Multi-party Computation from Any Linear Secret Sharing Scheme Unconditionally Secure against Adaptive Adversary: The Zero-Error Case

We consider a generalized adaptive and active adversary model for unconditionally secure Multi-Party Computation (MPC) in the zero error case. Cramer et al. proposed a generic approach to build a multiplicative Monotone Span Programs (MSP) – the special property of a Linear Secret Sharing Schemes (LSSS) that is needed to perform a multiplication of shared values. They give an efficient generic construction to build verifiability into every LSSS and to obtain from any LSSS a multiplicative LSSS for the same access structure. But the multiplicative property guarantees security against passive adversary only. For an active adversary a strong multiplicative property is required. Unfortunately there is no known efficient construction to obtain a strongly multiplicative LSSS yet. Recently Nikov et al. have expanded the construction of Cramer et al. using a different approach. Multiplying two different MSP M1 and M2 computing the access structures Γ1 and Γ2 a new MSPM called “resulting” is obtained. M computes a new access structure Γ ⊂ Γ1 (orΓ2). The goal of this construction is to enable the investigation of how the properties that Γ should fulfil are linked to the initial access structures Γ1 and Γ2. It is proved that Γ2 should be a dual access structure of Γ1 in order to have a multiplicative resulting MSP. But there are still not known requirements for initial access structures in order to obtain strongly multiplicative resulting MSP. Nikov et al. proved that to have unconditionally secure MPC the following minimal conditions for the resulting access structure should be satisfied (ΓA ] ΓA) ⊆ Γ . In this paper we assume that the resulting MSP could be constructed such that the corresponding access structure Γ satisfies the required ? The author was partially supported by IWT and Concerted Research Action GOAMEFISTO-666 of the Flemish Government