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

Abstract

We consider access structures over a set $\mathcal {P}$ of n participants, defined by a parameter k with $1 \le k \le n$ in the following way: a subset is authorized if it contains participants $i,i+1,\ldots ,i+k-1$, for some $i \in \{1,\ldots ,n-k+1\}$. 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 $\frac{2}{3}$. 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.