The complexity and randomness of linear multi-secret sharing schemes with non-threshold structures

In a linear multi-secret sharing scheme with non-threshold structures, several secret values are shared among n participants, and every secret value has a specified access structure. The efficiency of a multi-secret sharing scheme is measured by means of the complexity sigma and the randomness tau. Informally, the complexity sigma is the ratio between the maximum of information received by each participant and the minimum of information corresponding to every key. The randomness tau is the ratio between the amount of information distributed to the set of users U = {1, a <-, n} and the minimum of information corresponding to every key. In this paper, we discuss sigma and tau of any linear multi-secret sharing schemes realized by linear codes with non-threshold structures, and provide two algorithms to make sigma and tau to be the minimum, respectively. That is, they are optimal.