https://twitter.com/kanzure/status/1230561492254089224

paper: https://eprint.iacr.org/2019/1229.pdf

Our next speaker is Ben Fisch, part of the Stanford Applied Crypto team and he has done a lot of work that is relevant to the blockchain space. Proofs of replication, proofs of space, he is also one of the coauthors of a paper that defined the notion of verifiable delay functions. He has also worked on accumulators, batching techniques, vector commitments, and one of his latest works is exciting which is DARKs which he is going to talk about today.

Today I will be talking about our latest work on transparent SNARKs. This is work we put out around in the fall and we have made some improvements that I will mention throughout the talk but the bulk of the talk will focus on what we published in the Fall.

The concrete result is a new SNARK with no trusted setup. The SNARK is notable for having relatively practical proof sizes that are under 10 kilobytes for 1 million gates. We have other types of SNARKs that are based on FRI and you previously saw SNARKs with no trusted setup and the proofs were over 100 kilobytes. So the cool thing about this new SNARK is small proof size, and verification is under 100 ms, and the size of the proof and the verification time is logarithmic in the size or degree of the circuit being proved, with some other constant factors that also depend on the security parameter.

The main new tool that we're using here is a polynomial commitment scheme using groups of unknown order. This new tool is then plugged into other machinery that was developed by many other people over the course of years, and we applied that to get a new type of SNARK.

The bulk of my talk will not talk about the other machinery that we plug the polynomial commitment into, but rather the new polynomial commitment scheme itself. A polynomial commitment scheme allows one party to commit to a polynomial like over the field Fp of degree at most d with a small value c where c is going to be ideally constant-sized. Then later, be able to prove that the evaluation of this committed polynomial on some point z is equa l to some target point y. The idea would be to give a small proof ideally constant sized, although something polylogarithmic would also indeed be pretty good.

A polynomial commitment starts with the syntax of a normal commitment scheme where you have a setup procedure with some public parameters, a commitment to the polynomial which produces a small value, and then an opening ceremony that could open the entire polynomial. The additional thing it has is the eval protocol, which could be described as an interactive protocol called a public coin interactive protocol. Once you describe a pblic coin interactive protocol, then as a heuristic you can make it non-interactive by applying the fiat-shamir heuristic.

We want the commitments to be much smaller than the size of the original polynomial, ideally proportional to the security parameter and not to the size of the polynomial. The communication in the interactive eval protocol, we'd like to be sublinear in the size of the polynomial. If you compile that into a non-interactive proof, then that should turn into a proof size which is sublinear in the size of the original polynomial.

When we say sublinear, we would ideally like to build something polylogarithmic or even proportional to the security parameter, and in this work we get something that is logarithmic in the degree.

As for security, you want the commitment part to satisfy the standard part of commit and binding. You should not be able to open a commitment to two different polynomials. And you want to satisfy the commitment binding, meaning evaluation binding: I can't convince you that it evaluates to two different target points, there's only a single unique target point that this polynomial evaluates to.

A stronger property would be to say that if I'm able to run this eval protocol and convince you about the evaluation of the polynomial at any given point of your choice, then I must know any point. It's an argument of knowledge that I can't succeed unless I know the whole polynomial itself.

We would also like to have some hiding properties so we can plug it into a zero-knowledge scheme. The interactive protocol should be an interactive..... meaning you don't learn anything else about the polynomial.

What is a transparent setup? Well, remember there's this setup procedure at the beginning which generates some public parameters for the scheme. If that setup procedure requires some secrets that must be discarded after the process, then it's called a trusted setup which will often be implemented using multi-party computation to distribute that trust across multiple parties. It's the same concept from SNARKs where there's some SNARKs that have trusted setup with a ceremony to establish parameters at the beginning, like generating toxic waste- or secrets that must be discarded or destroyed. It's the same thing here. What we will see is that if we have a polynomial commitment scheme that doesn't require trusted setup, then we can build SNARKs that don't require trusted setup.

The most performant construction of polynomial commitment schemes previously required a trusted setup. It requires the parties doing the multi-part computation to compute powers of this group element G where this powers of s are secret and must be destroyed after the procedure. If anyone figures out what s is later, then the security of the protocol is broken. Our main challenge is to build a polynomial commitment scheme that doesn't require any trusted setup, but still has decent performance.

So we have transparent setup polynomial commitment and we get an evaluation argument that as I mentioned before scales logarithmically in the size of the polynomial, so the degree of the polynomial. There's also a generatlization to a multi-variant that I won't talk about. We apply this to build a transparent setup SNARK for arithmetic circuits where the size of the SNARK grows logarithmically in the size of the circuit.

Just to start with a rough comparison, what we end up with in the end with this transparent SNARK and how it compares in terms of its asymptotics to other SNARKs out there.... we have the most performant SNARKs which are not transparent and they don't even have something called universal setup like Groth16 which requires a new setup for each computation or circuit being proven, but those are the most performant. Then you have much more recently you have things like Supersonic, PLONK, BP (BBB+ 18), STARK.... PLONK has a universal setup and it doesn't have a transparent setup, but it performs competitively with Groth16. The verification time is basically just one pairing. It's actually, this is a little inaccurate, it's a couple pairings.... then you have STARKs which are also transparent setup, and you have many other things in the class of STARK like Fractal, but basically you have a class of FRI-based proof systems and they are not only transparent but quantum-secure but you have asymptotically much larger proof sizes that are scaling like log squared of the size of the computation and that's where you end up getting these... around 1 million gates you get around 100-200 in size of proof, versus 10 kilobytes in size in what we're calling Supersonic which sits in the middle in the sense that it's transparen,t it has a proof size that scales logarithmically in the degree, verification time is a logarithmic number of exponentiations but across schemes the exponentiations don't compare because they are in different groups. In Supersonic we're doing, it in class groups etc... Or in bulletproofs it's over prime groups. Bulletproofs ... is good on proof size, but it is linear in the circuit in terms of verification time, so very large circuits like 1 million gates it wouldn't scale at all.

Let's go through the main construction. We start with an integer encoding of this polynomial over this field Fp. We represent a polynomial over Fp as an integer polynomial simply by mapping the coefficients to representatives in the range 0 to p. Then we choose another integer q which is greater than p, and q must be substantilaly greater than p actually.... we output the encoding which is this ... of q which is going to be F hat of ... the encoding of the integer encoding of this polynomial will simply be this number 4213 for 4x^3 + 2x^2 + X + 3. We haven't done anything to make this succinct or a commitment scheme. It's still the same size polynomial f. We mapped it to integers, though, and now we're going to do integer commitments.

Let me first convince you that this is a valid encoding, meaning you can get back to the original polynomial from the encoding. The first fact to observe is that every integer in this range is uniquely decodable to a polynomial with positive coefficient, due to base decomposition of the number you got. You will be able to get the original coefficients back. We need a slightly fancier fact actually, because as a detail that comes out of the construction we're actually going to encode them as integers that could also be negative... but then it turns out they are decodable to polynomials with absolute value bounded by q/2.

Notice that this encoding has some homomorphic properties. If you add the encoding of f and the encoding of g, then you get the encoding of f + g as long as q is sufficiently large compared to the coefficients of the original polynomial. You are trying to prevent overflow. You need the coefficients of f + g to still be smaller than q, and we have to choose q to be substantially larger than the coefficients because we want the coefficients of the added polynomials to still be smaller than this q that we're going to be using.

We also have this property called monomial homomorphism.

Then we're going to use this key object called a group of unknown order, and a group of unknown order gives us the ability to commit to integers and the commitment is not only succinct it's just one group element but it's also homomorphic. So if I have the commitment g^x for an integer x, then g^x times g^y is g^(x+y). So we can get our integer encoding of the polynomial, and then just commit to this. This will inherent the same homomorphic property if I have this integer commitment and an integer commitment to a different polynomial, then I can multiply the commitments to get a commitment to both of them.

There are several different types of groups of unknown order. The most certain one is the RSA group. This is based on the RSA assumption. If you take N to be a large number, which is the multiple of two secret large primes p and q, then you get a group of integers co-prime with the order of the group which form a group under mod n. The only problem with this group of unknown order is that it requires someone to choose the secret p and q so it's not a trustless setup group of unknown order. If we were to use this for our scheme, then we would not have trustless setup, we would have something called universal setup. But the RSA groups are based on the most standard assumptions.

Class groups are believed to give us groups of unknown order. We believe it's hard to compute the order of the group, and it's hard to take odd prime roots in class groups. The really nice thing is that you need to specify the discriminant of the group and then you can start doing operations in the group. Nobody has to produce a trusted setup of secret values discarded later. For 1600 bits, we believe we can get 128 bit security, about equivalent to 3048-bit RSA and the discriminant size will also tell you the size of the group element rperesentation itself.

This appeared on IACR eprint the other day. It's a Jacobian group of genus 3 hyperelliptic curve, which has a group element size of 303 bits for 120-bit security which is the conjecture in the new work. This will require further study, but this would be extraordinarily exciting for our work that I'm about to show you.

I have shown you how to commit to polynomials, but how do you do the evaluation protocol? The intuition is that I am going to describe a recursive protocol where at every step I split the polynomial into a left part and a right part. I start a polynomial of degree d, I split it into two halves each with degree d, but if I split it into two halves with degree d each doesn't help. So instead I am going to tell you these are polynomials of degree d/2 where one has the left half and Fr has the right half of the coefficients. F(x) is Fl(x) + ... times Fr(x)... you can factor out the terms from the right part... So now I have two polynomials Fr and Fl which are degree over 2... The verifier is going to send a random challenge to the prover, and then the prover is going to -both the prover and the verifier are basically going to compute Fl(x) plus alpha times Fr(x) which gets replaced with F'(x)... and this thing is degree d/2 and then we recurse.

This protocol is not succinct since you're sending polynomials in the clear, but this was just to give you the information theoretic intuition about what's happening. But really we leverage the homomorphic properties over the commitments. So the prover sends a commitment to the left part and the right part, and then the verifier can use the homomorphism of the commitment to compute the linear combination of the two polynomials in the exponent and derive the values.

In order to convince you that it evaluates to a certain value, we also have to send some extra terms. Basically the prover is going to send the claim of the evaluation point y equals F(z) modulo p, and then the prover will send the evaluation of the left polynomial on this point z as well as the right polynomial and the verifier will be able to check on its own that y is consistent with an equivalency statement... if and only if the original polynomial evaluates to y at the point c.

There's only one part I haven't explained yet: how does the verifier check the consistency of the left commitment and the right commitment, with the original commitment? How does it check it's equal to the original c? Also note that this is leveraging the monomial homomorphism which is that we're able to compute the commitment to a polynomial ... by using... in the exponent.

This last part is going to use a proof-of-exponentiation trick, the same trick used for verifiable delay functions. The way we described it in the DARK paper was iwe were using Wesolowski 2018 exponentiation. This is a protocol to convince you that they are equal to some value... It would be too expensive for the verifier to run this exponentiation on their own, since it's size limited in the degree of the polynomial and we want a scheme that has logarithmic verification time. Without going into the details because I want to get to other materials, proof-of-exponentiation is a neat interactive protocol where the verifier is efficient and lets you do a proof of this form.

Every time we recurse, we multiply one of these polynomials by this value alpha which is chosen in the range 0 to p. The coefficients of the polynomials get larger and larger with each recursion. We have to set an encoding point q to be greater than q to the..... after levels of recursion, we need the homomorphic operations to still work.

When we get to the last step of recursion, we ned up with a degree 0 polynomial- just an integer. This gets sent to the verifier and the verifier will check if this is a consistent integer commitment, and also check a bound on the size- checking that F0 is less than p to the log d... will also imply that all of the polynomials had coefficients within the correct bound. This isn't a security proof, but there is a formal proof. The intuition is that this check at the end implies the polynomials above were encoded correctly, and therefore that they were binding commitments.

In the end, the proof size from all the recursion is going to be 2 log d field elements and 2 log d group elements. The verification time is around 2 log d, exponentiation is in G, and the prover time is like O(p d log d) and the verification time is 2 log d 256-bit exponentiation in G.

The security theorem is that the DARK evlauation protocol is an argument of knowledge based on low order assumption, strong RSA assumption, and adaptive root assumption all pertaining to groups of unknown order.

Here's a rough asymptotic comparison to other commitment schemes. I want to mention the work we've done since the fall when we released our DARK paper. We have a new method, DARKER. The highlights is that there are both security and performance improvements. The security is only based on the RSA assumption and not the other assumptions. Eval prover time is like square root of D with some other polylog terms which is a substantial improvement over getting much more practical prover times. Proving a commitment is still linear in the degree, but this means that eval time is going to be an insignificant part of the computation when we plug it into a SNARK. It is also of independent interest that we get an asymptotic square root of the .... The proof size increases to 3 log d instead of 2 log d.

To give concrete sizes, if you were using 3048-bit RSA then this is 23.8 KB. Using 1600-bit class group, you get 12.9 KB, and with 303-bit Jacobian you would get 3.2 KB but that's a really new proof of unknown order of course. We're excited to see whether this new group of unknown order will hold up.

The prototype was done by Findora- Phillipe Camacho, Fernando Krell. If we were to extrapolate the prover time for the evaluation protocol, using the new improvements that we have made since the fall, then we're looking at under 70 second time on evaluation for a polynomial degree going up to 1 million.

Basically, the modern paradigm is to construct a constraint system based on polynomial testing to build these information theoretic protocols that involve polynomials and then to replace evaluating polynomials on points, and to replace those polynomial evaluations by polynomial commitments and polynomial commitment evaluation protocols. Send oracles to the verifiers, and then the verifier is querying points- this gets replaced with sending polynomial commitments to the verifier and then running these evaluation protocols, the interactive evaluation protocols would be compiled with Fiat-Shamir to get non-interactive evaluation protocols, which then leads us to a SNARK. There's also some interaction as well in the polynomial oracle proof part, which gets included in the Fiat-Shamir compilation.

Here's a variation theorem on PLONK.... there's a three-round interactive oracle proof for any NP relation R with arithmetic circuit complexity n.... We can compile PIOP with DARK, we replace the oracles with polynomial commitments and the evaluations with the eval protocol... an important optimization is that we can open a commitment at multiple points by sending no field elements. For the DARK protocol, this is based on the homomorphic properties of the DARK protocol, leveraging the monomial homomorphism... the cost of opening at k different points is just one extra commitment and one eval on a 0-polynomial. This technique actually generalizes and this is explored in the Justin Drake and myself paper on IACR eprint. The optimization is only one eval required.

If we look at DARK PLONK, we get a proof size that has 5 field elements, 4 commitments, and 1 eval of degree 3n. With 1600-bit class groups, that's 10 kilobytes, with 1200-bit class groups that's 7.7 KB, and with 303-bit Jacobian this is going to be under 3 kilobytes. The verifier time is in milliseconds, and the prover time is relatively expensive. If you move to DARKER, you get slighlty larger proofs still for class groups it's around 14 KB and for the Jacobi it would be only 3.2 kilobytes. Based on our estimates, although take this with a grain of salt because we need to really run this more carefully, the prover time will be about 700 seconds, with 1 microsecond class group operations on the hardware.

It's really hard to do benchmarks between these systems and I don't want to give wrong numbers but here's a general picture of where things fit in. In terms of proof sizes, as I was saying over and over again, the universal setup and trusted setup SNARKs give you really small proof sizes like 3-4 kilobytes for universal setup and 0.2 kilobytes for Groth16. Bulletproofs are very small as well, but they have poor verifier time. If you look at the other end of the spectrum, then you're in the range of a few hundred kilobytes for STARKs. Jacobi DARK if it works will be exciting because it will be down in the range of universal setup SNARKs.

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Tweet: Transcript: "Transparent SNARKs from DARK compilers" https://diyhpl.us/wiki/transcripts/stanford-blockchain-conference/2020/transparent-snarks-from-dark-compilers/ @benafisch @CBRStanford #SBC20