On multiplicative linear secret sharing schemes

We consider both information-theoretic and cryptographic settings for Multi-Party Computation (MPC), based on the underlying linear secret sharing scheme. Our goal is to study the Monotone Span Program (MSP), that is the result of local multiplication of shares distributed by two given MSPs as well as the access structure that this resulting MSP computes. First, we expand the construction proposed by Cramer et al. for multiplying two different general access structures and we prove some properties of the resulting MSP. We prove that using two (different) MSPs to compute their resulting MSP is more efficient than building a multiplicative MSP. Next we define a (strongly) multiplicative resulting MSP and we prove that when one uses dual MSPs only all players together can compute the product. An analog of the algebraic simplification protocol of Gennaro et al. is presented. We show which conditions the resulting access structure should fulfill in order to achieve MPC secure, against an adaptive, active adversary in the zero-error case in both the computational and the information-theoretic model.