International Association
for Cryptologic Research

Messages in large-scale networks such as blockchain systems are typically disseminated using flooding protocols, in which parties send the message to a random set of peers until it reaches all parties. Optimizing the communication complexity of such protocols and, in particular, the per-party communication complexity is of primary interest since nodes in a network are often subject to bandwidth constraints. Previous flooding protocols incur a per-party communication complexity of $\Omega(l\cdot \gamma^{-1} \cdot (\log(n) + \kappa))$ bits to disseminate an $l$-bit message among $n$ parties with security parameter~$\kappa$ when it is guaranteed that a $\gamma$ fraction of the parties remain honest. In this work, we present the first flooding protocols with a per-party communication complexity of $O(l\cdot \gamma^{-1})$ bits. We further show that this is asymptotically optimal and that our protocols can be instantiated provably securely in the usual setting for proof-of-stake blockchains. To demonstrate that one of our new protocols is not only asymptotically optimal but also practical, we perform several probabilistic simulations to estimate the concrete complexity for given parameters. Our simulations show that our protocol significantly improves the per-party communication complexity over the state-of-the-art for practical parameters. Hence, for given bandwidth constraints, our results allow to, e.g., increase the block size, improving the overall throughput of a blockchain.

Many decentralized systems rely on flooding protocols for message dissemination. In such a protocol, the sender of a message sends it to a randomly selected set of peers. These peers again send the message to their randomly selected peers, until every network participant has received the message. This type of protocols clearly fail in face of an adaptive adversary who can simply corrupt all peers of the sender and thereby prevent the message from being delivered. Nevertheless, flooding protocols are commonly used within protocols that aim to be cryptographically secure, most notably in blockchain protocols. While it is possible to revert to static corruptions, this gives unsatisfactory security guarantees, especially in the setting of a blockchain that is supposed to run for an extended period of time. To be able to provide meaningful security guarantees in such settings, we give precise semantics to what we call $\delta$-delayed adversaries in the Universal Composability (UC) framework. Such adversaries can adaptively corrupt parties, but there is a delay of time $\delta$ from when an adversary decides to corrupt a party until they succeed in overtaking control of the party. Within this model, we formally prove the intuitive result that flooding protocols are secure against $\delta$-delayed adversaries when $\delta$ is at least the time it takes to send a message from one peer to another plus the time it takes the recipient to resend the message. To this end, we show how to reduce the adaptive setting with a $\delta$-delayed adversary to a static experiment with an Erdős–Rényi graph. Using the established theory of Erdős–Rényi graphs, we provide upper bounds on the propagation time of the flooding functionality for different neighborhood sizes of the gossip network. More concretely, we show the following for security parameter $\kappa$, point-to-point channels with delay at most $\Delta$, and $n$ parties in total, with a sufficiently delayed adversary that can corrupt any constant fraction of the parties: If all parties send to $\Omega(\kappa)$ parties on average, then we can realize a flooding functionality with maximal delay $\mathcal{O}\bigl(\Delta \cdot \log (n) \bigr)$; and if all parties send to $\Omega\bigl( \sqrt{\kappa n \log (n)} \bigr)$ parties on average, we can realize a flooding functionality with maximal delay $\mathcal{O}(\Delta)$.

In recent years, permisionless blockchains have received a lot of attention both from industry and academia, where substantial effort has been spent to develop consensus protocols that are secure under the assumption that less than half (or a third) of a given resource (e.g., stake or computing power) is controlled by corrupted parties. The security proofs of these consensus protocols usually assume the availability of a network functionality guaranteeing that a block sent by an honest party is received by all honest parties within some bounded time. To obtain an overall protocol that is secure under the same corruption assumption, it is therefore necessary to combine the consensus protocol with a network protocol that achieves this property under that assumption. In practice, however, the underlying network is typically implemented by flooding protocols that are not proven to be secure in the setting where a fraction of the considered total weight can be corrupted. This has led to many so-called eclipse attacks on existing protocols and tailor-made fixes against specific attacks. To close this apparent gap, we present the first practical flooding protocol that provably delivers sent messages to all honest parties after a logarithmic number of steps. We prove security in the setting where all parties are publicly assigned a positive weight and the adversary can corrupt parties accumulating up to a constant fraction of the total weight. This can directly be used in the proof-of-stake setting, but is not limited to it. To prove the security of our protocol, we combine known results about the diameter of Erdős–Rényi graphs with reductions between different types of random graphs. We further show that the efficiency of our protocol is asymptotically optimal. The practicality of our protocol is supported by extensive simulations for different numbers of parties, weight distributions, and corruption strategies. The simulations confirm our theoretical results and show that messages are delivered quickly regardless of the weight distribution, whereas protocols that are oblivious of the parties' weights completely fail if the weights are unevenly distributed. Furthermore, the average message complexity per party of our protocol is within a small constant factor of such a protocol.