Secret Sharing Schemes for (k, n)-Consecutive Access Structures

We consider access structures over a set P of n participants, defined by a parameter k with 1 <= k <= n in the following way: a subset is authorized if it contains participants i, i + 1, ... , i + k - 1, for some i is an element of {1, ... , n - k + 1}. We call such access structures, which may naturally appear in real applications involving distributed cryptography, (k, n)-consecutive. We prove that these access structures are only ideal when k = 1, n - 1, n. Actually, we obtain the same result that has been obtained for other families of access structures: being ideal is equivalent to being a vector space access structure and is equivalent to having an optimal information rate strictly bigger than 2/3. For the non-ideal cases, we give either the exact value of the optimal information rate, for k = n - 2 and k = n - 3, or some bounds on it.