Abstract

Algebraic geometric secret sharing schemes were proposed by Chen and Cramer so that the fundamental theorem in information-theoretically secure multiparty computation 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 ${\mathbb{𝔽}}_q$ are quasithreshold in the following sense: any subset of $u\le T-1$ players (nonqualified) has no information of the secret and any subset of $u\ge T+2g$ players (qualified) can reconstruct the secret. It is natural to ask how far from the threshold these quasithreshold secret sharing schemes are. How many subsets of $u\in [T,T+2g-1]$ players can recover the secret or have no knowledge of it?

We prove that if the size $q$ goes to infinity and $\lim <!--FUNCTION\; APPLICATION-->g∕q^{1∕2}=0$, then almost all subsets of $u\in [T,T+g-1]$ players have no information of the secret and almost all subsets of $u\in [T+g,T+2g-1]$ players can reconstruct the secret. 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 infinity.

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