Scalable RSA Modulus Generation with Dishonest Majority

Megan Chen

# Introduction

Groups of unknown order are interesting cryptographic primitives. You can commit to integers. If you commit to a very large integer, you have to do a lot of work to do the commitment so you have a sequentiality assumption useful for VDFs. But then you have this compression property where if you take a large integer with a lot of information in it, you can compress it down to something small. It also has a nice additive homomorphic commitment property where the commitment of the sum is the sum of the commitments, and you can do accumulators and SNARKs with this property. We're having this little mini revolution in cryptography with this blossoming of research around groups of unknown order. There's this key idea like "proofs of exponentiation" which started with verifiable delay functions but now they're used with accumulators, vector commitments, polynomial commitments, SNARKs, etc. We're going to see a talk about a trusted setup for MPC which is super-exciting for this space because it's breaking a number of records.

Diogenes: Lightweight scalable ...

Thank you everyone for waking up early. I'm a researcher at Lehero. I'll be talking about diogenes, our protocol for scalable RSA modulus generation. This is joint work with my colleagues. Okay, great.

# What is an RSA modulus

An RSA modulus is a product of two large primes p and q where they are kept secret. We call a bi-prime the product of exactly two primes and we want m to be a bi-prime. There's a long history of using RSA, starting in the 1970s for encryption and it has been used for other things since then. They are useful for verifiable delay functions and also a lot of blockchain networks want to use VDFs because they are a good strategy for generating randomness.

# VDF construction

Here's a popular construction of a VDF. It starts with the Rivest-SHamir-Wagner timelock puzzle from 1996 which is essentially doing t sequential squarings in a group mod n. So here the circled N is the RSA modulus. The security here relies on the fact that the factorization is kept secret, thus it makes sense to generate N with untrusted setup, which is the goal of our project.

# Goals

We want parties to interact and jointly sample a bi-prime modulus N. So everyone has a secret share, an additive share of N's factors p and q. Also we want 1024 parties to participate and achieve scalability in this project. Additionally, we want a dishonest majority or n-1 active security meaning we only require a single honest participant even though we allow malicious parties to deviate arbitrarly from the protocol.

# Previous work

This was originally introduced by Boneh and Franklin in 1997. They were the first to offer a solution. Notably to date, their framework still remains the basis for most of the follow-up work in this space. I'll highlight a few.

In our setting of having active security across N parties and a dishonest majority, there were only two previous works that gave proofs that satisfied this but unfortunately our analysis shows that they were not scalable to 1024 parties. Only two of these works actually implemented their protocols. They give active protocols in their papers, but their implementations remain passive.

# State of the art

FLOP18 wanted to generate an RSA modulus size of 2048 bits. Their protocol is for 2 parties and each party has 8 GB of RAM and 8 cores CPU and they have bandwidth of 40 Gbps. The most expensive part of the protocol requires at least 1.9 GB of online communication per party. Additionally, their protocol was timed and it lasted 35 seconds for an 8-thread implementation. Can we do better? We're pretty ambitious and we want scalability.

# Our goals

So we want to generate the same modulus size... and give an active implementation with identifiable abort. We want 1024 parties to participate in the computation. This avoids having a few parties, so it reduces the trust in any single party. Each party has only 2 GB of RAM and a single core of CPU. We limit the bandwidth to be 1 Mbps and 100 ms latency. Everyone has less than 100 MB of online communication. And also we want our protocol to terminate in less than 20 minutes. This is pretty crazy right? How can we achieve this?

# Protocol blueprint

First we designed a protocol secure against passive adversaries, then we compile it to something secure against active adversaries. This is a common paradigm when constructing protocols. So what's the best scalable passive protocol? Well, we used the Boneh-Franklin framework from 1997. Here's an overview of how it worked: parties randomly sample shares, p and q. Each of these values are random so there's no guarantee that the sum is prime. This step also includes distributed division checks to filter out values that are divisible by small primes, lots of works call this "trial division". Next, each party inputs their p and q into a distributed multiplication to get a candidate RSA modulus N. We have no idea if the sum is prime yet. We have done some initial filtering but this isn't enough. One way to check is to use a distributed bi-primality check. We can use the fact that N is now public, in conjunction with parties using their pi and qi shares we can test if N is the product of two primes.

# Prime candidate sampling and trial division

First we start with prime candidate sampling and trial division. Previous works pick q and p shares, then they do joint trial division, if they both pass then we multiply and get a candidate modulus N. How likely are we to get a prime here? If we're trying to generate a bi-prime, the probability is 1 in 5002. So we need about 250,000 samples to do this. Further down the line, we need to do large multiplications to construct N and by large multiplications I mean 1024 bit inputs. Is there more sophisticated way to do this? It turns out, there is.

# Chinese remainder theorem

This is an idea based on the chinese remainder theorem. Suppose we have a set of remainders taken... CRT says we can map this set of remainders to a unique value in the larger modulus which is specifically the product of all the smaller on the left, and this value is unique. So the mapping algorithm here is called CRT reconstruction. This mapped value will have the same remainders mod the moduli on the left. As a quick note, this only works if the small moduli on the left don't share any factors.

# Sieving trick

Based on CCD20 which will be published soon, we observe that..... we are able to yield a value that is not divisible by the t smaller moduli. If we set the modulus on the left to be exactly the first t small primes, this exactly works like trial division but we're instead picking pre-approved values and we call this process "sieving".

If we go back to our probabilities, we're still sampling the same size prime but we're sieving up to the 150th prime so the probability of actually sampling 1024-bit prime increases to 1/60 which is pretty good. Since we need two primes, the probability of getting a bi-prime is 1/(602) which means we need 3600 samples in expectation which is a huge reduction from 250,000. Further, we can construct N using a series of small multiplications by sampling two values, mod each of the small primes, we multiply them using a distributed multiplication to get these product residues, then we can use a local reconstruction to generate our candidate N which is pretty cool.

Next we have to do some multiplication so let's review how we build our multiplier. Let's start with secure multiplication where parties supply their additive shares as inputs. They input it into the functionality and receive additive shares of the product, and then that's great. We want to achieve the same functionality, but we want to do it with 1024 parties.

Our approach is to use threshold additive homomorphic encryption, which comes with key generation, encryption and distributed decryption. It's threshold, so each party gets a share of the secret key, and everyone needs to participate in order to decrypt a ciphertext. The scheme is additively homomorphic which means that if we add two encryptions then the sum of the encrypted values will be the same as the sum of the plaintext, and similar with scalar multiplication under encryption.

Another piece of our multiplication is introducing something called a "central coordinator" that lets us take advantage of a star topology. This coordinator is not trusted. They are not a part of the party count. This is external from the 1024 party count. Also it just does public operations, specifically we only want it to add ... separately. So quickly, for our implementation, this is the spec we set for our coordinator: it has 1 TB RAM, 128 core CPU, and 10 Gbps bandwidth. We do this so that we can move a lot of the heavier computation away from the parties and speed up the protocol. Remember, we want our protocol to be accessible to a wide range of parties that may not have high computational capacity.

Parties all have their pi qi additive shares. Then they work together to do a distributed key generation so everyone gets a public key of the scheme, as well as their share of the secret key. Everyone encrypts their pi and sends it to the coordinator, and it adds them all up, and sends them back to each party. This is additively homomorphic so.... then they multiply byqi, the coordinator adds them up, then receive enc(pq) from coordinator, and then you do a distributed decryption and now you have p times q.

# State of the art threshold additive homomorphic encryption

How do we instantiate threshold additively-homomorphic encryption? We settled with Ring-LWE because it supports AHE, has better parameters, and we can pack multiple plaintexts into a single ciphertext which is good for efficiency.

# BF97's primality test

Using our multiplication, we get our candidate modulus N. The next step is to do a bi-primality test to check if it's actually the product of two primes. We only filtered up to 150 at this point. We simply use the BF97 protocol. It's probabilistic and it's loosely based on the Miller-Rabin primality test. N is public right now, and you can do this test.

So we have a passive protocol now, and we want to have security against active adversaries. We compile it into this format using a generic method called GNW paradigm. Here's a passive protocol in my diagram. I show two parties here but it could be n parties. Each party has a secret input and their own internal randomness, and using this they send messages to each other and then do a protocol. What GNW says is that this is passively secure, but we can compile it up to something actively secure as follows: parties commit to their input and randomness at the beginning, each party then in subsequent steps validates all of their messages by sending a zero-knowledge proof along with their messages.

We don't exactly follow the GNW technique. In our protocol, parties don't have secret inputs into the protocol, they simply use their internal randomness so we commit to that at the beginning. This allows us to move the zero-knowledge proofs to until after the messages are generated.

# Zero-knowledge proof considerations

Next the question is what zero-knowledge proof system should we use? This is one of our main considerations. Our protocol requires operations in many mathematical structures. In lattices, we're doing operations in a ring listed on this slide. To generate our modulus candidates, we're doing operations in z mod p where p is prime. Lastly, we have this Jacobi test in Znstar (2048-bit number)

We want to commit to randomness for the AHE scheme, as well as the parties pi qi shares at the beginning. We want to run the entire passive protocol, and then we want to commit to randomness for the sigma protocol which we use for the Jacobi test, and then we send the proofs at the very end. These are all things that every party has to send to the coordinator. Lastly, we want the proof system to be memory efficient since our parties have memory constraints. It needs to support commit-and-prove because we're using GNW, and lastly we want this to be versatile.

# The proofs

We chose to use a combination of Liegro proofs and a Sigma protocol as well. Ligero is used for range proofs to put bounds on the noise for Ring-LWE. We are also proving the correctness of each party action afterwards. Also, sigma is used for correctness of the Jacobi test for bi-primality testing and this is a good proof when you need to do exponentiation.

# Coordinator security

Recall that the coordinator only adds things that it receives and it also has no inputs or any randomness. This means we can simply make it publish its transcripts online and thus it becomes publicly verifiable. We check that the messages it receives is consistent with the outputs it gets.

# Summary of the protocol

Key setup is based on generating threshold keys. We generate modulus candidates using sampling of pre-approved primes. We use TAHE to compute candidates. We have a bi-primality test. And one more thing.

# Performance

We implemented our passive protocol, and we were able to run a 10,000 party implementation and it took only 35 minutes. This also gives hope that we can do multi-party computation across all nodes in a blockchain so for ethereum there's roughly 10k nodes online at a given time.

For an active protocol, we ran this for 1024 parties last night and we got this timing. Just to generate an RSA modulus, this took about 5 minutes and 20 seconds. To generate all the proofs for the parties to send for verification, this took less than 7 minutes. Lastly, verification technically took longer than 8 seconds because what happens is that the parties are generating lots of proofs they are sending for verification. After generating each one, they send it immediately, so then the verification becomes parallelizable.

How did we achieve this performance metric? We opened nodes across AWS and we also had local participants with their own machines.

# VDF day trial run

We had a trial run at VDF day. Thanks to the VDF Alliance to give us an opportunity to test the protocol in the real world. We had 25 parties participating with all different kinds of computers. The coordinator was running on AWS. We were able to run it two times. Our passive protocol succeeded, meaning we successfully generated RSA moduli but our active protocol didn't complete. We used docker to distribute to participates.

# Identifiable abort

We got security with abort, but we didn't get identifiable abort. We would like to be able to kick out parties that are behaving maliciously and restart the protocol. Identifiable abort requires more rigorous testing. Up to now, no implemented MPC offers this. This is new territory for us. Lastly, I want to thank all the participants who came.

# Conclusion

We're on track, but we had to increase our online communication per party to 200 MB. But otherwise this has been a really exciting project, thanks everyone.