Paper 2024/974 Original: eprint.iacr.org/2024/974 ↗

Towards Optimal Parallel Broadcast under a Dishonest Majority

Abstract

The parallel broadcast (PBC) problem generalizes the classic Byzantine broadcast problem to the setting where all $n$ nodes broadcast a message and deliver $O(n)$ messages. PBC arises naturally in many settings including multi-party computation. The state-of-the-art PBC protocol, TrustedPBC, is due to Tsimos, Loss, and Papamanthou (CRYPTO 2022), which is secure under an adaptive adversary assuming $f < (1 - \epsilon)n$, where $f$ is the number of Byzantine failures and $\epsilon \in (0,1)$. TrustedPBC focuses on single-bit inputs and achieves $\tilde{O}(n^2 \kappa^4)$ communication and $O(\kappa\log n)$ rounds. In this work, we propose three PBC protocols for $L$-bit messages, for any size $L$, that significantly improve TrustedPBC. First, we propose a new extension protocol that uses a $\kappa$-bit PBC as a black box and achieves i) communication complexity of $O(L n^2 + n^3\kappa+\mathcal{P}(\kappa))$, where $\mathcal{P}(\kappa)$ is the communication complexity of the $\kappa$-bit PBC, and ii) round complexity same as the $\kappa$-bit PBC. By comparison, the state-of-the-art extension protocol for regular broadcast (Nayak et al., DISC 2020) incurs $O(n)$ additional rounds of communication. Next, we propose a protocol that is secure against a static adversary, for $\kappa$-bit messages with $O(n^2 \kappa^{1+K} + n\kappa^3 + \kappa^4)$ communication and $O(\kappa)$ round complexity, where $K$ is an arbitrarily small constant such that $0<K<1$. Finally, we propose an adaptively-secure protocol for $\kappa$-bit messages with $\tilde{O}(n^2\kappa^2 + n\kappa^3)$ communication overhead and $O(\kappa \log{n})$ round complexity. Notably, our latter two protocols are $\tilde{O}(\kappa^{2 - K})$ and $O(\kappa^2)$ times more communication-efficient, respectively, than the state-of-the-art protocols while achieving the same round complexity.