Threshold Secret Sharing Through Multivariate Birkhoff Interpolation

Abstract

Secret sharing schemes have been well studied and widely used in different aspects of real life applications. The original secret sharing scheme was proposed by Adi Shamir in 1979. A similar scheme was also invented independently in the same year by George Blakley. Shamir’s scheme is based on Lagrange interpolation while Blakley’s approach uses principles of hyperplane geometry. In 2007, Tamir Tassa proposed a hierarchical secret sharing scheme through univariate Birkhoff interpolation (a generalization of Lagrangian and Hermitian interpolation). In the contribution at hand we investigate the idea of generalizing Tassa’s scheme through multivariate Birkhoff interpolation. We consider the problem of finding secret sharing schemes with multilevel structures and partially ordered sets of levels of participants. In order to ensure that our scheme meets the necessary requirements, we use totally nonsingular matrices.