I am going to tell you about zero-knowledge proofs for Bitcoin scalability. Even though this is an overview talk, many of the results I will be telling you about are based on work done by people in the slides and also here at the conference.

First I will give a brief introduction to zero knowledge proofs. Then I will try to convince you that if you care about Bitcoin scalability, then you should care about zero-knowledge proofs and you should keep them on your radar.

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The instance here is that a circuit and a witness. The circuit output is 1 then it's an input. Both prover and verifier know the circuit, and the prover knows the witness. Finally, the verifier gets the final answer from the prover. It either accepts or rejects. We want the following three properties for this. If the prover didn't know the valid witness, and finally and most interestingly, we wanted just given this interaction, the verifier learns nothing about it other than the witness exists. The verifier would not learn anything about what made the circuit accept. So those are interactive zero-knowledge proofs.

They become more useful when they are non-interactive provers. The prover sends just one message, a proof to the verifier, and the verifier either accepts or rejects, just like before. It's very useful because the prover can post a proof, and later go away, and nobody needs to interact with the prover again, and everyone can verify the proof for the rest of the time. This is a formal notion. What I have told you is a little bit of a lie, however it becomes possible if you let a trusted pre-computation, just one time, and how to make a common reference string.

Here's how it works. There needs to be a trusted generator, this is required only once, the prover and verifier can use the CRS to prove statements and verify them respectively. This might sound very abstract, but I will moderate this example for how this is useful for Bitcoin scalability.

My claim is that there are many Bitcoin scalability problems that can be traced back to questions about privacy. If all transactions are public, then receiving the wrong coin could be disastrous because it could taint all of your other bitcoin. Solvency is another example. Exchanges could claim that proving solvency is a privacy liability, well you could use zero-knowledge proofs. Mining centralization could also push back issues of privacy.

Zero-knowledge proofs could be helpful for all of the above. I am going to prove my claim by example. I will choose fungibility and solvency examples, to show how you can use zero-knowledge proofs from the previous slides, to solve this. I will show you how to develop a currency that is completely private. This is based on a scheme by Sanderson Tosha.

Suppose you have the Bitcoin blockchain, and you want to have anonymous currency sitting right... anonymous coin with coin commitment. So CM is just going to be a hash of a serial number, and a note. You can go on like this and create wonderful anonymous coins like this, and it's not really useful is it-- so what you do next is you need some way to spend those coins, presumably in a private way. How can you spend a coin? Just choose one of your anonymous coins from the nice bubble and publish it ... (stream problems) .... is publishing, is doing the following trick. You can construct a merkle tree over all of your coin commitments in your wonderful bubble, then you publish a serial number plus a zero knowledge proof. ((streaming))... There exists a nonce such that when you compute the serial number plus the nonce, is indeed in the merkle tree. You can see that this is private, the serial number does not reveal anything about where the CM came from. The proof doesn't reveal this either since it's zero knowledge. So the transaction is completely unlinkable, yet a zero knowledge proof ensures integrity. There exists a coin in this anonymous bubble, and more over you can only spend this coin once, since you tried to spend this coin in the bubble twice, you would spend the same serial number twice and you would be caught.

This is the basic idea behind zerocash, at a high level. Zerocash adds divisibility and some other concepts. The basic idea remains the same. You have a ledger note of transactions, but a ledger of proofs that the transactions are valid. So that's one example of zero-knowledge proofs.

Another example is how to use zero-knowledge for privacy-preserving proofs of solvency. Here by solvency I just mean the simple claim that my assets are greater than my liabilities, and privacy-preserving should mean that the exchange not reveal anything about its fees or balances. All of this is 10,000 feet overview of Provisions, which works as follows. You need to publish two commitments, one to all your assets, one commitment to your liabilities, then three kinds of zero-knowledge proofs. You want to claim that the exchange is solvent, which means that the asset commitment is greater than the liabilities commitment. This is not enough because anything could be in those two commitments. So what you need next is you prove that the liability commitment is correctly formd, meaning

https://eprint.iacr.org/2015/1008.pdf zk proof solvency

... I will publish commitmnts to all balancs, t, htn I will go tach of my users and prove that each user will... and finally prove that all liabilities are consistent, and all the user balance commitments are complete. And that's how I prove liability. Of course, finally, I need to prove that I can control the set amount of asset. I will do this first by choosing a large anonymity set, say the entire set of Bitcoin public keys on the blockchain and their associated balances. Then I would try to prove knowledge of private keys for a subset, that can open up at least the minimum amount of bitcoin required. A zero knowledge proof is quite involved, but you should believe this is doable. There is a way to prove solvency here.

These examples have demonstrated to us that zero-knowledge is something really useful to us. We should go and seek can we implement zero-knowledge in practice. I'll tell you about that.

First, I'll tell you about two kinds of zero-knowledge proofs. There's classical non-interactive zero-knowledge proofs, such as Schnoor proofs or range proofs. These proofs have been implemented and even deployed, but classical NIZKs ... the proofs can be long, as long as time to decide membership in relation and hard to verify.

There's a different cryptographic primitive called SNARKs that address this inefficiency of NIZKs. These are proofs that are short in constant size, they are easy to verify in practically milliseconds. This succinctness comes from somewhere, and this somewhere is cryptographic assumptions. NIZKs require more traditional cryptographic assumptions, while SNARKs rely on stronger cryptographic assumptions. NIZKs and SNARKs differ in how you generate the common reference strong. For NIZKs, the common reference string just outputs random coins and you can get randomness from hash functons and sun spots or whatever, while for SNARKs the generator gives a lot of structure but it's really complex and you need someone to do it. But we will see later how to address this.

We like the efficiency of SNARKs because they are orders-of-magnitude more efficient, like for NP statements. So for the rest of the talk I am going to be focusing on this cryptographic primitive. They are strong, powerful, do they exist? Yes, we have a beautiful line of research that have used theoretical constructions starting from the 1990s. Some of the constructions are in working prototypes from groups around the world. Most of those prototypes have full source code available online. You can go to those links and get implemenations.

They are feasible for certain applications such as zerocash or (inaudible). SNARKs do exist, I can go get their source code. How do I program a SNARK? I have a relation in mind, what do I do? There's a bit of a challenge, which is representation. The relation I have in mind might be high-level, like hashes and merkle trees and signatures, while the implemented constructions of SNARKs understand something like circuit satisfyability. How do I get from my relation on paper to an actual circuit gate-by-gate that the SNARK prover can check?

This is a solved problem. There are three solutions on how to do that. Number one, there's the snarks-for-c approach which mimics the early days of computing. You first pick a CPU, then you write the transition function of the CPU in a circuit. You can write a universal circuit that can decide any circuit written for this CPU. Finally, you can compile the program down to assembly and assembly is something that your universal circuit can eat and all is happy. This approach is probably the only approach that I know that can give you universality or bootstrapping or what not, but it can also be quite inefficient because for each step of computation you need the entire CPU. So, if your computation is sufficiently specialized, you can use a different approach called program analysis. You write your program that decides the relation in a restricted subset of C, your program must have all of its memory accesses there. If this is the case, then (... buffering ....) .... usually writing subcomponents for your relation, by hand, a circuit and then composing them together. You can imagine being an electrical engineer and placing your breadboard for circuits, a component for SNARK verify or a component for elliptic curve verifying. This may seem crazy to write all of this by hand, but this only needs to be done once. libsnark has a lot of these components at this time. You could probably use a gadget approach.

Okay so there are 3 approaches, that's quite a slide. Which one should I choose in practice? In fact, what is the practical efficiency for SNARKs? The verifier efficiency only depends on the input of your relation. Usually when you structure your relation write, it's milliseconds. The bottleneck seems to be prover performance which is essentially base SNARK performance, applied to the size of the circuit. You can write a small circuit and that would be very nice, but what does it mean in concrete numbers? Well, there's a paper about non-outsourceable puzzle proofs, which are benchmarks for provers, like a half-million gate circuit. One circuit takes about 20 minutes on this system, or anothe rtime on another system.

.. which is a useful primitive to have in merkle trees, and if you write it donw in a snarks-for-c approach for say vntinyram, you can get 0.8 conversion factor. If you just use the circuit generator, you would get maybe 8.1, and if you write it down with gadgets, you could probably .....

The crucial thing for efficient use of SNARKs is what we call relation engineering, meaning that your relation should really be checking different properties. You should use non-determinism. I am going to tell you about the most exciting part. Which is applying NIZKs and SNARKs.

Let's say you have a NIZK system, and you have implemented it and you can prove and verify. The question is, can it be deployed? Systems no longer confined to a single.. so who in practice generates a common reference string? Who is the person who is trusted by absolutely everyone? There is no such person. This does not mean that SNARKs cannot be deployed. Zero-knowledge still holds, transactions are still private. Unfortunately some of this breaks, then you could start making false statements. Some SNARKs usage is not consenus critical.

petertodd has a proposal for sending blockheader plus proof. If you have a bad common reference string, then you have a few lost blocks at worst. Can a consensus-critical SNARK be deployed? The answer is yes, and I am going to show you that. Ideally, we would like, there's a wonderful SNARK generator that generates a CRS, and then it goes away without leaking. We can't have this one person trusted by everyone. We're going to have a decentralized SNARK setup that has multiple parties in it, those parties will pass messages and out comes the CRS. Our protocol will produce the same thing an idealized generator would have produced, up to M-1 corruption which means that the CRS is fully secure as long as one of the participants is honest. We have designed a protocol that does this, it's reasonably efficient, not ultimately efficient, but it's still something.

As long as you trust at least one of them to not collude, you're fine. To conclude, many scalability problems can be traced back to privacy, and zero-knowledge proofs can help. Some people say that zero-knowledge is 10 years away, but I think this is false, it was 10 years away 10 years ago, today it can be deployed. If you want to be part of this, one of the things working on zero-knowledge is libsnark. We are putting out a call for contributions. We need review and contributions and visit our web page and help us out reach a way for Bitcoin scalability.

Have Edward Snowden as an MPC participant.

Q: ...

A: The gadget approach can be combined with program analysis, so you can do the heavy lifting of hash functions and gadgets. You can write them in C. The team from U of Maryland is about to release a compiler that does this.

Q: ...

A: If I understand correctly, if all the transactions are private, who stores information about values? All of the individual participants would only know the values that they control. The consensus part is only that the validity was maintained.

Q: How well does SNARK verify in your gadget work?

A: This was crypto that ... peter is asking for recursive SNARKs, do we have verifiers for some? The answer is yes, and you can do bootstrapping with recursive SNARKs. Oops I misspoke, it was not Peter. It was Pindar.

Q: ...

A: We hvne't tried implementing that. ....

Q: To prove solvency with zero-knowledge proofs, is that where you have to burn the bitcoin to prove you have the private keys? If you burn them, then how does that connect to proof your solvency, did this just move your BTC off to DCMs (what?).

A: .....

Last question.

Q: bitcoin mixers; so why snarks?

A: There is a tradeoff between the anonymity you get and the protocol modifications you need. For mixers, you are not fully anonymous, there is still some knowledge about where the coins came from. For a system wth SNARKs, the anonymity set is everyone in the system. If you need this kind of anonymity or fungibility, the cost needs to be put somewhere. I guess lesser anonymity could be obtained through other mechanisms.

One last question.

Q: Is there any way to update the CRS?

A: ...

https://github.com/scipr-lab/libsnark

http://libsnark.org/help