Linear-Communication Asynchronous Complete Secret Sharing with Optimal Resilience

Secure multiparty computation (MPC) allows a set of $n$ parties to jointly compute a function on their private inputs. In this work, we focus on the information-theoretic MPC in the asynchronous network setting with optimal resilience ($t<n/3$). The best-known result in this setting is achieved by Choudhury and Patra [J. Cryptol '23], which requires $O(n^4\kappa)$ bits per multiplication gate, where $\kappa$ is the size of a field element.

An asynchronous complete secret sharing (ACSS) protocol allows a dealer to share a batch of Shamir sharings such that all parties eventually receive their shares. ACSS is an important building block in AMPC. The best-known result of ACSS is due to Choudhury and Patra [J. Cryptol '23], which requires $O(n^3\kappa)$ bits per sharing. On the other hand, in the synchronous setting, it is known that distributing Shamir sharings can be achieved with $O(n\kappa)$ bits per sharing. There is a gap of $n^2$ in the communication between the synchronous setting and the asynchronous setting.

Our work closes this gap by presenting the first ACSS protocol that achieves $O(n\kappa)$ bits per sharing. When combined with the compiler from ACSS to AMPC by Choudhury and Patra [IEEE Trans. Inf. Theory '17], we obtain an AMPC with $O(n^2\kappa)$ bits per multiplication gate, improving the previously best-known result by a factor of $n^2$. Moreover, with a concurrent work that improves the compiler by Choudhury and Patra by a factor of $n$, we obtain the first AMPC with $O(n\kappa)$ bits per multiplication gate.