  A general theory for constructing linear secret sharing schemes over a finite
field $\Fq$ from toric varieties is introduced. The number of players can be as
large as $(q-1)^r-1$ for $r\geq 1$. We present general methods for obtaining
the reconstruction and privacy thresholds as well as conditions for
multiplication on the associated secret sharing schemes.
  In particular we apply the method on certain toric surfaces. The main results
are ideal linear secret sharing schemes where the number of players can be as
large as $(q-1)^2-1$. We determine bounds for the reconstruction and privacy
thresholds and conditions for strong multiplication using the cohomology and
the intersection theory on toric surfaces.
