A general formula of the (T, n)-threshold visual secret sharing scheme

Abstract

This paper provides a new method for construction of the generating (or basis) matrices of the (t, n)-threshold visual secret sharing scheme ((t, n)-VSSS) for any n ≥ 2 and 2 ≤ t ≤ n. We show that there exists a bijection between a set of generating matrices of the (t, n)- VSSS and a set of homogeneous polynomials of degree n satisfying a certain property. We also show that the set of homogeneous polynomials is identified with a set of lattice points in a linear space of dimension n − t + 1 with explicitly expressed bases. These results yields a general formula of the generating matrices of the (t, n)-VSSS. The formula is not only theoretically of interest but also enables us to obtain efficient generating matrices that have been unknown.