Algebraic Geometric Secret Sharing Schemes over Large Fields Are Asymptotically Threshold

TL;DR

This paper demonstrates that algebraic geometric secret sharing schemes over large finite fields become asymptotically threshold schemes as the field size increases, with most subsets either reconstructing the secret or having no information.

Contribution

It proves that over large fields, these schemes are asymptotically threshold, extending understanding of their threshold properties in the limit.

Findings

Almost all subsets of size between T and T+g-1 have no information about the secret.

Almost all subsets of size between T+g and T+2g-1 can reconstruct the secret.

Schemes become asymptotically threshold as the field size q approaches infinity with g/√q → 0.

Abstract

In Chen-Cramer Crypto 2006 paper \cite{cc} algebraic geometric secret sharing schemes were proposed such that the "Fundamental Theorem in Information-Theoretically Secure Multiparty Computation" by Ben-Or, Goldwasser and Wigderson \cite{BGW88} and Chaum, Cr\'{e}peau and Damg{\aa}rd \cite{CCD88} 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 ${\bf F}_q$ is quasi-threshold in the following sense, any subset of $u \leq T-1$ players (non qualified) has no information of the secret and any subset of $u \geq T+2g$ 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 $u \in [T, T+2g-1]$ players can recover the secret or have no information of the secret? In this paper it is proved that almost all subsets of $u \in [T,T+g-1]$ players have no information of the secret and almost all subsets of $u \in [T+g,T+2g-1]$ players can reconstruct the secret when the size $q$ goes to the infinity and the genus satisfies $\lim \frac{g}{\sqrt{q}}=0$. 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.