A Linear Construction of Secret Sharing Schemes

In this paper, we will generalize the vector space construction due to Brickell. This generalization, introduced by Bertilsson, leads to secret sharing schemes with rational information rates in which the secret can be computed efficiently by each qualified group. A one to one correspondence between the generalized construction and linear block codes is stated, and a matrix characterization of the generalized construction is presented. It turns out that the approach of minimal codewords by Massey is a special case of this construction. For general access structures we present an outline of an algorithm for determining whether a rational number can be realized as information rate by means of the generalized vector space construction. If so, the algorithm produces a secret sharing scheme with this information rate.