Paper 2025/2135

Improved Robust Secret-Sharing for Honest-Majority Preprocessing-MPC

Abstract

We design a novel robust secret-sharing scheme in the honest-majority setting for $n$ parties that allows constant communication per opening per party when amortized over $\Theta(n)$ simultaneous openings. We attain this scheme by a novel application of list decoding of folded Reed-Solomon codes. This was stated as an open problem by Escudero and Fehr (2023), who could obtain this amortized complexity only for $\Theta(n^2)$ simultaneous openings. Additionally, the dealing of such shares for a random value in a finite field $\mathbb{F}$ can itself be done using an honest-majority MPC protocol using only $O(n^2)$ multiplications between Shamir-shared values. This allows us to build an offline-online fully-secure (GOD) honest-majority MPC protocol, which has a fast online stage with amortized communication cost only $O(n)$ per multiplication gate (assuming the circuit has width $\Omega(n)$) and with no additional overhead, and an offline preprocessing stage with incremental communication cost only $O(n^3)$ per multiplication gate. This improves the previous best protocol in this setting by Escudero and Fehr (2023), which required the circuit to have width $\Omega(n^2)$. Further, the earlier scheme had an incremental preprocessing cost of $O(n^4)$ per multiplication gate. Due to requiring non-responsive rounds, the online stage of these protocols may still be concretely far less efficient than the online stage of the offline-online BGW protocol. Since the BGW protocol only provides security against fewer than $n/3$ malicious parties, we also devise a novel MPC protocol that boosts the BGW protocol's correctness threshold to $n/2$, while maintaining privacy and robustness thresholds at $n/3$, and with minimal online overhead. This is useful in applications, such as elections and auctions, where certification of a correct result is paramount. To achieve this correctness against fewer than $n/2$ malicious parties and to keep the pre-processing cost low, we use our novel robust honest-majority secret-sharing scheme, but only once during a final SPDZ-style attestation.