Secret Sharing Schemes with a large number of players from Toric Varieties

A general theory for constructing linear secret sharing schemes over a finite field $\text{Fq}$ from toric varieties is introduced. The number of players can be as large as $(q-1{)}^r-1$ for $r\ge 1$. 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.