On the information ratio of non-perfect secret sharing schemes

The final publication is available at Springer via http://dx.doi.org/10.1007/s00453-016-0217-9A secret sharing scheme is non-perfect if some subsets of players that cannot recover the secret value have partial information about it. The information ratio of a secret sharing scheme is the ratio between the maximum length of the shares and the length of the secret. This work is dedicated to the search of bounds on the information ratio of non-perfect secret sharing schemes and the construction of efficient linear non-perfect secret sharing schemes. To this end, we extend the known connections between matroids, polymatroids and perfect secret sharing schemes to the non-perfect case. In order to study non-perfect secret sharing schemes in all generality, we describe their structure through their access function, a real function that measures the amount of information on the secret value that is obtained by each subset of players. We prove that there exists a secret sharing scheme for every access function. Uniform access functions, that is, access functions whose values depend only on the number of players, generalize the threshold access structures. The optimal information ratio of the uniform access functions with rational values has been determined by Yoshida, Fujiwara and Fossorier. By using the tools that are described in our work, we provide a much simpler proof of that result and we extend it to access functions with real values.Peer ReviewedPostprint (author's final draft

A Note on Extension Properties and Representations of Matroids

We discuss several extension properties of matroids and polymatroids and their application as necessary conditions for the existence of different matroid representations, namely linear, folded linear, algebraic, and entropic representations. Iterations of those extension properties are checked for matroids on eight and nine elements by means of computer-aided explorations, finding in that way several new examples of non-linearly representable matroids. A special emphasis is made on sparse paving matroids on nine points containing the tic-tac-toe configuration. We present a clear description of that family and we analyze extension properties on those matroids and their duals