# What are access structures in Attribute-Based Encryption?

I'm reading this article, but I could'nt figure out Definition 1 (Access Structure) on page 3, section 3.1. It says:

Let $$\{P_1, P_2,...,P_n\}$$ be a set of parties. A collection $$\mathbb{A} \subseteq 2^{\{P_1, P_2,...,P_n\}}$$ is monotone if $$\forall B,C:$$ if $$B \in \mathbb{A}$$ and $$B \subseteq C$$ then $$C \in \mathbb{A}$$. An access structure (respectively, monotone access structure) is a collection (respectively, monotone collection) $$\mathbb{A}$$ of non-empty subsets of $$\{P_1, P_2,...,P_n\}$$, i.e., $$\mathbb{A} \subseteq 2^{\{P_1, P_2,...,P_n\}}-\{\emptyset\}$$. The sets in $$\mathbb{A}$$ are called the authorized sets, and the sets not in $$\mathbb{A}$$ are called the unauthorized sets.

First, what do they mean by access structure? And what is the power of a set of parties ($$\mathbb{A} \subseteq 2^{\{P_1, P_2,...,P_n\}}$$)? Is it meant as a power set? Could you give me an explanation about that, and/or give some examples?

An access structure is the set of sets that should have access. For example, if the set of parties is $$\{P_1, P_2, P_3, P_4\}$$ and the access structure, $$\mathbb{A}$$, is $$\{\{P_1, P_2\}, \{P_3, P_4\}, \{P_1, P_2, P_3\}, \{P_1,P_2,P_4\},\{P_1, P_3, P_4\}, \{P_2, P_3, P_4\}, \{P_1,P_2,P_3,P_4\}\}$$ then $$P_1$$ and $$P_2$$ together have access, $$P_3$$ and $$P_4$$ together have access, but $$P_2$$ and $$P_3$$ together don't. Note that this access structure is monotonic. Since $$\{P_1, P_2\}$$ has access, $$\{P_1, P_2, P_3\}$$, $$\{P_1, P_2, P_4\}$$, and $$\{P_1, P_2, P_3, P_4\}$$ all have access.
Yes, it is meant to be the power set. $$\mathbb{A}$$ is a set of sets of parties.