Asymptotically Good Multiplicative LSSS over Galois Rings and Applications to MPC over $$\mathbb {Z}/p^k\mathbb {Z} $$

We study information-theoretic multiparty computation (MPC) protocols over rings \(\mathbb {Z}/p^k\mathbb {Z} \) that have good asymptotic communication complexity for a large number of players. An important ingredient for such protocols is arithmetic secret sharing, i.e., linear secret-sharing schemes with multiplicative properties. The standard way to obtain these over fields is with a family of linear codes C, such that C, \(C^\perp \) and \(C^2\) are asymptotically good (strongly multiplicative). For our purposes here it suffices if the square code \(C^2\) is not the whole space, i.e., has codimension at least 1 (multiplicative).