General Error Decodable Secret Sharing Scheme and Its Application

Consider a model of secret sharing schemes with cheaters. We say that a secret sharing scheme is error decodable if we can still recover the secret s correctly from a noisy share vector (share₁', ..., sharen'). In this paper, we first prove that a perfect secret sharing scheme is error decodable if and only if the adversary structure Γ satisfies a certain condition called Q³. Next, for such Γ , we show a scheme such that the decoding algorithm runs in polynomial-time in |S | and the size of a linear secret sharing scheme which realizes Γ. We finally show an application to 1-round perfectly secure message transmission schemes (PSMT).