Proof-of-Stake Longest Chain Protocols Revisited
Speakers: David Tse
Date: February 20, 2020
Transcript By: Bryan Bishop
Thanks. I am the last talk of the session so I better make it interesting, and if not interesting then short at least. I am going to talk about proof-of-stake. This is collaboration with a few people. Vivek Bagaria, … et al.
The starting point of this project was to come up with a proof-of-stake version of prism, which we just saw in the last talk. Prism has good latency and very high throughput and it also inherits the simplicity of the longest-chain protocol because its security is maintained by the longest chain protocol on each of the 1000 voting chains. The only potential drawback for some people is that it’s proof-of-work and not proof-of-stake. So can we convert it from proof-of-work to proof-of-stake?
The central problem is that because of the modular structure of the Prism protocol, the task is to convert each of the voting chains from the longest-chain protocol from proof-of-work to proof-of-stake. So the problem reduces to solving the problem of a longest-chain protocol using proof-of-stake instead of proof-of-work.
Proof-of-stake longest chain protocol
The problem with designing a proof-of-stake longest chain protocol is not a new problem. It has been worked on by several different groups of people including from Cornell, like Snow White, and from the Cardano project we have a sequence of Ouroboros papers, which all deal with the same problem of proof-of-stake longest chain protocols. There’s a lot of papers.
You might ask, so, since we already have these papers, why don’t we just take one of those systems and replace the proof-of-work longest chain in Prism by this new thing? Then we’re done and I can step off the stage and go to a coffee break, right? Maybe the problem is solved.
Is the problem solved? The problem is how to design a proof-of-stake longest chain protocol. I claim today that the problem is not solved, there’s still work to be done and I’ll show some results.
I’ll just focus on one of the papers, OUroboros Praos. Let’s understand how it works and what the drawbacks are of this protocol. Starting with genesis, in a proof-of-stake protocol a very important thing is to keep a source of randomness around which is used to select participants that can propose blocks into the blockchain. This randomness is very important.
You start a genesis block with some initial randomness. Ouroboros Praos divides time into epochs. The epoch is a long time. Within this epoch, this same randomness is used to pick all the participants. So in other words, the set of participants is fixed already in the very beginning of the epoch because all of this is driven by a single source of randomness or 0. These participants are allowed to build the blockchain along this epoch.
When an epoch is over, you move to another epoch and another source of randomness is generated. This r1 value is generated as a function of the blocks in the earlier epoch which had already reached consensus. r1 is then consensus among nodes, and it’s used to select another set of participants for the next epoch and they can continue to build the blockchain. You repeat this process, and you go on forever.
But here’s a question: how long is this epoch? Anyone? It’s 5 days. Okay. This epoch is 5 days long. Okay. In the current implementation of Ouroboros project, in Cardano project, it’s 5 days long. So what happens with such a long epoch?
The attack is a bribery attack. Here’s how it works. In this epoch, all the participants that are selected are already known. They already know that they are going to participate in building the blockchain during this epoch. To an adversary, they will know- they will post on a public website and says whoever wants to take a bribe can come and join me. Some of the individuals can take the bribe. What are they supposed to do by getitng the bribe is they are supposed to participate in a double-spend 51% attack.
The participants that didn’t take the bribe will continue building the blockchain as normal. However, some of the other participants do not build the blockchain until a transaction is confirmed. Once it is confirmed, the bribed participants are now going to build an alternative chain and this is a double spend, removing that original transaction from history. These participants are now working together to make a double spending attack.
You only need to bribe k+1 individuals. That’s all you need to build the longest chain. So that’s pretty serious because there’s actually many participants over a 5 day period and just bribing k+1 of them is enough.
There’s no explicit double spending because they haven’t done anything; they only sign once. VRF doesn’t help. VRF says that the adversary can’t not know who are the participants. However, the participants themselves know that they can participate so they can go and take a bribe. VRF doesn’t help to solve this.
So this is pretty serious. The whole issue is because this epoch window is so long. This is really a prediction window because it is allowing the participants themselves to know 5 days in advance that hey I can participate and therefore I have plenty of time to go and take this bribe. So this is really bad.
The natural solution is to shorten this prediction window, in other words shorten the epoch time to very few blocks so that this attack cannot be easily conducted. What’s the problem with this? When the epoch size is small, then there’s not enough time to get consensus on the randomness. The ourboros protocol design philosophy is about reaching consensus first on randomness to drive the next epoch. So this breaks the whole design and the analysis for Ourboros breaks down.
So here’s the situation… on the x-axis we have the prediction window, and on the y-axis we have the security threshold which is how much adversary power is needed to attack the system. What ouroboros shows, what the paper showed, is that as a strong secure guarantee against a 50% adaptive adversary… so very strong security guarantee, however, because this prediction window is 5 days is so long, it’s subject to this bribery attack which is outside their model. So that’s the problem.
Another proof-of-stake protocol
A natural question now is whether one can design another proof-of-stake protocol with a much shorter window, hopefully, and keep the security threshold as close to 50% as possible, and at the same time we can provide formal guarantee just as rigorous as Ouroboros? The main result of this talk is that yes we can achieve it, and this is the curve that is achieved by the protocol we designed.
The point is that this protocol is proven-secure with these thresholds, even without consensus on randomness. No consensus on randomness, and still proven rigorously secure. The performance is very close to 50% except when c is very small. But even where c=1, you can get a security threshold of about 27%.
Let me explain what happens in the c=1 case. The innovation here is realy that it’s a new approach to doing security analysis. I’ll briefly talk about the whole curve.
Let’s look at the protocol. Instead of an epoch window of 5 days, let’s go to something with every 20 second when a new block is generated, we update the randomness. This is a totally different extreme, 5 days for Ourboros and 20 seconds for this protocol. Every time a block picks a leader, a proposer and a participant, based on the randomness r0, and then the randomness gets abated as a function of the key of this individual and the original randomness r0. You get r1, etc. etc. So participants are picked one at a time, no longer known in advance. Every 20 seconds. Every 20 seconds. One block every 20 seconds. Alright. Okay. There’s no consensus on randomness.
What’s the problem here? The protocol is “mine on the longest chain always”. This is a longest-chain protocol. No consensus on the randomness. But we have a problem. This is the reason why Ourboros doesn’t want to go there, but we’re going there.
Okay, so, what is the problem? The problem is that we have independent randomness for each of the different blocks. Okay? And that gives a lot of opportunity for the adversary to try many different blocks, to try to find an advantage over the honest guy who just keeps growing on the longest chain. This is a version of the “nothing at stake” attack.
Private attack analysis
“Nothing at stake” is the biggest problem of this protocol. Several parties have used this protocol already, it’s not new. A paper that was presented a year ago at SBC 2019 analyzed this protocol for c=1 where randomness is updated at every block. Unfortunately, they did not give a formal security analysis. What they did is a private attack analysis, in other words they looked at one particular attack, and they analyzed the security protocol. In a private attack, you give the adversary- the adversary mines- the honest guy mines on the longest chain, the adversary mines on every block which means that it can grow a tree because it can mine every block and since there’s so many blocks the opportunity for you to increase, amplify the growth of this tree beyond stake of the adversary. What you get is that the growth rate of the honest…. the total growth rate… is he adversary’s stake fraction, okay, and lambda times 1 - beta is therefore the honest growth rate. This is the honest growth rate. Blue. Blue is honest growth rate. Now the adversary growth rate should be lambda beta right because beta is the adversary stake. But because of the “nothing at stake” tree, it has amplification factor of e and somehow e always shows up at appropriate times or inappropriate times. e times lambda beta. Okay, so you can do the math, this means that if beta is less than 1/(1+e) then this growth rate is slower than this one, and therefore the private attack has failed, which means that if you look at the threshold on beta, then, that means you have power staked more than that, then forget it. The system is insecure there. Now the leftover issue from the paper is what happens on the left hand side? Given less than 1/(1+e), what happens? What they showed is that the private attack would fail. But what about other attacks? Can they succeed?
To understand this issue about private attack vs other possible attack, let’s go back to the history of the literature. Let’s go back to proof-of-work, okay? Longest chain. Nakamoto proposed a protocol 12 years ago in two oh oh eight. In that paper, he showed that the Nakamoto proof-of-work longest chain is secure against the private attack. A specific attack. Six years later, a beautiful paper by … they show that actually it’s secure against all attacks. “The bitcoin backbone protocoL: Analysis and application”. It takes how much effort to do this? Let’s measure work by number of pages in the paper…. Nakamoto paper had 9 pages long. Everybody can read that. The backbone manuscript was 46 pages long, so it must be more complicated right? Indeed, it took us 46 days to read this damn paper. You might say, well these guys just wanted to publish at some good conference but nobody cares…. Nakamoto is okay, right?
GHOST and balance attacks
It turns out that these pages are necessary because if you try to modify a design and make a new protocol, and you think that hey like Nakamoto as long as I’m secure against the private attack then I’m okay…. like this protocol GHOST by these people where several … it turns out, that there’s an attack. This attack is a pretty famous work called a “balance attack”. It takes a different type of attack, where it builds two chains to balance it. It’s not secure against the balance attack. So therefore, the 46 pages are necessary. You need the 46 pages. It’s a lot of work and sweat, but you have to do the work and do the sweat.
Formal security analysis
Now we’re going to sweat it out… We’re not Greek, but we’re still going to do it. We’re going to do a formal security analysis on the proof-of-stake protcol we talked about earlier. “Nothing at stake” provides a difficulty that is not there in proof-of-work where the argument goes as follows: 46 pages but let me summarize in 10 seconds. Basically the paper says that where there’s a private attack or balance attack or whatever attack, the endpoint is that to be successful you need to build two chains, and to build the two chains the adversary needs to match the honest nodes block-by-block. If the honest chain has one-block, then the adversary needs one block. The number of adversary blocks is less than the number of nodes? Then the attack isn’t possible, which implies 50% security. So the security is obtained basically by 46 pages– it’s clever counting of the number of adversary blocks. This whole argument doesn’t work for proof-of-stake with nothing at stake (NaS). Look at the tree: there’s an exponential number of adversarial blocks over honest blocks. If you follow the argument, you will get a trivial result: nothing at stake has 0% security. Is it true?
Nothing at stake, and adversary-proof convergence
Is nothing at stake 0% security, or is it 1/(1+e) or somewhere in between? We resolve the problem by not using the Cardano framework but we invent a totally new approach which we call adversary-proof convergence. Here’s how it works.
The blue blocks are honest. Okay? They may not be on the chain, but these are all the honest blocks that have been generated so far. Let’s think about what are the adversarial blocks that can disrupt my security? Well, these are all the trees that can be grown from the previous …. So before we were looking at the private attack with only one tree, but now we have to deal with all kinds of attacks, we have to look at all the trees which looks scary. So the problem becomes a race between the blue blocks, and all the trees. A blue block is growing at lambda 1 beta, and the red guy is each growing at e lambda beta. If you look at the race between these guys and the trees that are far back, that’s easy because e lambda beta on this side is going to be less than this number, so they can’t catch up which is the essence of the private attack. But what about the other trees? A guy close by can definitely catch up. By randomness they can catch up. Most of the time, there will be some trees that can catch up to me. However, the point though is that as I run along, I will get at some point lucky by random times that all the trees behind me will not be able to catch up just by sheer waiting long enough this will happen. So that’s why we call it adversarial-proof convergence. At this point, the green block, everybody can’t catch up and at that point I reach consensus on the longest chain up to that point. Every now and then this will happen, and you will converge, and nobody can attack you, the longest chain gets frozen, and then you move forward. That’s how we show that we’re okay here.
This analysis shows the power of the longest-chain protocol because it’s able to win the race against all these trees.
Improving the security threshold
So what about improving the threshold? Humans are very greedy. Before, we were worried that “nothing at stake” would kill us and give us 0% security but then we got 27% with 1/(1+e). Now we can go to Hawaii and relax, right? No, we’re going to work harder. We don’t want to be less than Ouroboros 1/2 right, we’re here to win. You want to get to 1/2. How do we do that? That’s the question.
Actually, one idea fom Fan & Zhou was hey, you know, the adversary is growing everywhere… Hey, I can, I’m honest but I can grow everywhere as well. Let me grow at a few places and then call these protocols g-greedy or d-distance greedy, fighting nothing-at-stake with nothing-at-stake. Sounds prety cool, but it turns out to be no good because the analysis is based only on the private chain analysis, meaning comparative growth rate of two different trees. In that sense, it’s a good idea. But for the other attacks- we show that a balance attack, because of the confusion among yourself, will kill you.
Instead, our idea instead is to introduce correlation in the randomness. So instead of c=1, we introduce correlation and we allow some correlation of the randomness across levels and we can control the level with a parameter in our protocol. With zero correlation, everything is uncorrelated and you get amplification rate e. You do increasing correlation, then the chance of you having opportunity is reduced and the growth rate slows down etc.
You can see that compared to Ouroboros, only very small c will already get you to very close to 50%. This operating point allows us to get a very short prediction window, and at the same time very secure against a traditional adaptive adversary. The best of both worlds. Okay.
Let me say a few words. 30 seconds. What’s the story here? Just before my talk, we heard a talk on Prism and the goal of prism is to achieve vertical scaling and scaling the performance of blockchain up to physical limits of the nodes on the network. What I just talked about is proof-of-stake which is to achieve energy scaling and reduce the energy requirement of proof-of-work.
We have one more talk tomorrow about coded merkle trees which is a data availability method for allowing to solve horizontal scaling, that is using sharding to scale across many nodes.
We’re starting a startup called https://www.trifectachain.com/ to commercialize these efforts. Independently, I need to advertise for my students as well. There’s a paper tomorrow by Joachim Neu on a layer 2 solution called Boomerang which is completely separate from the company and the other projects, but still very interesting.
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