Algebraic Geometric Secret Sharing Schemes over Large Fields Are Asymptotically Threshold
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In Chen-Cramer Crypto 2006 paper <cit.> algebraic geometric secret sharing schemes were proposed such that the "Fundamental Theorem in Information-Theoretically Secure Multiparty Computation" by Ben-Or, Goldwasser and Wigderson <cit.> and Chaum, Crépeau and Damgård <cit.> can be established over constant-size base finite fields. These algebraic geometric secret sharing schemes defined by a curve of genus g over a constant size finite field F_q is quasi-threshold in the following sense, any subset of u ≤ T-1 players (non qualified) has no information of the secret and any subset of u ≥ T+2g players (qualified) can reconstruct the secret. It is natural to ask that how far from the threshold these quasi-threshold secret sharing schemes are? How many subsets of u ∈ [T, T+2g-1] players can recover the secret or have no information of the secret? In this paper it is proved that almost all subsets of u ∈ [T,T+g-1] players have no information of the secret and almost all subsets of u ∈ [T+g,T+2g-1] players can reconstruct the secret when the size q goes to the infinity and the genus satisfies limg/√(q)=0. Then algebraic geometric secret sharing schemes over large finite fields are asymptotically threshold in this case. We also analyze the case when the size q of the base field is fixed and the genus goes to the infinity.
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