Hybrid Multiplicative Secret Sharing

A secret-sharing scheme is $d$-multiplicative if it allows the players to multiply $d$ (rather than two) shared secrets (without recovering them) by {\em locally} converting their shares into an additive sharing of the product. In this work, the $d$-multiplicative secret-sharing (MSS) scheme is extended to a hybrid MSS (HMSS), which is mainly designed for $d$ secrets shared against different access structures. A necessary and sufficient condition for $n$-player $d$-HMSS schemes to exist is presented. The sufficiency is constructively proven based on the replication-based secret-sharing scheme. The necessity for any general case is reduced to the impossibility of a minimum case. The results holds for arbitrary (possibly inefficient or even nonlinear) secret-sharing schemes.