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 Fq 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 limqg=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