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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?

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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.

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    $\begingroup$ You convey most of the idea, however with one misstatement; ABE allows access only if a single party has all the attributes; two different parties can't conspire to decrypt a document (by combining attribute keys) that either one can't decrypt by themselves. $\endgroup$
    – poncho
    Jan 6, 2021 at 20:29
  • $\begingroup$ I've update the example to be monotonic. Hopefully it gets the point across better $\endgroup$ Jan 6, 2021 at 20:51

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