Access tree $\mathcal{T}$. Let $\mathcal{T}$ be a tree representing an access structure. Each non-leaf node of the tree represents a threshold gate, described by its children and a threshold value. If $num_x$ is the number of children of a node $x$ and $k_x$ is its threshold value, then $0 < k_x ≤ num_x$. (That is, if any $k_x$ or more children of node $x$ are satisfied, node $x$ is satisfied.)
When $k_x = 1$, the threshold gate is an OR gate and when $k_x = num_x$, it is an AND gate. Each leaf node $x$ of the tree is described by an attribute and a threshold value $k_x = 1$.
To facilitate working with the access trees, we define a few functions. We denote the parent of the node $x$ in the tree by $\text{parent}(x)$. The function $\text{att}(x)$ is defined only if $x$ is a leaf node and denotes the attribute associated with the leaf node $x$ in the tree. The access tree $\mathcal{T}$ also defines an ordering between the children of every node, that is, the children of a node are numbered from $1$ to $num$. The function $\text{index}(x)$ returns such a number associated with the node $x$. Where the index values are uniquely assigned to nodes in the access structure for a given key in an arbitrary manner.
Reference:
KP-ABE 2006 Goyal et al. CP-ABE 2007 Bethencourt et al.
