Monotonic Access Structure

Question

I got confused with the monotonic access structure definition.

Following definition is one of the answers to the following question:

What are monotonic and non monotonic access structures in ABE ?

Let $\{1,2,...,n\}$ be a set of indices. An access structure is a collection $\mathbb{A}$ of non-empty subsets of $\{1,2,3,...,n\}$. We say a collection (or an access structure) $\mathbb{A} \subseteq 2^{\{1,2,...,n\}}$ is monotonic if for any $B,C \in 2^{\{1,2,...,n\}}$, if $B \in \mathbb{A}$ and $B \subseteq C$ then $C \in \mathbb{A}$.

According to the above definition, every access structure $\mathbb{A}$ which not include $\{1,2,3,...,n\}$ is not monotonic!, because every arbitrary $B \in 2^{\{1,2,...,n\}}$ always is subset of ${\{1,2,...,n\} }$

Could you please clarify what I am going wrong?

Answer 1

The access structure itself need not be non-empty, so the universe doesn't have to be in the structure. Of course, any non-empty access structure will have $\{1,2,\dots,n\}$ as an element.

In the context of ABE, it suffices to consider monotone access structures if the scheme supports an expressive policy set, because any subset $\mathbb{B}$ of $2^{\{1,2,\dots,n\}}$ (i.e., any not necessarily monotone access structure of universe $\{1,2,\dots,n\}$) can be encoded as a monotone access structure $\mathbb{A}$ of the universe $\{\pm1,\pm2,\dots,\pm n\}$ as follows: $$\begin{aligned}\mathbb{A}'&{}=\Bigl\{{\{i\,|\,i\in X\}\cup\{-j\,|\,j\notin X\}}\,\Big|\,X\in\mathbb{B}\Bigr\},\\\mathbb{A}&{}=\Bigl\{A\,\Big|\,\mathbb{A}'\ni A'\subseteq A\subseteq\{\pm1,\pm 2,\dots,\pm n\}\Bigr\}.\end{aligned}$$ In English, this means you prepare positive/negative literals for every member of $\{1,2,\dots,n\}$, then encode any subset of it by explicitly putting negative literals for non-members into it, which gives you $\mathbb{A}'$. Then you take the smallest monotone access structure $\mathbb{A}$ containing $\mathbb{A}'$.

Monotone access structure gives rise to secret-sharing schemes, which are an important ingredient in many ABE constructions. Note that secret-sharing schemes can only handle monotone access structures, because gaining more shares only makes reconstruction easier.

Answer 2

There isn’t anything wrong. In most access structure applications if a group of users (say subset $B$) are authorized for access this means we don’t mind giving larger sets $C$ containing $B$ access as well.

So this property is used to define a monotone access structure.