guarantee that fraud that is detected actually gets propagated around the network? And if all that fails… then what?

The nightmare scenario in that kind of system is some miner successfully gets away with fraud for awhile, possibly creating hundreds of millions of dollars worth of bitcoins out of thin air. Those fake coins could easily “taint” a significant fraction of the economy, making rollback impossible and shaking faith in the value of the currency. Right now in Bitcoin this is pretty much impossible because everyone can run a full node to validate the chain for themselves, but in a sharded system that’s far harder to guarantee.

Now, suppose we can guarantee validity. zk-SNARKS are basically a way of mathematically proving that you ran a certain computer program on some data, and that program returned true. Recursive zk-SNARKS are simply zk-SNARKS where the program can also recursively evaluate that another zk-SNARK is true. With this technology a miner could prove that the shard they’re working on is valid, solving the problem of fake coins. Unfortunately, zk-SNARKS are bleeding edge crypto, (if zerocoin had been deployed the entire system would have been destroyed by a recently found bug that allowed fake proofs to be created) and recursive zk-SNARKS don’t exist yet.

The closest thing I know of to recrusive zk-SNARKS that actually does work without “moon-math” is an idea I came up with for treechains called coin history linearization. Basically, if you allow transactions to have multiple inputs and outputs, proving that a given coin is valid requires the entire coin history, which has quasi-exponential scaling - in the Bitcoin economy coins are very quickly mixed such that all coins have pretty much all other coins in their history.

Now suppose that rather than proving that all inputs are valid for a transaction, what if you only had to prove that one was valid? This would linearize the coin history as you only have to prove a single branch of the transaction DAG, resulting in O(n) scaling. (with n <= total length of the blockchain chain)

Let’s assume Alice is trying to pay Bob with a transaction with two inputs each of equal value. For each input she irrevocable records it as spent, permanently committing that input’s funds to Bob. (e.g. in an irrevocable ledger!) Next she makes use of a random beacon - a source of publicly known random numbers that no-one can influence - to choose which of the two inputs’ coin history’s she’ll give to Bob as proof that the transaction is real. (both the irrevocable ledger and random beacon can be implemented with treechains, for example)

If Alice is being honest and both inputs are real, there’s a 100% chance that she’ll be able to successfully convince Bob that the funds are real. Similarly, if Alice is dishonest and neither input is real, it’ll be impossible for her convince prove to Bob that the funds are real.

But what if one of the two inputs is real and the other is actually fake? Half the time the transaction will succeed - the random beacon will select the real input and Bob won’t know that the other input is fake. However, half the time the fake input will be selected, and Alice won’t be able to prove anything. Yet, the real input has irrevocably been spent anyway, destroying the funds! If the process by which funds are spent really is irrevocable, and Alice has absolutely no way to influence the random beacon, the two cases cancel out. While she can get away with fraud, there’s no economic benefit for her to do so. On a macro level, this means that fraud won’t result in inflation of the currency. (in fact, we want a system that institutionalizes this so-called “fraud” - creating false proofs is a great way to make your coins more private) (FWIW the way zk-SNARKS actually work is similar to this simple linearization scheme, but with a lot of very clever error correction math, and the hash of the data itself as the random beacon)

An actual implementation would be extended to handle multiple transaction inputs of different sizes by weighing the probability that an input will be selected by it’s value - merkle-sum-trees work well for this. We still have the problem that O(n) scaling kinda sucks; can we do better?

Yes! Remember that a genesis transaction output has no history - the coins are created out of thin air and its validity is proven by the proof of work itself. So every time you make a transaction that spends a genesis output you have a chance of reducing the length of the coin validity proof back to zero. Better yet, we can design a system where every transaction is associated with a bit of proof-of-work, and thus every transaction has a chance of resetting the length of the validity proof back to zero. In such a system you might do the PoW on a per-transaction basis; you could outsource the task to miners with a special output that only the miner can spend. Now we have O(1) scaling, with a k that depends on the inflation rate. I’d have to dig up the calculations again, but IIRC I sketched out a design for the above that resulted in something like 10MB or 100MB coin validity proofs, assuming 1% inflation a year. (equally you can describe that 1% inflation as a coin security tax) Certainly not small, but compared to running a full node right now that’s still a huge reduction in storage space. (recursive zk-SNARKS might reduce that proof to something like 1kB of data)

Regardless of whether you have lightweight zk-SNARKS, heavyweight linearized coin history proofs, or something else entirely, the key advantage is that validation can become entirely client side. Miners don’t even need to care whether or not their own blocks are “valid”, let alone other miners’ blocks. Invalid transactions in the chain are just garbage data, which gets rejected by wallet software as invalid. So long as the protocol itself works and is implemented correctly it’s impossible for fraud to go undetected and destroy the economy the way it can in a sharded system.

However we still have a problem: censorship. This one is pretty subtle, and gets to the heart of how these systems actually work. How do you prove that a coin has validly been spent? First, prove that it hasn’t already been spent! How do you do that if you don’t have the blockchain data? You can’t, and no amount of fancy math can change that.

In Bitcoin if everyone runs full nodes censorship can’t happen: you either have the full blockchain and thus can spend your money and help mine new blocks, or that alternate fork might as well not exist. SPV breaks this as it allows funds to be spent without also having the ability to mine - with SPV a cartel of miners can prevent anyone else from getting access to the blockchain data required to mine, while still allowing commerce to happen. In reality, this type of cartel would be more subtle, and can even happen by accident; just delaying other miners getting blockchain data by a few seconds harms those non-cartel miners’ profitability, without being obvious censorship. Equally, so long as the cartel has >30% of hashing power it’s profitable in the long run for the cartel if this happens.

In sharded systems the “full node defense” doesn’t work, at least directly. The whole point is that not everyone has all the data, so you have to decide what happens when it’s not available.

Altcoins provide one model, albeit a pretty terrible one: taken as a whole you can imagine the entire space of altcoins as a series of cryptocurrency shards for moving funds around. The problem is each individual shard - each altcoin - is weak and can be 51% attacked. Since they can be attacked so easily, if you designed a system where funds could be moved from one shard to another through coin history proofs every time a chain was 51% attacked and reorged you’d be creating coins out of thin air, destroying digital scarcity and risking the whole economy with uncontrolled inflation. You can instead design a system where coins can’t move between shards - basically what the altcoin space looks like now - but that means actually paying someone on another “shard” requires you to sell your coins and buy their coins - a inefficient and expensive logistical headache. (there’s a reason the Eurozone was created!)

If you want to transfer value between shards with coin history proofs, without risking inflation, you need all the shards to share some type of global consensus. This is the idea behind treechains: every part of the tree is linked to a top-level timestamp chain, which means we have global consensus on the contents of all chains, and thus spending a coin really is an immutable one-time act.

Let’s go into a bit more detail. So what is a coin in a treechains system? First and foremost it’s a starting point in some part of the tree, a specific subchain. When Alice wants to prove to Bob that she spent a coin, giving it to Bob, she inserts into that subchain the data that proves that someone could have spent that coin - a valid signature and the hash of the transaction output it was spending. But the actual proof that she gives to Bob isn’t just that spend data, but rather proof that all the blocks in that chain between the starting point and the spend did not have a valid spend in them. (easiest way to do that? give Bob those blocks) That proof must link back to the top-level chain; if it doesn’t the proof is simply not valid.

Now suppose Alice can’t get that part of the subchain, perhaps because a cartel of miners is mining it and won’t give anyone else the data, or perhaps because everyone with the data suffered a simultaneous harddrive crash. We’ll also say that higher up in the tree the data is available, at minimum the top-level chain. As with Bitcoin, as long as that cartel has 51% of the hashing power, Alice is screwed and can’t spend her money.

What’s interesting is what happens after that cartel disbands: how does mining restart? It’s easy to design a system where the creation of a block doesn’t require the knowledge of previous blocks, so new blocks can be added to extend the subchain. But Alice is still screwed: she can’t prove to Bob that the missing blocks in the subchain didn’t contain a valid spend of her coin. This is pretty bad, on the other hand the damage is limited to just that one subchain, and the system as a whole is unaffected.

There’s a tricky incentives problem here though: if a miner can extend a subchain without actually having previous blocks in that chain, where’s the incentive for that miner to give anyone else the blocks they create? Remember that exclusive knowledge of a block is potentially valuable if you can extort coin owners for it. (Bitcoin suffers from this problem right now with validationless “SPV” mining, though the fact that a block can be invalid in Bitcoin helps limit its effects)

Part of the solution could be mining reward; in Bitcoin, coinbase outputs can’t be spent for 100 blocks. A similar scheme could require that a spend of a coinbase output in a subchain include proof that the next X blocks in that subchain were in fact linked together. Secondly make block creation dependent on actually having that data to ensure the linkage actually means something, e.g. by introducing some validity rules so blocks can be invalid, and/or using a PoW function that requires hashers to have a copy of that data.

Ultimately though this isn’t magic: like it or not lower subchains in such a system are inherently weaker and more dangerous than higher ones, and this is equally true of any sharded system. However a hierarchically sharded system like treechains can give users options: higher subchains are safer, but transactions will expensive. The hierarchy does combine the PoW security of all subchains together for the thing you can easily combine: timestamping security.

There’s a big problem though: holy !@#$ is the above complex compared to Bitcoin! Even the “kiddy” version of sharding - my linearization scheme rather than zk-SNARKS - is probably one or two orders of magnitude more complex than using the Bitcoin protocol is right now, yet right now a huge % of the companies in this space seem to have thrown their hands up and used centralized API providers instead. Actually implementing the above and getting it into the hands of end-users won’t be easy.

On the other hand, decentralization isn’t cheap: using PayPal is one or two orders of magnitude simpler than the Bitcoin protocol.