Efficient Algorithm to Construct Perfect Secret Sharing Scheme for a Given Access Structure

The threshold scheme, the monotone circuit construction, and the vector space construction are some of the well-known secret sharing schemes in cryptography. The threshold and monotone circuit secret sharing schemes are fairly easy to construct for any given access structure Γ. The construction of a secret sharing scheme realizing a given access structure Γ with Vector Space Construction requires the existence of a function φ from a set of participants into a vector space, that is, φ: P → (Zp)d. This function φ must satisfy certain conditions in order to recover the secret key. There is no known algorithm to construct such a function φ in general. Constructions are mainly done by trial and error. In this paper, we develop polynomial algorithm to construct such φ function(s) for given access structures. Using the φ function, we also give an algorithm to construct secret sharing scheme for the access structures.