A secret-sharing scheme realizes a graph if every two vertices connected by an edge can reconstruct the secret while every independent set in the graph does not get any information on the secret. Similar to secret-sharing schemes for general access structures, there are gaps between the known lower bounds and upper bounds on the share size for graphs. Motivated by the question of what makes a graph "hard" for secret-sharing schemes (that is, require large shares), we study very dense graphs, that is, graphs whose complement contains few edges. We show that if a graph with n vertices contains n 2 − n 1+β edges for some constant 0 ≤ β < 1, then there is a scheme realizing the graph with total share size of O(n 5/4+3β/4 ). This should be compared to O(n 2 / log n) -the best upper bound known for the share size in general graphs. Thus, if a graph is "hard", then the graph and its complement should have many edges. We generalize these results to nearly complete k-homogeneous access structures for a constant k. To complement our results, we prove lower bounds for secret-sharing schemes realizing very dense graphs, e.g., for linear secret-sharing schemes we prove a lower bound of Ω(n 1+β/2 ) for a graph with n 2 − n 1+β edges.

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Abstract. In a multisecret sharing scheme, several secret values are distributed among a set of n users, and each secret may have a different associated access structure. We consider here unconditionally secure schemes with multithreshold access structures. Namely, for every subset P of k users there is a secret key that can only be computed when at least t of them put together their secret information. Coalitions with at most w users with less than t of them in P cannot obtain any information about the secret associated to P . The main parameters to optimize are the length of the shares and the amount of random bits that are needed to set up the distribution of shares, both in relation to the length of the secret. In this paper, we provide lower bounds on this parameters. Moreover, we present an optimal construction for t = 2 and k = 3, and a construction that is valid for all w, t, k and n. The models presented use linear algebraic techniques.

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