Asymptotically-Good Arithmetic Secret Sharing over \(\mathbb {Z}/p^{\ell }\mathbb {Z}\) with Strong Multiplication and Its Applications to Efficient MPC

This paper studies information-theoretically secure multiparty computation (MPC) over rings \(\mathbb {Z}/p^{\ell }\mathbb {Z}\). In the work of [Abs+19a, TCC’19], a protocol based on the Shamir secret sharing over \(\mathbb {Z}/p^{\ell }\mathbb {Z}\) was presented. As in the field case, its limitation is that the share size grows as the number of players increases. Then several MPC protocols were developed in [Abs+20, Asiacrypt’20] to overcome this limitation. However, (i) their offline multiplication gate has super-linear communication complexity in the number of players; (ii) the share size is doubled for the most important case, namely over \(\mathbb {Z}/2^{\ell }\mathbb {Z}\) due to infeasible lifting of self-orthogonal codes from fields to rings; (iii) most importantly, the BGW model could not be applied via the secret sharing given in [Abs+20, Asiacrypt’20] due to lack of strong multiplication.