Secure Multi-Party Computation from any Linear Secret Sharing Scheme

We present a general treatment of non-cryptographic (i.e. information-theoretically secure) multi-party computation, based on underlying linear secret sharing scheme. This general approach gives pure linear-algebra conditions on the linear mappings describing the scheme. The approach establishing the minimal conditions for security, can lead to design of more efficient Multi-Party Computation (MPC) schemes for general adversary structures. Our first goal is to study the Monotone Span Programs (MSP), which is the result of local multiplication of shares distributed by two given MSPs as well as the access structure that this result MSP computes. Second, we expand the definition for multiplicative MSP from [4] and prove that when we use dual MSPs only all players together can compute the product. The knowledge of the result MSP and the access structure it computes allows us to build an analog of the Genaro et al. algebraic simplification protocol [9]. Using this fact and the homomorphic commitments an efficient general MPC protocol in the computational model for general adversary structures can be build, as described in [9, 4].