Abstract We consider access structures over a set of n participants, defined by a parameter k with $$1 \le k \le n$$ 1 ≤ k ≤ n in the following way: a subset is authorized if it contains at least k consecutive participants. Depending on whether we consider the participants placed in a line (that is, participant 1 is not next to participant n) or in a circle, we obtain two different families, that we call (k, n)-line-consecutive and (k, n)-circle-consecutive access structures, respectively. Such access structures can appear in real-life situations involving distributed cryptography, which makes it more interesting to look for the best secret sharing schemes that can realize them. For both families, we characterize which are the configurations (k, n) that admit ideal secret sharing schemes. For the non-ideal (k, n)-consecutive access structures, we give both upper and lower bounds on the information ratio of the best secret sharing schemes that can realize them. Some of these bounds are obtained after proving relations between the information ratios of access structures in the two considered families.

In a distributed key distribution scheme, a set of servers help a set of users in a group to securely obtain a common key. Security means that an adversary who corrupts some servers and some users has no information about the key of a non-corrupted group. In this work we formalize the security analysis of one of such schemes \\cite{DHPS02}, which was not considered in the original proposal. We prove the scheme secure in the random oracle model, assuming that the Decisional Diffie-Hellman problem is hard to solve. We also detail a possible modification of that scheme and the one in \\cite{NPR99}, which allows to prove the security of the schemes without assuming that a specific hash function behaves as a random oracle. As usual, this improvement in the security of the schemes is at the cost of an efficiency loss.

We study the information rate of secret sharing schemes whose access structure is bipartite. In a bipartite access structure there are two classes of participants and all participants in the same class play an equivalent role in the structure. We characterize completely the bipartite access structures that can be realized by an ideal secret sharing scheme. Both upper and lower bounds on the optimal information rate of bipartite access structures are given. These results are applied to the particular case of weighted threshold access structure with two weights.