Secret sharing schemes for (k, n)-consecutive access structures

Abstract

We consider access structures over a set P of n participants, defined by a parameter k with (Formula Presented) in the following way: A subset is authorized if it contains participants (Formula Presented), for some (Formula Presented). 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 (Formula Presented). 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.