2014/10/16 by Johan P. Hansen, Hansen, Johan P.
Computer Science · Mathematics · #14M25 #94A60 #94A62 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #Cryptography and Data Security #FOS: Mathematics #Polynomial and algebraic computation #math.AG #msc:14M25 #msc:94A60 #msc:94A62
paper · pdf · doi:10.48550/arxiv.1410.4378
15 pages, 4 figures. arXiv admin note: text overlap with arXiv:1203.4544
openalex publication_date 2014/10/16 · arxiv created 2016/03/13 · arxiv updated 2016/03/15 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28
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≥ 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.