Tight proofs of space and and replication

Ben Fisch (Stanford University)

https://twitter.com/kanzure/status/1091064672831234048

paper: https://eprint.iacr.org/2018/702.pdf

# Introduction

This talk is about proofs of space and tight proofs of space.

# Proof of Space

A proof-of-space is an alternative to proof-of-work. Applications have been proposed like spam prevention, DoS attack prevention, sybil resistance in consensus networks which is most relevant to this conference. In proof-of-space, it's an interactive protocol between a miner and a prover who has some disk storage and claims that hey I am storing 1 gigabyte of data. The proof for this to be an interesting type of protocol should be very small compared to the size of the data.

We can make this protocol non-interactive using the fiat-shamir trick. Instead of having it an interactive protocol, the prover chooses the id and the commitment and then generates a challenge as the hash of the first message and then sends the proof which the verifier should be convinced of. At this particular moment in time, the prover should have been using at least that much storage in order to produce a successful proof.

# Proof of persistent space

You move the non-interactive protocol into an initialization phase where the prover initializes his storage, fills up his drive with a gigabyte of data and he says I'm storing a lot of data on my disk and then sends a commitment to that and then later during a repetitive online phase, receive challenges from the verifier and if the prover passes then it indicates the prover is still storing that same one gigabyte of data. The reason why we split it up like this is because the initialization phase is a relatively expensive operation whereas the online operations are cheap to check that the prover is still storing the same data.

If the prover deletes its data, then it will not be able to regenerate the same proof within the allotted time. Alternatively, you can think of it as the prover will have to (and some constructions do not have this sequential time property) the prover will have to expend a lot of computational work in order to pass the online challenge if it is not storing the data, which incurs a time-space tradeoff. We will talk later about what it means for a proof of space to be secure.

# How tight is a proof of space?

Tightness here refers to how much space can an adversarial prover save and still pass the protocol. Can the adversarial prover pass with only 1 - epsilon gigabytes of data? The answer should be no, epsilon should be as close to 1 as possible. Closer to 1 means tighter.

# Proof of space security

The strongest notion for security here is parallel security. The proof of space is epsilon tight if any online prover who stores less than 1 - epsilon gigabytes then will fail to respond within the time limit T, except with negligible probability. We need T to be large, proportional to the storage size. The itghter the proof is, that would mean 1 - epsilon is as close to 1 as possible.

A weaker notion is a time-space tradeoff. Imagine the prover could pass in the allotted time, while storing less than 1 - epsilon gigabytes of data. Well, the prover still has to do a large amount of work proportional to the storage size. An even weaker notion would be a time-space tradeoff where there isn't a threshold where a prover has to do a tremendous amount of work, but rather that there's some kind of equation that gives a time-space tradeoff where the requirements for storage might be if S if the total storage and T is the total work then S to the K times T would be greater than (1 - epsilon) times N^{k}. Can never be too "tight" as prover would use...

# Tight proof of space

Arbitrarily tight would mean that I can prove the protocol parameters so that it is an epsilon-tight proof of space. I can tune hte parameters of the protocol so that if I want to make sure even adversary doesn't have to save 1% of data or 0.1% of data, I can tune hte protocol parameters so that that is the case.

A tight proof of space construction would maintain efficiency even as epsilon is made arbitrarily small, and ideally proportional to 1/epsilon. If you make it smaller, you do have to increase communication or the proof size, and the best you can get is 1/epsilon.

This work achieves tightness of space that is the ideal tightness of space.

# Previous work

Before this work, what was the scope of proof-of-space? There were no provably tight proofs of space. The original proof of space was proposed in 2015 and it had 1 - epsilon of less than 1/512. Perhaps the protocol was more secure, but that was as much we could prove about the protocol. We could not prove that the adversary was storing more than 1/512th fraction of the data. It's only storing 1/512th gigabytes of the previous example.

Ren-Devadas 2016 moved this closer to 1. In their protocol, you could only prove that 1 - epsilon is less than 1/2 which is a theoretical upperbound. The protocol becomes impractical at 1/3rd. You cannot get practical parameters and then prove the adversary is storing more than 1/3rd of the data.

AACKPR 2017 introduced a construction that had weaker time-space property, but it was 1 - epsilon < 1 / log(N). It uses the weakest notion of proof of space which has the time-space tradeoff and theoretically cannot be a tight proof of space.

Pietrzak 2018 had something that was "quasi-tight", proof size increases proportional to log(n)/epislon^{2}. We don't have any practical instantiation of this construction, even heuristically. From a theoretical perspective, it is tight, it doesn't maintain efficiency proportional to 1/epsilon which is the ideal, but we don't have any practical instantiations of this construction because it requires a certain special type of depth robust graphs (DRGs) which we don't have practically or experimentally.

# Why do tight proofs of space matter?

Well, we get better provable security for proofs of space. If we are basing a blockchain on this, then we want to know the bounds are tight on how much space the adversaries are really using. But another reason and the motivation for my owrk was that a tight proof of space is necessary for another thing called proof of replication (PoRep).

# Proof of replication

Instead of storing a lot of data, let's say the prover or miner is encoding a specific movie on their disk. Say this is a movie that a client would be interested in. Then they go through the dance of the protocol. There's a challenge, there's a proof, the same as proof of space. The verifier then gets convinced somehow that the prover is storing not only a lot of data, but a specific piece of interesting data.

It turns out though that you can't really achieve this perfectly cryptographically. The best possible security you can achieve for this is some notion of "rational security", which I call epsilon rational security. The adversary cannot save more than an epsilon fraction by not storing the actual movie. So he might as well just store the actual data instead.

# epsilon-rational security

In connection to epsilon-tight proof of space, if something is epsilon-rational secure then you.... it means it's tight, for small epsilon. These notions are kind of equivalent. So in order to construct this protocol that is epsilon-rational secure, then you need an epsilon-tight proof space. They are equivalent notions and can be used to construct each other. They are equivalent from the perspective of security.

# Extraction

But there's an extra consideration for replication. This is interesting data that you might want to extract. The verifier might want the whole data. A protocol with efficient extraction is harder.

Efficient extraction means that the protocol is not asynchronously composable. If the prover is producing different proofs over time, for independent movies and claiming to use storage to store both of these movies, then if the verifier is not proving to the actual movie that it is storing, just verifying the space, then the prover can generate the encoding of the first file and then use the output of that as sort of the input to the next proof-of-space. By this property of efficient extraction, the prover only needs to store the data it's using for the second proof of replication and can efficiently extract from it the data it needs for the first proof of replication protocol. This is prevented by focusing on protocols where the prover commits to the total amount of storage or total amount of files it is going to store for a single period of time, and then the security would hold.

The way you could use this in a system if you want efficient data extraction, you make the prover commit to all the movies it wants to store at a period of time. And then for each epoch, make them reinitialize all the movies they want to store at any point in time, and then challenge them on this. You allow the prover to lazily add committed movies over time, then you can't have both security and efficient extraction which has serious practical implications.

# Construction description

Let's skip to how we actually construct these things. It's a basic technique using labeled directed acyclic graphs. In proofs of space, you store specially encoded data by labeling the nodes of a graph. Every label is like a 32-byte label. So say you have n labels overall for the total amount of storage of 32n bytes. So you are going to derive a special label on every node in the graph. How do we label the graph? It's going to be a directed acyclic graph and for the labeling we're going to use a collision-resistant hash function salted by the id for the proof. If you want to do this for two different proofs, then they would be completely independent.

# Graph labeling

Let me walk you through how labeling works in this simple example of 5 nodes. The first label is derived just as the hash of one, the number one. Then the next label will be derived as a hash of the input index 2 but also any labels that it depends on, meaning that labels that belong to nodes that are direct parents of this particular node. This continues so and so forth. This forces you to do sequential work to derive the labeling, which is an important property that leads to some security. If you delete labels, then you forcibly have to do a lot of sequential work to re-derive them.

So how do you use this labeling in the proof of space framework we described before? You send a commitment to the graph labels you derive, and you get a challenge set chosen, you open the labels of the nodes in the challenge set, and their parents, and the verifier checks that all the hash relationships between labels and nodes and their parents were correct in how they were derived.

# Depth robust graph

Another main tool we will use is called a depth robust graph. It's a graph, a directed acyclic graph on N nodes, but it has a special property that any node-- it's a graph where if you delete or look at a subset of 80% of the graph, so you delete only say 20% of the graph and keep 80%, and the remaining 80% of nodes contains a long path. What do I mean by long? I mean basically, proportional to N. At least (1 - delta) * N. It's going to contain a long path. If you have to recompute the labels along a long path, it requires a lot of sequential work.

The building block for the type of proof of space that we're going to develop in the next few slides is a proof-of-space based on the depth robust graph alone. This is joint work with Joe Beannu and I rpesented it last year at BPASE 2018 and we were using it for proof of replication. There, you basically just label a depth robust graph. Why does this give you a proof of space? If the prover deletes 80% of the labels, then it will have to do a-- since 80% of the labels are deleted contains a long path, it wil lhave to do a lot of sequential hashing ot re-derive those labels if challenged to produce those labels in an online challenge space. This isn't tight because it allows the prover to delete 80% of the data.

# Alpha beta bipartite expander graph

In this graph, there are two sets called the sources and the sinks. There are directed edges between nodes in these two sets. Any alpha fraction of A are connected to at least a beta fraction of B. We call this an expander graph when B is noticably larger than A. When you look at the total number of nodes that are connected, there's an expansion factor of beta over alpha.

# Construction

So we combine these two different graph tools together and we're going to stack depth robust graphs, such that at every layer of n nodes, you put the edges of a depth robust graph that satisfy this minimal condition of being depth robust if you look at an 80% subset of the nodes. Then you use bipartite expander edges to connect the nodes, and finally we apply the method I described earlier to label the graph and derive labels on the final graph and we only store the labels from the last level.

If I delete these nodes, I need labels on the previous level too, which in this example I have not actually stored. The dependencies keep expanding and blowing up with every level because we're using the expander graph. On the first level, where we have the edges of a depth robust graph, you basically have to re-derive basically all the nodes, and since that requires a lot of sequential work due to the depth robust property, you basically have a proof-of-space.

When we want to prove this has a proof-of-space, we can't assume adversaries are only storing labels on the last layer, we can allow it to store labels sprinkled throughout the layers. We can prove that the total storage the adversary is using is less than 1 - epsilon over n, then we can find a blow-up in the dependencies of the things he is missing.

From this, you can build a proof of replication by taking the labels on the last level and you use this as a one-time encryption pad, you XOR it with your movie file, and you get an encoded movie data file. Extraction in this very generic construction is inefficient, and it requires you to rederive the deterministic labels on the last layer to decode your movie.

So how could we build something with efficient data extraction, given the caveats about composability from before? We start with the same construction, we tweak it, and we absorb the expander edges into the layers so that between the layers there's just one edge between nodes. A node in a given layer, does not have a direct dependency on any nodes in the previous layer, it just has an edge from the node with the same index in the prior level. The edges that were there before were absorbed into the ... from the same level. Then we reverse the direction of the edges on every level, and I'll explain the intuition for why we are doing this. The dashed edges basically correspond or indicate where you would XOR the higher level label with a key derived from the parents on the next level. We're deriving an encoding of each label, so each label encodes the same index label on the previous level. The final level of nodes contains an encoding of the data inputs, and the data can be extracted efficiently by reversing this process. When we reverse this process, it can be completely parallelized so that it's much faster.

So why do we still have this property of a proof-of-space? Say we forget one of the labels. Consider both its targets in this level, and also its dependencies. If we move it to the level right before, the targets of this node sort of become the dependency of the node on hte previous level. In order to re-derive this label, we needed still to have a label c9 because c14 encodes c9. It's the XOR of c9. But in order to re-derive c9, we need c10 which was you know part of the target set of c14 in the last level. The targets become dependencies of the node that we need to re-derive on the previous level, and the dependencies then become targets. We keep on switching the dependencies and targets. Why? The way we chose the edges came from the expander graph construction, which has the expander property, which applies to the size of the target and the dependencies of any deleted set of labels inherits this expansion property. The whole construction is in the paper.