Accumulators for blockchains

Benedikt Bunz



I am going to be talking about batching techniques for accumulators with applications to interactive oracle proofs and blockchains. I'll maybe also talk about how to make these proofs even smaller. The main thing that we are going to focus on today is the bitcoin state and this application of accumulators to bitcoin and blockchain.

Bitcoin UTXO states

The bitcoin state is the UTXO set. It's the number of coins that are currently unspent. It turns out that this set is the more people use bitcoin and the more transactions happen with small dust, the larger and larger the dust grows. All of the nodes need to have the state so they can verify whether the coin has been spent or not. The solution to the stateless consensus problem is something we've heard a lot about in this conference, it's proofs. When you propose a state transition, you should provide a proof that the state transition is correct. All the validators and miners don't store the state, just have a small commitment to the state, and they can check the proof and verify the state transition is correct. The work has been shifted from miners and validators to the users who store and maintain these proofs. What do these proofs look like? How are they generated? Do they require state? Who generates them? But what this general paradigm does- where transactions prove their own correctness-- removes dependence in consensus, I can participate in consensus, and we want many nodes to participate in consensus and it removes this burden to store the entire UTXO state.


Let's look at a concrete example of the consensus system of mainly something like bitcoin and I'll talk about other systems later in the talk. Let's focus on bitcoin. In bitcoin, if I want to commit to the UTXO set, I can do that with something called an accumulator which is a commitment to a set. I have a set of UTXOs and then in every transaction I provide a proof that the UTXO I'm spending is in fact in the UTXO set so it's in fact unspent. And the set needs to be updated with the new coins that were created and remove the old ones which were spent.


In general, the accumulator has the following functionalities. You can add thing to the accumulator, you can remove things, and you can provide inclusion proofs and the miner or validator can just take the transaction and the inclusion proof and the accumulator and verify that these proofs are short. The important thing is that both the accumulator state and the inclusion proof are short. One concrete example is merkle trees, but I'll also talk about RSA accumulators which has the benefit that inclusion proofs are constant size.

UTXO commitments

The blockchain design would look like this... you have something that commits to the state. Merkle trees have nice properties. The merkle root would be stored in the block, and the inclusion proofs are logarithmic in size, and you can do nonmembership proofs if you sort the tree. The problem with merkle trees is that if we have all of these merkle tree witnesses which need to be communicated then this really increases the communication a lot. For 100 million UTXOs, this would increase communication to 100 gigabytes and these things need to be stored with the users. The miners don't need to store much, but we've increased communication by a lot.

Ideal properties

So what we would want is basically something where-- is something efficiently updateable, efficient verification, and maybe even aggragetable witnesses. Could we combine these proofs or verify multiple proofs faster than verifyuing one proof?

RSA accumulators

RSA accumulators have been around since the early 2000s. They could be used as a drop-in replacement for a merkle tree if the merkle tree was being used as an accumulator. These accumulators they are working in RSA groups. What we have to do is we have to choose an RSA modulus n which is the product of two primes and it's important that these two prime factors are thrown away. This is a trusted setup. The other thing is that I need a hash function that maps my elements to primes. Then I initialize the accumulator with a generator of the group.

How do I add something to the accumulator? It's simple. I use my current accumulator state, I map an element to a prime, and then I raise my current accumulator to the element and then I get my new accumulator state.

How do I do deletion? Deletion is, you take the element's root and do the inverse operation. It looks trivial, but we need a property where taking roots is hard.

What is the state of the accumulator after doing a bunch of additions? I will omit the hash function for the rest of the talk. If I have my set S of elements, then the state of the accumulator is just the product of these elements, the accumulator base element of the accumulator raised to these... It's commutative, it doesn't matter the order that they are added. Multiplication is commutative.

Accumulator proofs

I can make accumulator inclusion proofs. It's just the state of the accumulator without the element. I remove the element from the accumulator. I can compute this with a trapdoor or by reconstructing the accumulator. Verifying is just testing- you add the element again and test whether you get back to the original accumulator. If htat holds, then it worked. The inclusion proofs are constant sized. Merkle tree inclusion proofs grow logarithmically with the set size, like 20 hashes for a million size set and 30 hashes for a billion size set.

I can also exclusion proofs or nonmembership proofs. There's these integers a and b such that ax + bu is equal to 1. These integers always exist, if and only if x and u are co-prime. But if you think about it, this is why we map everything to a prime. If x is not in the set u, if x does not divide u, then they have to be co-prime because they are distinct primes and u is a product of other primes. So this is how you can do an exclusion proof and you can also update these proofs very efficiently without knowing what is in the set using LiLiXue07.

RSA accumulator state of the art

You have constant size inclusion proofs, it's better than merkle tree for set sizes beyond 4000. It has dynamic stateless adds, which can add elements without knowing the set, and it has decentralized storage. But we want aggregate/batch inclusion proofs, stateless deletes, faster batch verification, and it would be good to be trapdoor free.

Aggregate membership witnesses

Say I have two elements for two distinct elements. I can use Shamir's trick to combine these proofs. It turns out that I can combine these proofs to be the xy root of A. It's an inclusion proof of xy. But if you think about it, you would only be able to construct this proof only if both of the elements are in the set. This combined proof is still just one. I took two proofs and combined it into one, and can repeat it over and over again, and a miner could take all the inclusion proofs and all the membership proofs and crunch them together into one single aggregated membership witnesses. This is similar to the aggregation for BLS signatures where you can non-interactively combine all of the signatures. If you use RSA accumulators, it would just be about 3000 bits per block. There's even aggregation that you could do across blocks if you wanted.

RSA requires trusted setup?

I need a value N which has these unknown factors p and q. Also we want efficient delete which requires a trapdoor. You can find Ns in the wild ("RSA number assumption") where someone has promised to throw away the values.... But there's another proposal, using class groups of imaginary quadratic fields and these are just mathematical objects that basically function in a very similar way to these RSA groups and it's also hard to take roots in these groups and I don't know the so-called order of the group. If I want to achieve 128-bit security, then they are actually even a bit smaller. [BW88, L12]. They don't require trusted setup, I can generate them from a publicly known string.

Stateless deletion

How do I delete if I don't have the trapdoor? When do I want to delete? I want to delete a coin from the UTXO set from when it's being spent. Before it's being spent, I need to receive an inclusion proof. I only really need to delete if we have an inclusion proof. But the inclusion proof was the accumulator value without the element. So the inclusion proof has the accumulator with my element deleted. Deleting is simple if yo have an accumulator.

It gets more complex when I have two things that I want to delete from the same accumulator at the same time. Multiple transactions are spending from the same accumulator. It turns out the same trick to combine inclusion proofs can serve us again. We can combine all of the proofs together and this will be exactly the accumulator with all of the items removed. This operation we already wanted to do, to batch the inclusion proofs, helps us delete elements from the accumulator.

Stateless means I can delete without having any idea what else is in the accumulator other than the deleted elements. It doesn't matter if there's a billion other things in the accumulator. So no state is required.

Verification of witnesses is too slow?

These accumulators are sort of expensive. Maybe I can do about 600 operations/second on my computer. With class groups, I don't have the results but there's been a competition on implementing them efficiently for verifyable delay functions. But these things are slow. Many people have to check correctness too.

Can we somehow outsource the computation? There's this beautiful proof designed by Wesolowski 2018 for verifyable delay functions. It's a proof that x raised to a gigantic number is equal to y... and the whole point of this proof is just efficiency. It's kind of like, it's not about zero-knowledge. Here it is about achieving efficiency. How could I prove to you that x raised to a gigantic number, is equal to y?

Well it turns out you can run his protocol and then the verifier can verify htis protocol in only log(t) steps. It's very efficient. Computing it takes a million steps, but verification takes 20 steps. This verification is very nice, but it only works if the exponent is 2t a so-called smooth number.

We consider the problem, what do we have to do to check in these accumulators? It's our accumulator raised to some arbitrary number. It's not smooth, it doesn't look nice and beautfiul. It turns out the same protocol again still works.. however now the efficiency is log of alpha, so asymptotically it's not necessarily better but it turns out that in practice with this proof I can get a 5000x factor speedup.

PoE efficiency

There's a user, a miner who creates this accumulator and adds a bunch of things to the accumulator and I want to check that his work was done correctly. He will provide a proof of exponentiation. I will have to do only 1/5000th of the work that the miner will have to do on an expensive AWS instance and I can maybe run on a smartphone or something. This factor of 5000 matters.

Fast block verification

What does this whole system look like? I am stateless but still want performance. I have a bunch of updates coming in, and I use the BatchDelete property, and I add the transactions with the new coins that I created. So my block spends a bunch of coins, creates a bunch of coins, has some signatures, and then some updates to the accumulators and attached is a proof of exponentiation.

The verifier needs to check all the signatures. You could use SNARKs and recursive SNARKs to solve that problem I guess. We are more concerned about the check that the coins are unspent. Here the verifier only has to check the proof of exponentiation which is really fast.


Merkle trees are very fast and you can do 100,000 transactions per second. The most expensive part is mapping things to primes and doing primality checks. There might be some interesting optimizations like maybe batch primality checks or something... that still takes up most of the time but it takes like 20,000 transactions/second which could be improved with a better implementation.

Accounting systems

I have talked a lot about UTXOs and set commitments and accumulators as a commitment to a set. But that's only for simple things like bitcoin's UTXO system. A lot of systems out there are more like accounts, and each one stores information about accounts. The way I would use this is using something called a vector commitment. The index of the vector commitment... I prove to you that the value of the commitment is some other thing, it's a positional proof. The simplest vector commitment that you already know is a merkle proof. I can prove to you that the ith element is some value. I can only prove to you that there's no other thing that could convince you is at that position. The vector commitment has the functionality of opening the vector commitment.

There's also RSA-based vector commitments that are constnat sized, but I need those large parameters which are a problem. With accounts, I really want a massive number of accounts. I want an exponential number of accounts, most of which haven't been created. I want future accounts that have not yet been created.

So we used our accumulator to create a new vector commitment which has a constant sized setup. The idea is simply that for every index, I have a prime, and it's a bit commitment. I only commit to 1 or 0 at the index. If the prime is in the accumulator, it's a 1 at the index. And if it's not, then it's 0. Then I want to prove to you that a bunch of indices, so maybe a whole word or world... I need to give you a batch inclusion proof and a batch exclusion proof. A big part of the paper is focused on batch exclusion proof which uses some new interesting techniques.

With this vector commitment, I can prove to you that large messages are in this again with a constant sized proof.

With Catalone Fiore vector commitments, we can setup many many more accounts than the alternative. But because we only have the bit commitments, our prover is more expensive.

Short interactive oracle proofs

I can open you a proof at certain points and give you merkle tree inclusion proofs. But this is just a vector commitment. I could use a vector commitment like mine to commit to these and then using our batching techniques I don't have to send you one proof per index but only one proof overall. Some back of the envelope calculations show that this reduces the proof size by a factor of 3 so maybe this can push DEXes from 10,000 transactions to maybe 10 million transactions or something.

One downside is that the commitments are slower to construct, and they are not quantum secure.

Takeaway points

For blockchain, I can shift away work from consensus to the user and in fact the users can split up storing the state into different ways. Not all users need to store the entire state. The user can store just part of the state they care about, or they can outsource their work to other parties and you can load balance the work and the storage much better with this separation of consensus and state.