# Building Bulletproofs

Speakers: Henry de Valence, Cathie Yun

*Transcript By: Bryan Bishop*

Tags: Zero knowledge

Category: Conference

https://twitter.com/kanzure/status/1090668746073563136

# Introduction

There are two parts to this talk. The first part of the talk will be building, like all the pieces we put into making the implementation of bulletproofs. We’ll talk about the group we used, the Ristretto group and ristretto255 group. I’ll talk about parallel lelliptic curve arithmetic, and Merlin transcripts for zero-knowledge proofs, and how all these pieces fit together.

Cathie will talk in part two about constraint system proofs, cloak for a confidential assets protocol, ZkVM for zero-knowledge contracts, and aggregation multi-party computation with session types to aggregate proofs and have faster verification.

# Ristretto

Before I talk about ristretto, let’s talk about the problem we’re trying to solv.e Many prtocols you might want to implement use some prime order group like “Let G denote a cyclic group of prime order p” from the Bulletproofs paper. As an implementer, you’re supposed to have one ready to use to instantiate the protocol. But how do you implement that?

Probably you want to use an elliptic curve. But which one could yo uuse? Maybe a Weierstrass curve like secp256k1, or an Edwards curve like curve25519 or FourQ. The advantage of a Weierstrass curve is that the curve gives you a prime order group which is the abstraction you want, but it has some downsides. Edwards curves have faster algorithms that are complete without special cases to deal with, and it’s easy to implement them in constant time. Those are pretty big implementation advantages.

# What’s wrong with having a small cofactor?

The security proofs for the abstract protocol don’t then apply to your implementation. Having full validation that some input point lies on the subgroup requires multiplying by that order, which negates most of the speedup yo uget. You could ad-hoc tweaks like multiplying by a cofactor in an appropriate place, but how do you pick that place? And also, are there any subtle effects on your protocol by doing that? The ansewr is probably yes.

# Example of cofactor problems

Ed25519 has different behavior between single and batch verification. Two implementations are freely allowed to disagree about which signatures are valid, which might be a problem for some kind of blockchain.

Tor had an issue like this, where onion service addreses in tor had to add extra validation, the cofactor problem had 8 addresses for the same server.

Monero had a critical vulnerability due to cofactors where having a cofactor 8 meant that you could spend 8 times. See https://www.getmonero.org/2017/05/17/disclosure-of-a-major-bug-in-cryptonote-based-currencies.html for more details.

Edwards curves are simpler, but they push complexity into the upwards into the protocol. As a curve designer, that’s great for you, but implementation now becomes more complexity.

# Decaf and ristretto

Decaf and Ristretto is a construction to help this. Decaf lets you construct a prime order group from a non-prime-order curve. Ristretto is a variant of decaf that works with certain curves like ed25519. Ristretto255 group gives you a prime order group, canonical encoding, decoding, hash-to-group built in, you can use curve25519 internally so it’s easy to extend an existing implementation but because of a really magical way that it is constructed then an implementation can swap out curve25519 for a faster curve with no problems for wire compatibility and there’s now a draft for a spec for this.

Because you’re using an Edwards curve internally, you can implement the elliptic curve operations using parallel formulas. The fastest formulas for doing elliptic curve operations were published in 2008. The paper that describes them (Hisil, Wong, Carter, Dawson 2008) mentions a scenario where you have 4 processors working in parallel which is impractical to implement. If you look more closely at the formulas, the expensive steps of the formula are all uniform so you can do a symbian implementation. You could use IFMA or AVX2.

The IFMA implementation is just barely slower than FourQ even using the patented homomorphism speedup. We also show libsecp with and without endomorphisms. Depending on how you compare, you get a prime order group that is up to 4x faster, without using assembly it’s all written in rust. So that’s nice.

# Merlin transcripts

There’s this thing called the Fiat-Shamir hueristic. The idea is that if you have an argument where there’s a prover and a verifier and the verifier is sending random challenges to the prover, then as long as you believe in random oracles, you can replace a verifier’s random challenge with the hash of the prover’s message. But there’s weird complications if you might want to do this in practice. You might forget to feed data into the hash, what if your data is ambuguously encoded in the hash? How are you structuring the data put into the hash? If you have a multi-round protocol, you need a challenge input and put that into the hash of the next round and stuff. If you want to have domain separators so different applications have proofs that aren’t interchangeable, so nobody can take a proof from one part and stick it into another part… A lot of edge cases.

What if there was a first-class transcript objcet? In the paper when it says the prover sends something to the verifier, you could just have transcript commit something commit L commit R or whatever, and the verifier can just do a challenge scalar so the implementation can match the paper.

Merlin is built on STROBE. It allows you to implement your ptocool as if it iwas interactive, passing a transcript parameter. The fiat-shamir transform is done in software, not by hand. It does automatic message framing so you don’t have any problems with amibiguous encodings. You can do domain separation because in the transcripts you have to give a label, and commit arbitrarily structured data into the transcript. Also you can do automatic sequential composition of proofs.

There’s one more thing that this lets you go.

# Case study: bad entropy for Schnorr signatures

In Schnorr signatures, you have a signer (like a prover) and at some point they have to generate a nonce which is used as a blinding factor. If they leak the blinding factor then that’s bad because their secret key is recoverable. Sony had this problem with the Playstation keys. Also, leaking just a few bits of blinding over many signatures is fatal. This class of attacks presumably also applies to more complex zero-knowledge proofs.

You can get this defense by using a transcript. You can provide a transcript RNG, constructed by cloning the transcript state (binds to the public data), rekeying with witness data (binds to prover’s secrets), and rekeying with external entropy (non-deterministic). So you have at least as much entropy as your secrets. Rekeying with external entropy prevents problems with deterministic signatures. This is like a synthetic nonce.

When we put all these pieces together, our implementation is pretty good and fast. Using IFMA, we are 3x faster than libsecp256k1, 7x faster than Monero. Using AVX2, the speedup is 2x faster than libsecp256k1, and 4.6x faster than Monero. This was with SIMD backends in curve25519-dalek.

# Building on bulletproofs

After we have those building blocks, we could make bulletproofs.

# Constraint system proofs

The first thing we did is we made a constraint system proofs over the paper. I’ll tell you what it is, and show you how we built a constraint system proof.

A constraint system is a collection of two different kinds of constraints, such as multiplicative constraints where one secret times another secret is a third secret. We also have linear constraints (a linear combination of secrets with cleartext weights) that you set to zero.

With these two kinds of constraints, we can represent any efficiently verifiable program. We could do smart contract validation too. A constraint system proof is a proof that all the constraints are satisfied by certain constraint system inputs.

So we implemented constraint systems, and then we allowed reuse of challenges. Bulletproofs have this nice property where you can make constraint systems with no setups and you can build a constraint system on the fly. THis allows you to get and use random challenge scalars from commitments to variables. This uses the transcript protocol that Henry talked about earlier, where you make commitments and put them into the transcript and get challenge scalars back from this.

This allows us to make smaller and more efficient constraint systems. I’ll show a compact one. This is currently under research.

# Shuffle gadget

A gadget is a term we use for a collection of constraints. You can build multiple gadgets together into one constraint system and make one proof over it. An example is a shuffle gadget where you want to prove that the outputs of the shuffle gadget are a valid output of the inputs. We could do this many ways, but we use a random challenge scalar and then use equality of polynomials when roots are permuted. If the equation holds for random x then the inputs must equal the others in any order. What we want to do is to represent this constraint using our constraint system API. I’ll walk you through how we do that using the API we put in place (two_shuffle).

Full sample code: https://github.com/interstellar/spacesuit/blob/2-shuffle/src/gadgets/two_shuffle.rs

# Cloak: Confidential assets with bulletproofs

Shuffle gadget isn’t that useful on its own. But if we use multiple other gadgets, we can make a confidential assets protocol using this constraints system. We use shuffle, merge, split and range to make confidential assets protocol. Merge proves that the outputs are either a merge or a move of the inputs into two values. The range is a check that the value is not negative, which is similar to the rangeproof in the bulletproofs paper but constructed using the constraint system API. Shuffle does secret reordering of values.

Both the quantity and the asset type of all the inputs and outputs are kept secret. To an external observer, you can’t tell what type or how many input amounts are, but you can verify that it is a valid transaction and that the inputs and outputs add up. As a prover, you have to make individual assets move between the shuffle, merge and splits and rangeproofs. I’ll walk through what the prover generates to show you exactly how this transaction works.

Suppose you’re a prover, and you want to make a proof that your inputs and outputs align. First you want to prove there’s a random shuffle where you reorder the input values by asset types. So you group the dollars together here. Then you want to merge all the assets of the same type together. This allows you to take the sum over each asset type. Then you shuffle again to move the non-zero items to the top. Then you split the amounts into the target amounts for the outputs. In the last shuffle, we prove that the output ordering is a valid permutation of the input ordering to that shuffle. It gets shuffled into the output ordering that we expect. Then the rangeproof shows that none of these outputs are negative, which is bad where you can create money out of nothing. This is the walkthrough that the prover does, and then you make a proof that all of these inputs and outputs are valid for the inputs to the gadget. You make that proof, and you give it to a verifier, and it just sees a 3-3 cloaked transaction which is indistinguishable from any other transaction. You can’t tell what happened, but you can tell it is a valid transaction.

https://github.com/interstellar/spacesuit

The majority of the cost of running the cloak protocol is the rangecheck, and it requires vastly more multiplications than shuffle, merges or splits. This is not much more expensive than a typical confidential transaction proof for which you’d have to do a rangecheck for anyway.

# zkVM: Zero-knowledge smart contracts

It would be great if we could do more than just confidential assets. We’re working on zkVM, a zero-knowledge smart contract language. Last year I presented on TxVM, which aims to have deterministic results, a safe execution environment. The zero-knowledge virtual machine zkVM takes a lot of concepts from TxVM but then adds confidentiality to it, which I think is pretty cool. We take concepts from TxVM like having values and contracts that are first-class types that have a law of conservation where you can’t create or destroy value without satisfying strict checks. Also there’s a deterministic transaction log so you can reason about what effects your transaction will have. Also, we have the ability to do encrypted values and contract parameters with bulletproofs, and contracts built with custom constraints, and we can protect asset flows with the cloak protocol.

https://github.com/interstellar/zkvm

# Aggregated proofs with bulletproofs using session types

An aggregated proof is smaller and faster to verify than the underlying proofs that it aggregates. The difficulty is that this requires a multi-party computation protocol with multiple parties and a dealer in order to create an aggregated proof. This can be complicated because when we implemented the aggregator ptotocol, here’s the chart of all the messages being passed around and it’s hard to track state changes and it’s important to do it in order and only once otherwise you might leak secrets.

So we use session types where we encode protocol states into the rust type system, and you can make functions that consume the previous state for each party and dealer state and then output the next state. This actually statically ensures correct state transitions and you can eliminate multiple-evaluation attacks that leak secrets.

# Q&A

Q: Is this all open source?

A: Yes. The pieces are meant to be reusable.

Q: What’s your plan for applying this in the world?

A: We’re still talking about that. The confidential assets protocol is working and in its own library. We encourage people to use that. It’s ready.

Q: If anyone has a zero-knowledge statement they want to prove, they could use this system?

A: We designed a language similar to TxVM. We designed it to work with transaction value flows. It’s not with the goal for doing any potential proof of any statement ever, although you can do that with constraints. But this is designed for proofs over values and it restricts how you can create or destroy value. We prioritized safety and ability to reason about creation and destruction of value, over language flexibility. So this is designed specifically for financial transaction.

Q: Who are the players in multi-party computation protocols?

A: For the smart contract use case, I think probably the idea would be that since the proofs- you get log compression on the proof size. If you have a bunch of people who all want to have a proof of their contract execution included into some block or something, you could imagine having an untrusted dealer service where people can just register like if you want to submit a transaction instead of submitting to the chain you would phone up the aggregation service and it would group you together with other people who wanted to make transactions at the same time. Depending on the rate of requests, it could adjust the aggregation size and you could have pre-compression or filtering before you commit onchain data.

Q: If I have a constraint system, and I built something for SNARKs or something, could I use that then? Is it plug and play? What if I wanted to change my SNARK constraints?

A: There’s no ability to plug in play a SNARK into a bulletproof constraint system. But we’re talking with folks about a way to do a potentially shared common layer. Some SNARK use an arm-to-cs conversion, or some higher level language. We’d like to be interoperable with SNARKs in the longer run.

Q: For aggregation for multi-party computation protocol, the dealer is assumed to be centralized?

A: Yeah, there’s actually multiple ways you can do proof aggregation. In Benedikt’s paper, he outlined multiple ways. The most straightforward we thought was to have a centralized dealer. You don’t have to trust the dealer to not leak your secrets, because the dealer only gets your commitments and the only thing you have to trust the dealer to do is to not go offline.

https://github.com/dalek-cryptography/bulletproofs/issues/198