If one wants to set up a decentralized cryptographic enforcement scheme in which the level of 'confidentiality' of information is marked by its position in a poset, why not use Diffie-Hellman to derive the key of a node from its immediate parents?
This is as opposed to the tree-based scheme described in this paper.
In more detail:
You have a poset $X$ from which you want to build a CES. You have a public one-way function $f$. The poset has a top element $r$ and a randomly generated key $k(r)$. Here is how you derive $k(x)$ for all $x \in X$: Let $u \in X$. We can assume one of only two possibilities (we can easily generalize to more than these):
- $u$ has one parent $x$. Then $k(u) = f(k(x))$.
- $u$ has two parents $x_1$ and $x_2$. Then choose a random DDH-hard cyclic group $G$ and a generator $g \in G$ and publish $(G, g, g^{k(x_1)}, g^{k(x_2)})$. $k(u) = g^{k(x_1)k(x_2)}$.
[edit]
When sending a message, a chain can be used to communicate its position in the poset. Let's say that if a node $u$ has one parent, its 'name' will be denoted $u$ (which will be random). Let's say if it has 2 parents, its name will be the $(G, g, g^{k(x_1)}, g^{k(x_2)})$. Then you publish the chain.